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PUNCT iajs-3349 22 1 the the DET iajs-3349 22 2 gamma gamma NOUN iajs-3349 22 3 - - PUNCT iajs-3349 22 4 kumaraswamy kumaraswamy NOUN iajs-3349 22 5 gfamily gfamily ADV iajs-3349 22 6 of of ADP iajs-3349 22 7 distributions distribution NOUN iajs-3349 22 8 was be AUX iajs-3349 22 9 introduced introduce VERB iajs-3349 22 10 received receive VERB iajs-3349 22 11 19 19 NUM iajs-3349 22 12 march march NOUN iajs-3349 22 13 2023 2023 NUM iajs-3349 22 14 accepted accept VERB iajs-3349 22 15 14 14 NUM iajs-3349 22 16 may may PROPN iajs-3349 22 17 2023 2023 NUM iajs-3349 22 18 published publish VERB iajs-3349 22 19 20 20 NUM iajs-3349 22 20 april april PROPN iajs-3349 22 21 2024 2024 NUM iajs-3349 22 22 statistical statistical ADJ iajs-3349 22 23 propoerties propoertie NOUN iajs-3349 22 24 and and CCONJ iajs-3349 22 25 application application NOUN iajs-3349 22 26 for for ADP iajs-3349 22 27 [ [ X iajs-3349 22 28 0,1 0,1 NOUN iajs-3349 22 29 ] ] PUNCT iajs-3349 22 30 truncated truncated ADJ iajs-3349 22 31 nadarajahhaghighi nadarajahhaghighi PROPN iajs-3349 22 32 exponential exponential ADJ iajs-3349 22 33 distribution distribution NOUN iajs-3349 22 34 doi.org/10.30526/37.2.3349 doi.org/10.30526/37.2.3349 NOUN iajs-3349 22 35 https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ PROPN iajs-3349 22 36 https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042 https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042 PROPN iajs-3349 22 37 https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407 https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407 VERB iajs-3349 22 38 https://orcid.org/0009-0005-1045-2777 https://orcid.org/0009-0005-1045-2777 PROPN iajs-3349 22 39 mailto:khalaf.h.khalaf35385@st.tu.edu.iq mailto:khalaf.h.khalaf35385@st.tu.edu.iq PROPN iajs-3349 22 40 https://orcid.org/0000-0001-8827-3748 https://orcid.org/0000-0001-8827-3748 NOUN iajs-3349 22 41 mailto:mun880088@tu.edu.iq mailto:mun880088@tu.edu.iq NOUN iajs-3349 22 42 https://orcid.org/0000-0003-3430-2464 https://orcid.org/0000-0003-3430-2464 PROPN iajs-3349 22 43 mailto:almof1hm@cmich.edu mailto:almof1hm@cmich.edu PROPN iajs-3349 22 44 ihjpas ihjpa VERB iajs-3349 22 45 . . 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PUNCT iajs-3349 25 1 the the DET iajs-3349 25 2 marshal marshal ADJ iajs-3349 25 3 - - PUNCT iajs-3349 25 4 olkinweibull olkinweibull NOUN iajs-3349 25 5 - - PUNCT iajs-3349 25 6 h h NOUN iajs-3349 25 7 family family NOUN iajs-3349 25 8 was be AUX iajs-3349 25 9 introduced introduce VERB iajs-3349 25 10 by by ADP iajs-3349 25 11 [ [ X iajs-3349 25 12 11 11 NUM iajs-3349 25 13 ] ] PUNCT iajs-3349 25 14 . . PUNCT iajs-3349 26 1 the the DET iajs-3349 26 2 odd odd ADJ iajs-3349 26 3 - - PUNCT iajs-3349 26 4 chen chen PROPN iajs-3349 26 5 family family NOUN iajs-3349 26 6 was be AUX iajs-3349 26 7 introduced introduce VERB iajs-3349 26 8 by by ADP iajs-3349 26 9 [ [ X iajs-3349 26 10 12 12 NUM iajs-3349 26 11 ] ] PUNCT iajs-3349 26 12 . . PUNCT iajs-3349 27 1 researchers researcher NOUN iajs-3349 27 2 have have AUX iajs-3349 27 3 derived derive VERB iajs-3349 27 4 a a DET iajs-3349 27 5 truncated truncate VERB iajs-3349 27 6 distribution distribution NOUN iajs-3349 27 7 from from ADP iajs-3349 27 8 a a DET iajs-3349 27 9 parent parent NOUN iajs-3349 27 10 distribution distribution NOUN iajs-3349 27 11 , , PUNCT iajs-3349 27 12 such such ADJ iajs-3349 27 13 as as ADP iajs-3349 27 14 a a DET iajs-3349 27 15 normal normal ADJ iajs-3349 27 16 or or CCONJ iajs-3349 27 17 exponential exponential ADJ iajs-3349 27 18 distribution distribution NOUN iajs-3349 27 19 , , PUNCT iajs-3349 27 20 by by ADP iajs-3349 27 21 bounding bound VERB iajs-3349 27 22 the the DET iajs-3349 27 23 random random ADJ iajs-3349 27 24 variable variable NOUN iajs-3349 27 25 from from ADP iajs-3349 27 26 below below ADV iajs-3349 27 27 , , PUNCT iajs-3349 27 28 from from ADP iajs-3349 27 29 above above ADV iajs-3349 27 30 , , PUNCT iajs-3349 27 31 or or CCONJ iajs-3349 27 32 from from ADP iajs-3349 27 33 both both PRON iajs-3349 27 34 . . PUNCT iajs-3349 28 1 [ [ X iajs-3349 28 2 13 13 NUM iajs-3349 28 3 ] ] PUNCT iajs-3349 28 4 the the DET iajs-3349 28 5 authors author NOUN iajs-3349 28 6 discussed discuss VERB iajs-3349 28 7 the the PRON iajs-3349 28 8 [ [ X iajs-3349 28 9 0 0 NUM iajs-3349 28 10 , , PUNCT iajs-3349 28 11 1 1 NUM iajs-3349 28 12 ] ] PUNCT iajs-3349 28 13 truncated truncate VERB iajs-3349 28 14 inverted invert VERB iajs-3349 28 15 inverted inverted ADJ iajs-3349 28 16 gamma gamma NOUN iajs-3349 28 17 distribution distribution NOUN iajs-3349 28 18 as as ADP iajs-3349 28 19 a a DET iajs-3349 28 20 special special ADJ iajs-3349 28 21 case case NOUN iajs-3349 28 22 of of ADP iajs-3349 28 23 the the DET iajs-3349 28 24 cdf cdf NOUN iajs-3349 28 25 , , PUNCT iajs-3349 28 26 moments moment NOUN iajs-3349 28 27 , , PUNCT iajs-3349 28 28 mean mean VERB iajs-3349 28 29 , , PUNCT iajs-3349 28 30 variance variance NOUN iajs-3349 28 31 , , PUNCT iajs-3349 28 32 skewness skewness NOUN iajs-3349 28 33 , , PUNCT iajs-3349 28 34 kurtosis kurtosis NOUN iajs-3349 28 35 , , PUNCT iajs-3349 28 36 median median NOUN iajs-3349 28 37 , , PUNCT iajs-3349 28 38 and and CCONJ iajs-3349 28 39 characteristic characteristic ADJ iajs-3349 28 40 function function NOUN iajs-3349 28 41 . . PUNCT iajs-3349 29 1 following follow VERB iajs-3349 29 2 the the DET iajs-3349 29 3 same same ADJ iajs-3349 29 4 method method NOUN iajs-3349 29 5 , , PUNCT iajs-3349 29 6 [ [ X iajs-3349 29 7 14 14 NUM iajs-3349 29 8 ] ] PUNCT iajs-3349 29 9 introduced introduce VERB iajs-3349 29 10 the the DET iajs-3349 29 11 [ [ X iajs-3349 29 12 0,1]-truncated 0,1]-truncated ADJ iajs-3349 29 13 inverse inverse ADJ iajs-3349 29 14 weibull weibull PROPN iajs-3349 29 15 family family NOUN iajs-3349 29 16 . . PUNCT iajs-3349 30 1 more more ADV iajs-3349 30 2 recently recently ADV iajs-3349 30 3 , , PUNCT iajs-3349 30 4 [ [ X iajs-3349 30 5 15 15 NUM iajs-3349 30 6 ] ] PUNCT iajs-3349 30 7 introduced introduce VERB iajs-3349 30 8 and and CCONJ iajs-3349 30 9 studied study VERB iajs-3349 30 10 a a DET iajs-3349 30 11 new new ADJ iajs-3349 30 12 distribution distribution NOUN iajs-3349 30 13 called call VERB iajs-3349 30 14 [ [ X iajs-3349 30 15 0 0 NUM iajs-3349 30 16 , , PUNCT iajs-3349 30 17 1]-truncated 1]-truncated NUM iajs-3349 30 18 exponential exponential ADJ iajs-3349 30 19 gompertz gompertz NOUN iajs-3349 30 20 distribution distribution NOUN iajs-3349 30 21 . . PUNCT iajs-3349 31 1 in in ADP iajs-3349 31 2 addition addition NOUN iajs-3349 31 3 , , PUNCT iajs-3349 31 4 many many ADJ iajs-3349 31 5 researchers researcher NOUN iajs-3349 31 6 have have AUX iajs-3349 31 7 introduced introduce VERB iajs-3349 31 8 new new ADJ iajs-3349 31 9 extensions extension NOUN iajs-3349 31 10 to to ADP iajs-3349 31 11 the the DET iajs-3349 31 12 exponential exponential ADJ iajs-3349 31 13 distribution distribution NOUN iajs-3349 31 14 , , PUNCT iajs-3349 31 15 such such ADJ iajs-3349 31 16 as as ADP iajs-3349 31 17 gupta gupta PROPN iajs-3349 31 18 et et PROPN iajs-3349 31 19 . . PUNCT iajs-3349 32 1 al al PROPN iajs-3349 32 2 . . PROPN iajs-3349 33 1 ( ( PUNCT iajs-3349 33 2 2001 2001 NUM iajs-3349 33 3 ) ) PUNCT iajs-3349 34 1 [ [ X iajs-3349 34 2 16 16 NUM iajs-3349 34 3 ] ] PUNCT iajs-3349 34 4 , , PUNCT iajs-3349 34 5 generalized generalize VERB iajs-3349 34 6 exponential exponential ADJ iajs-3349 34 7 distribution distribution NOUN iajs-3349 34 8 . . PUNCT iajs-3349 35 1 [ [ X iajs-3349 35 2 17 17 NUM iajs-3349 35 3 ] ] X iajs-3349 35 4 , , PUNCT iajs-3349 35 5 a a DET iajs-3349 35 6 new new ADJ iajs-3349 35 7 generalization generalization NOUN iajs-3349 35 8 of of ADP iajs-3349 35 9 the the DET iajs-3349 35 10 exponential exponential ADJ iajs-3349 35 11 pareto pareto ADJ iajs-3349 35 12 distribution distribution NOUN iajs-3349 35 13 [ [ X iajs-3349 35 14 18 18 NUM iajs-3349 35 15 ] ] PUNCT iajs-3349 35 16 , , PUNCT iajs-3349 35 17 the the DET iajs-3349 35 18 weibull weibull PROPN iajs-3349 35 19 exponential exponential ADJ iajs-3349 35 20 distribution distribution NOUN iajs-3349 35 21 [ [ X iajs-3349 35 22 19 19 NUM iajs-3349 35 23 ] ] PUNCT iajs-3349 35 24 and and CCONJ iajs-3349 35 25 the the DET iajs-3349 35 26 beta beta ADJ iajs-3349 35 27 exponential exponential ADJ iajs-3349 35 28 distribution distribution NOUN iajs-3349 35 29 . . PUNCT iajs-3349 36 1 [ [ X iajs-3349 36 2 20 20 NUM iajs-3349 36 3 ] ] PUNCT iajs-3349 36 4 . . PUNCT iajs-3349 37 1 we we PRON iajs-3349 37 2 use use VERB iajs-3349 37 3 the the DET iajs-3349 37 4 modified modify VERB iajs-3349 37 5 exponential exponential ADJ iajs-3349 37 6 distribution distribution NOUN iajs-3349 37 7 ( ( PUNCT iajs-3349 37 8 [ [ X iajs-3349 37 9 0,1 0,1 NUM iajs-3349 37 10 ] ] PUNCT iajs-3349 37 11 ) ) PUNCT iajs-3349 37 12 centered center VERB iajs-3349 37 13 on on ADP iajs-3349 37 14 the the DET iajs-3349 37 15 nadarajah nadarajah PROPN iajs-3349 37 16 - - PUNCT iajs-3349 37 17 haghighi haghighi PROPN iajs-3349 37 18 distribution distribution NOUN iajs-3349 37 19 to to PART iajs-3349 37 20 generate generate VERB iajs-3349 37 21 a a DET iajs-3349 37 22 new new ADJ iajs-3349 37 23 family family NOUN iajs-3349 37 24 of of ADP iajs-3349 37 25 distributions distribution NOUN iajs-3349 37 26 and and CCONJ iajs-3349 37 27 achieve achieve VERB iajs-3349 37 28 greater great ADJ iajs-3349 37 29 flexibility flexibility NOUN iajs-3349 37 30 than than ADP iajs-3349 37 31 the the DET iajs-3349 37 32 existing exist VERB iajs-3349 37 33 submodels submodel NOUN iajs-3349 37 34 . . PUNCT iajs-3349 38 1 the the DET iajs-3349 38 2 article article NOUN iajs-3349 38 3 is be AUX iajs-3349 38 4 organized organize VERB iajs-3349 38 5 as as SCONJ iajs-3349 38 6 follows follow VERB iajs-3349 38 7 : : PUNCT iajs-3349 38 8 section section NOUN iajs-3349 38 9 2 2 NUM iajs-3349 38 10 presents present VERB iajs-3349 38 11 a a DET iajs-3349 38 12 useful useful ADJ iajs-3349 38 13 [ [ X iajs-3349 38 14 0 0 NUM iajs-3349 38 15 , , PUNCT iajs-3349 38 16 1 1 NUM iajs-3349 38 17 ] ] PUNCT iajs-3349 38 18 truncated truncate VERB iajs-3349 38 19 k k PROPN iajs-3349 38 20 - - PROPN iajs-3349 38 21 md md PROPN iajs-3349 38 22 . . PROPN iajs-3349 39 1 section section NOUN iajs-3349 39 2 3 3 NUM iajs-3349 39 3 introduces introduce VERB iajs-3349 39 4 the the DET iajs-3349 39 5 [ [ X iajs-3349 39 6 0 0 NUM iajs-3349 39 7 , , PUNCT iajs-3349 39 8 1 1 NUM iajs-3349 39 9 ] ] PUNCT iajs-3349 39 10 truncated truncated ADJ iajs-3349 39 11 nadarajah nadarajah PROPN iajs-3349 39 12 - - PUNCT iajs-3349 39 13 haghighi haghighi PROPN iajs-3349 39 14 - - PUNCT iajs-3349 39 15 m m PROPN iajs-3349 39 16 family family NOUN iajs-3349 39 17 of of ADP iajs-3349 39 18 distributions distribution NOUN iajs-3349 39 19 . . PUNCT iajs-3349 40 1 in in ADP iajs-3349 40 2 addition addition NOUN iajs-3349 40 3 , , PUNCT iajs-3349 40 4 section section NOUN iajs-3349 40 5 4 4 NUM iajs-3349 40 6 contains contain VERB iajs-3349 40 7 the the DET iajs-3349 40 8 statistical statistical ADJ iajs-3349 40 9 properties property NOUN iajs-3349 40 10 of of ADP iajs-3349 40 11 the the PRON iajs-3349 40 12 [ [ X iajs-3349 40 13 0 0 NUM iajs-3349 40 14 , , PUNCT iajs-3349 40 15 1 1 NUM iajs-3349 40 16 ] ] PUNCT iajs-3349 40 17 tnhe tnhe VERB iajs-3349 40 18 distribution distribution NOUN iajs-3349 40 19 . . PUNCT iajs-3349 41 1 to to PART iajs-3349 41 2 estimate estimate VERB iajs-3349 41 3 the the DET iajs-3349 41 4 parameters parameter NOUN iajs-3349 41 5 of of ADP iajs-3349 41 6 this this DET iajs-3349 41 7 new new ADJ iajs-3349 41 8 distribution distribution NOUN iajs-3349 41 9 , , PUNCT iajs-3349 41 10 the the DET iajs-3349 41 11 mle mle PROPN iajs-3349 41 12 method method NOUN iajs-3349 41 13 is be AUX iajs-3349 41 14 used use VERB iajs-3349 41 15 , , PUNCT iajs-3349 41 16 which which PRON iajs-3349 41 17 is be AUX iajs-3349 41 18 presented present VERB iajs-3349 41 19 in in ADP iajs-3349 41 20 section section NOUN iajs-3349 41 21 5 5 NUM iajs-3349 41 22 . . PUNCT iajs-3349 42 1 in in ADP iajs-3349 42 2 section section NOUN iajs-3349 42 3 6 6 NUM iajs-3349 42 4 , , PUNCT iajs-3349 42 5 the the DET iajs-3349 42 6 estimates estimate NOUN iajs-3349 42 7 are be AUX iajs-3349 42 8 validated validate VERB iajs-3349 42 9 by by ADP iajs-3349 42 10 the the DET iajs-3349 42 11 simulation simulation NOUN iajs-3349 42 12 process process NOUN iajs-3349 42 13 of of ADP iajs-3349 42 14 the the DET iajs-3349 42 15 tnhe tnhe NOUN iajs-3349 42 16 distribution distribution NOUN iajs-3349 42 17 . . PUNCT iajs-3349 43 1 in in ADP iajs-3349 43 2 section section NOUN iajs-3349 43 3 7 7 NUM iajs-3349 43 4 , , PUNCT iajs-3349 43 5 a a DET iajs-3349 43 6 real real ADJ iajs-3349 43 7 data datum NOUN iajs-3349 43 8 set set VERB iajs-3349 43 9 is be AUX iajs-3349 43 10 used use VERB iajs-3349 43 11 to to PART iajs-3349 43 12 illustrate illustrate VERB iajs-3349 43 13 the the DET iajs-3349 43 14 effectiveness effectiveness NOUN iajs-3349 43 15 of of ADP iajs-3349 43 16 the the DET iajs-3349 43 17 tnhe tnhe NOUN iajs-3349 43 18 distribution distribution NOUN iajs-3349 43 19 . . PUNCT iajs-3349 44 1 the the DET iajs-3349 44 2 conclusions conclusion NOUN iajs-3349 44 3 are be AUX iajs-3349 44 4 presented present VERB iajs-3349 44 5 in in ADP iajs-3349 44 6 section section NOUN iajs-3349 44 7 8 8 NUM iajs-3349 44 8 . . NOUN iajs-3349 45 1 2 2 NUM iajs-3349 45 2 . . PUNCT iajs-3349 46 1 [ [ X iajs-3349 46 2 0,1 0,1 NUM iajs-3349 46 3 ] ] PUNCT iajs-3349 46 4 truncated truncate VERB iajs-3349 46 5 k k PROPN iajs-3349 46 6 - - PROPN iajs-3349 46 7 md md PROPN iajs-3349 46 8 in in ADP iajs-3349 46 9 this this DET iajs-3349 46 10 paper paper NOUN iajs-3349 46 11 , , PUNCT iajs-3349 46 12 we we PRON iajs-3349 46 13 have have AUX iajs-3349 46 14 generated generate VERB iajs-3349 46 15 a a DET iajs-3349 46 16 new new ADJ iajs-3349 46 17 family family NOUN iajs-3349 46 18 of of ADP iajs-3349 46 19 continues continue VERB iajs-3349 46 20 distributions distribution NOUN iajs-3349 46 21 based base VERB iajs-3349 46 22 on on ADP iajs-3349 46 23 [ [ X iajs-3349 46 24 0,1 0,1 NUM iajs-3349 46 25 ] ] PUNCT iajs-3349 46 26 truncated truncate VERB iajs-3349 46 27 cdf cdf PROPN iajs-3349 46 28 k k PROPN iajs-3349 46 29 - - PROPN iajs-3349 46 30 m m PROPN iajs-3349 46 31 , , PUNCT iajs-3349 46 32 named name VERB iajs-3349 46 33 [ [ X iajs-3349 46 34 0,1 0,1 NUM iajs-3349 46 35 ] ] X iajs-3349 46 36 k k NOUN iajs-3349 46 37 - - NOUN iajs-3349 46 38 m m VERB iajs-3349 46 39 as as SCONJ iajs-3349 46 40 noted note VERB iajs-3349 46 41 by by ADP iajs-3349 46 42 [ [ X iajs-3349 46 43 0,1 0,1 NUM iajs-3349 46 44 ] ] X iajs-3349 46 45 tk tk PROPN iajs-3349 46 46 - - PROPN iajs-3349 46 47 m m PROPN iajs-3349 46 48 distributions distribution NOUN iajs-3349 46 49 . . PUNCT iajs-3349 47 1 suppose suppose VERB iajs-3349 47 2 that that SCONJ iajs-3349 47 3 𝑀(π‘₯ 𝑀(π‘₯ NUM iajs-3349 47 4 ) ) PUNCT iajs-3349 47 5 , , PUNCT iajs-3349 47 6 π‘š(π‘₯ π‘š(π‘₯ NOUN iajs-3349 47 7 ) ) PUNCT iajs-3349 47 8 is be AUX iajs-3349 47 9 any any DET iajs-3349 47 10 continuous continuous ADJ iajs-3349 47 11 cdf cdf PROPN iajs-3349 47 12 and and CCONJ iajs-3349 47 13 pdf pdf NOUN iajs-3349 47 14 , , PUNCT iajs-3349 47 15 respectively respectively ADV iajs-3349 47 16 of of ADP iajs-3349 47 17 the the DET iajs-3349 47 18 random random ADJ iajs-3349 47 19 variable variable NOUN iajs-3349 47 20 𝑋 𝑋 PROPN iajs-3349 47 21 , , PUNCT iajs-3349 47 22 and and CCONJ iajs-3349 47 23 assume assume VERB iajs-3349 47 24 that that SCONJ iajs-3349 47 25 𝐾 𝐾 PROPN iajs-3349 47 26 ( ( PUNCT iajs-3349 47 27 . . PUNCT iajs-3349 47 28 ) ) PUNCT iajs-3349 47 29 , , PUNCT iajs-3349 47 30 π‘˜ π‘˜ PROPN iajs-3349 47 31 ( ( PUNCT iajs-3349 47 32 . . PUNCT iajs-3349 47 33 ) ) PUNCT iajs-3349 47 34 , , PUNCT iajs-3349 47 35 respectively respectively ADV iajs-3349 47 36 represent represent VERB iajs-3349 47 37 the the DET iajs-3349 47 38 cdf cdf PROPN iajs-3349 47 39 and and CCONJ iajs-3349 47 40 pdf pdf NOUN iajs-3349 47 41 of of ADP iajs-3349 47 42 any any DET iajs-3349 47 43 continuous continuous ADJ iajs-3349 47 44 distribution distribution NOUN iajs-3349 47 45 on on ADP iajs-3349 47 46 the the DET iajs-3349 47 47 interval interval NOUN iajs-3349 47 48 [ [ X iajs-3349 47 49 0 0 NUM iajs-3349 47 50 , , PUNCT iajs-3349 47 51 ∞ ∞ PROPN iajs-3349 47 52 ) ) PUNCT iajs-3349 47 53 . . PUNCT iajs-3349 48 1 the the DET iajs-3349 48 2 suggested suggest VERB iajs-3349 48 3 general general ADJ iajs-3349 48 4 formula formula NOUN iajs-3349 48 5 of of ADP iajs-3349 48 6 cdf cdf PROPN iajs-3349 48 7 for for ADP iajs-3349 48 8 this this DET iajs-3349 48 9 class class NOUN iajs-3349 48 10 depends depend VERB iajs-3349 48 11 on on ADP iajs-3349 48 12 the the DET iajs-3349 48 13 synthesis synthesis NOUN iajs-3349 48 14 of of ADP iajs-3349 48 15 k k PROPN iajs-3349 48 16 with with ADP iajs-3349 48 17 m m PROPN iajs-3349 48 18 is be AUX iajs-3349 48 19 𝐹(π‘₯)π‘‡πΎβˆ’π‘€ 𝐹(π‘₯)π‘‡πΎβˆ’π‘€ X iajs-3349 48 20 = = SYM iajs-3349 48 21 𝐾[𝑀(π‘₯ 𝐾[𝑀(π‘₯ NOUN iajs-3349 48 22 ) ) PUNCT iajs-3349 48 23 ] ] PUNCT iajs-3349 49 1 βˆ’ βˆ’ PROPN iajs-3349 49 2 𝐾(0 𝐾(0 NUM iajs-3349 49 3 ) ) PUNCT iajs-3349 49 4 𝐾(1 𝐾(1 NUM iajs-3349 49 5 ) ) PUNCT iajs-3349 49 6 βˆ’ βˆ’ PROPN iajs-3349 49 7 𝐾(0 𝐾(0 NUM iajs-3349 49 8 ) ) PUNCT iajs-3349 49 9 ( ( PUNCT iajs-3349 49 10 1 1 X iajs-3349 49 11 ) ) PUNCT iajs-3349 49 12 now now ADV iajs-3349 49 13 , , PUNCT iajs-3349 49 14 let let VERB iajs-3349 49 15 𝐾(0 𝐾(0 PUNCT iajs-3349 49 16 ) ) PUNCT iajs-3349 49 17 = = SYM iajs-3349 50 1 0 0 NUM iajs-3349 50 2 , , PUNCT iajs-3349 50 3 then then ADV iajs-3349 50 4 the the DET iajs-3349 50 5 cdf cdf PROPN iajs-3349 50 6 in in ADP iajs-3349 50 7 ( ( PUNCT iajs-3349 50 8 1 1 NUM iajs-3349 50 9 ) ) PUNCT iajs-3349 50 10 can can AUX iajs-3349 50 11 be be AUX iajs-3349 50 12 written write VERB iajs-3349 50 13 as as ADP iajs-3349 50 14 : : PUNCT iajs-3349 50 15 𝐹(π‘₯)π‘‡πΎβˆ’π‘€ 𝐹(π‘₯)π‘‡πΎβˆ’π‘€ X iajs-3349 50 16 = = PUNCT iajs-3349 50 17 𝐾[𝑀(π‘₯ 𝐾[𝑀(π‘₯ NOUN iajs-3349 50 18 ) ) PUNCT iajs-3349 50 19 ] ] PUNCT iajs-3349 51 1 𝐾(1 𝐾(1 NUM iajs-3349 51 2 ) ) PUNCT iajs-3349 51 3 ( ( PUNCT iajs-3349 51 4 2 2 X iajs-3349 51 5 ) ) PUNCT iajs-3349 51 6 and and CCONJ iajs-3349 51 7 its its PRON iajs-3349 51 8 associated associated ADJ iajs-3349 51 9 pdf pdf NOUN iajs-3349 51 10 will will AUX iajs-3349 51 11 be be AUX iajs-3349 51 12 𝑓(π‘₯)π‘‡πΎβˆ’π‘€ 𝑓(π‘₯)π‘‡πΎβˆ’π‘€ NOUN iajs-3349 51 13 = = SYM iajs-3349 51 14 π‘˜[𝑀(π‘₯)]π‘š(π‘₯ π‘˜[𝑀(π‘₯)]π‘š(π‘₯ NOUN iajs-3349 51 15 ) ) PUNCT iajs-3349 51 16 𝐾(1 𝐾(1 NUM iajs-3349 51 17 ) ) PUNCT iajs-3349 51 18 ( ( PUNCT iajs-3349 51 19 3 3 X iajs-3349 51 20 ) ) PUNCT iajs-3349 51 21 3 3 NUM iajs-3349 51 22 . . PUNCT iajs-3349 52 1 [ [ X iajs-3349 52 2 0,1 0,1 NUM iajs-3349 52 3 ] ] PUNCT iajs-3349 52 4 truncated truncated ADJ iajs-3349 52 5 nadarajah nadarajah PROPN iajs-3349 52 6 - - PUNCT iajs-3349 52 7 haghighi haghighi PROPN iajs-3349 52 8 - - PUNCT iajs-3349 52 9 m m PROPN iajs-3349 52 10 family family NOUN iajs-3349 52 11 here here ADV iajs-3349 52 12 we we PRON iajs-3349 52 13 will will AUX iajs-3349 52 14 propose propose VERB iajs-3349 52 15 a a DET iajs-3349 52 16 new new ADJ iajs-3349 52 17 family family NOUN iajs-3349 52 18 of of ADP iajs-3349 52 19 [ [ X iajs-3349 52 20 0,1 0,1 NUM iajs-3349 52 21 ] ] PUNCT iajs-3349 52 22 truncated truncate VERB iajs-3349 52 23 based base VERB iajs-3349 52 24 on on ADP iajs-3349 52 25 nadarajh nadarajh PROPN iajs-3349 52 26 - - PUNCT iajs-3349 52 27 haghighi haghighi PROPN iajs-3349 52 28 distribution distribution NOUN iajs-3349 52 29 . . PUNCT iajs-3349 53 1 the the DET iajs-3349 53 2 nadarajh nadarajh PROPN iajs-3349 53 3 - - PUNCT iajs-3349 53 4 haghighi haghighi PROPN iajs-3349 53 5 distribution distribution NOUN iajs-3349 53 6 was be AUX iajs-3349 53 7 introduced introduce VERB iajs-3349 53 8 by by ADP iajs-3349 53 9 [ [ X iajs-3349 53 10 21 21 NUM iajs-3349 53 11 ] ] PUNCT iajs-3349 53 12 as as ADP iajs-3349 53 13 an an DET iajs-3349 53 14 extention extention NOUN iajs-3349 53 15 of of ADP iajs-3349 53 16 the the DET iajs-3349 53 17 exponential exponential ADJ iajs-3349 53 18 distribution distribution NOUN iajs-3349 53 19 . . PUNCT iajs-3349 54 1 the the DET iajs-3349 54 2 n n CCONJ iajs-3349 54 3 - - PUNCT iajs-3349 54 4 h h NOUN iajs-3349 54 5 distribution distribution NOUN iajs-3349 54 6 has have VERB iajs-3349 54 7 the the DET iajs-3349 54 8 cdf cdf PROPN iajs-3349 54 9 and and CCONJ iajs-3349 54 10 pdf pdf NOUN iajs-3349 54 11 , , PUNCT iajs-3349 54 12 as as ADP iajs-3349 54 13 folllows folllow NOUN iajs-3349 54 14 : : PUNCT iajs-3349 54 15 𝐾(𝑋 𝐾(𝑋 ADJ iajs-3349 54 16 ) ) PUNCT iajs-3349 54 17 = = SYM iajs-3349 55 1 1 1 NUM iajs-3349 55 2 βˆ’ βˆ’ NOUN iajs-3349 55 3 𝑒1βˆ’[1+𝑏π‘₯]π‘Ž 𝑒1βˆ’[1+𝑏π‘₯]π‘Ž NOUN iajs-3349 55 4 π‘₯ π‘₯ NOUN iajs-3349 55 5 > > X iajs-3349 55 6 0 0 NUM iajs-3349 55 7 , , PUNCT iajs-3349 55 8 π‘Ž π‘Ž X iajs-3349 55 9 , , PUNCT iajs-3349 55 10 𝑏 𝑏 PROPN iajs-3349 55 11 > > X iajs-3349 55 12 0 0 PUNCT iajs-3349 55 13 ( ( PUNCT iajs-3349 55 14 4 4 NUM iajs-3349 55 15 ) ) PUNCT iajs-3349 55 16 π‘˜(π‘₯ π‘˜(π‘₯ NOUN iajs-3349 55 17 ) ) PUNCT iajs-3349 55 18 = = SYM iajs-3349 56 1 π‘Žπ‘[1 π‘Žπ‘[1 PROPN iajs-3349 56 2 + + NUM iajs-3349 56 3 𝑏π‘₯]π‘Žβˆ’1 𝑏π‘₯]π‘Žβˆ’1 ADJ iajs-3349 56 4 𝑒1βˆ’[1+𝑏π‘₯]π‘Ž 𝑒1βˆ’[1+𝑏π‘₯]π‘Ž NOUN iajs-3349 56 5 π‘₯ π‘₯ NOUN iajs-3349 56 6 > > X iajs-3349 56 7 0 0 NUM iajs-3349 56 8 , , PUNCT iajs-3349 56 9 π‘Ž π‘Ž X iajs-3349 56 10 , , PUNCT iajs-3349 56 11 𝑏 𝑏 PROPN iajs-3349 56 12 > > X iajs-3349 56 13 0 0 PUNCT iajs-3349 57 1 ( ( PUNCT iajs-3349 57 2 5 5 NUM iajs-3349 57 3 ) ) PUNCT iajs-3349 57 4 ihjpas ihjpa NOUN iajs-3349 57 5 . . PUNCT iajs-3349 58 1 37 37 NUM iajs-3349 58 2 ( ( PUNCT iajs-3349 58 3 2 2 NUM iajs-3349 58 4 ) ) PUNCT iajs-3349 58 5 2024 2024 NUM iajs-3349 58 6 365 365 NUM iajs-3349 58 7 then then ADV iajs-3349 58 8 , , PUNCT iajs-3349 58 9 𝐾[𝑀(π‘₯ 𝐾[𝑀(π‘₯ PROPN iajs-3349 58 10 ) ) PUNCT iajs-3349 58 11 ] ] PUNCT iajs-3349 59 1 = = SYM iajs-3349 59 2 1 1 NUM iajs-3349 59 3 βˆ’ βˆ’ PROPN iajs-3349 59 4 𝑒1βˆ’[1+𝑏𝑀(π‘₯)]π‘Ž 𝑒1βˆ’[1+𝑏𝑀(π‘₯)]π‘Ž NOUN iajs-3349 59 5 , , PUNCT iajs-3349 59 6 π‘˜[𝑀(π‘₯ π‘˜[𝑀(π‘₯ NOUN iajs-3349 59 7 ) ) PUNCT iajs-3349 59 8 ] ] PUNCT iajs-3349 60 1 = = PUNCT iajs-3349 60 2 π‘Žπ‘[1 π‘Žπ‘[1 PROPN iajs-3349 60 3 + + NUM iajs-3349 60 4 𝑏π‘₯]π‘Žβˆ’1 𝑏π‘₯]π‘Žβˆ’1 ADJ iajs-3349 60 5 𝑒1βˆ’[1+𝑏𝑀(π‘₯)]π‘Ž 𝑒1βˆ’[1+𝑏𝑀(π‘₯)]π‘Ž NOUN iajs-3349 60 6 so so SCONJ iajs-3349 60 7 that that SCONJ iajs-3349 60 8 equations equation NOUN iajs-3349 60 9 ( ( PUNCT iajs-3349 60 10 2 2 NUM iajs-3349 60 11 ) ) PUNCT iajs-3349 60 12 , , PUNCT iajs-3349 60 13 ( ( PUNCT iajs-3349 60 14 3 3 X iajs-3349 60 15 ) ) PUNCT iajs-3349 60 16 can can AUX iajs-3349 60 17 be be AUX iajs-3349 60 18 rewritten rewrite VERB iajs-3349 60 19 as as SCONJ iajs-3349 60 20 follows follow VERB iajs-3349 60 21 : : PUNCT iajs-3349 60 22 𝐹(π‘₯ 𝐹(π‘₯ X iajs-3349 60 23 ) ) PUNCT iajs-3349 60 24 = = SYM iajs-3349 60 25 1βˆ’ 1βˆ’ NUM iajs-3349 60 26 𝑒1βˆ’[1+𝑏𝑀(π‘₯,πœ‘)]π‘Ž 𝑒1βˆ’[1+𝑏𝑀(π‘₯,πœ‘)]π‘Ž PROPN iajs-3349 60 27 1βˆ’ 1βˆ’ NUM iajs-3349 60 28 𝑒1βˆ’[1+𝑏]π‘Ž 𝑒1βˆ’[1+𝑏]π‘Ž NOUN iajs-3349 60 29 ( ( PUNCT iajs-3349 60 30 6 6 NUM iajs-3349 60 31 ) ) PUNCT iajs-3349 60 32 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3349 60 33 ) ) PUNCT iajs-3349 60 34 = = SYM iajs-3349 60 35 π‘Žπ‘[1+𝑏𝑀(π‘₯,πœ‘)]π‘Žβˆ’1 π‘Žπ‘[1+𝑏𝑀(π‘₯,πœ‘)]π‘Žβˆ’1 X iajs-3349 60 36 𝑒1βˆ’[1+𝑏𝑀(π‘₯,πœ‘)]π‘Ž 𝑒1βˆ’[1+𝑏𝑀(π‘₯,πœ‘)]π‘Ž PROPN iajs-3349 60 37 π‘š(π‘₯,πœ‘ π‘š(π‘₯,πœ‘ NOUN iajs-3349 60 38 ) ) PUNCT iajs-3349 60 39 1βˆ’ 1βˆ’ NUM iajs-3349 60 40 𝑒1βˆ’[1+𝑏]π‘Ž 𝑒1βˆ’[1+𝑏]π‘Ž NOUN iajs-3349 60 41 ( ( PUNCT iajs-3349 60 42 7 7 NUM iajs-3349 60 43 ) ) PUNCT iajs-3349 60 44 the the DET iajs-3349 60 45 equations equation NOUN iajs-3349 60 46 ( ( PUNCT iajs-3349 60 47 6 6 NUM iajs-3349 60 48 ) ) PUNCT iajs-3349 60 49 , , PUNCT iajs-3349 60 50 ( ( PUNCT iajs-3349 60 51 7 7 NUM iajs-3349 60 52 ) ) PUNCT iajs-3349 60 53 , , PUNCT iajs-3349 60 54 respactively respactively ADV iajs-3349 60 55 represent represent VERB iajs-3349 60 56 the the DET iajs-3349 60 57 cdf cdf PROPN iajs-3349 60 58 , , PUNCT iajs-3349 60 59 pdf pdf NOUN iajs-3349 60 60 of of ADP iajs-3349 60 61 [ [ X iajs-3349 60 62 0,1 0,1 NUM iajs-3349 60 63 ] ] PUNCT iajs-3349 60 64 truncated truncated ADJ iajs-3349 60 65 nadarajahhaghighi nadarajahhaghighi PROPN iajs-3349 60 66 - - PUNCT iajs-3349 60 67 m m PROPN iajs-3349 60 68 family family NOUN iajs-3349 60 69 of of ADP iajs-3349 60 70 distributions distribution NOUN iajs-3349 60 71 , , PUNCT iajs-3349 60 72 where where SCONJ iajs-3349 60 73 𝑀(π‘₯ 𝑀(π‘₯ NOUN iajs-3349 60 74 , , PUNCT iajs-3349 60 75 πœ‘ πœ‘ NOUN iajs-3349 60 76 ) ) PUNCT iajs-3349 60 77 , , PUNCT iajs-3349 60 78 π‘š(π‘₯ π‘š(π‘₯ NOUN iajs-3349 60 79 , , PUNCT iajs-3349 60 80 πœ‘ πœ‘ PRON iajs-3349 60 81 ) ) PUNCT iajs-3349 60 82 are be AUX iajs-3349 60 83 the the DET iajs-3349 60 84 cdf cdf PROPN iajs-3349 60 85 and and CCONJ iajs-3349 60 86 pdf pdf NOUN iajs-3349 60 87 of of ADP iajs-3349 60 88 the the DET iajs-3349 60 89 baseline baseline ADJ iajs-3349 60 90 distribution distribution NOUN iajs-3349 60 91 with with ADP iajs-3349 60 92 vector vector NOUN iajs-3349 60 93 of of ADP iajs-3349 60 94 parameters parameter NOUN iajs-3349 60 95 πœ‘. πœ‘. VERB iajs-3349 60 96 the the DET iajs-3349 60 97 exponential exponential ADJ iajs-3349 60 98 distribution distribution NOUN iajs-3349 60 99 was be AUX iajs-3349 60 100 introduced introduce VERB iajs-3349 60 101 by by ADP iajs-3349 60 102 the the DET iajs-3349 60 103 random random ADJ iajs-3349 60 104 variable variable NOUN iajs-3349 60 105 𝑋 𝑋 NOUN iajs-3349 60 106 provided provide VERB iajs-3349 60 107 that that SCONJ iajs-3349 60 108 the the DET iajs-3349 60 109 following follow VERB iajs-3349 60 110 cdf cdf PROPN iajs-3349 60 111 and and CCONJ iajs-3349 60 112 pdf pdf NOUN iajs-3349 60 113 : : PUNCT iajs-3349 60 114 𝑀(π‘₯ 𝑀(π‘₯ NUM iajs-3349 60 115 , , PUNCT iajs-3349 60 116 πœ† πœ† NOUN iajs-3349 60 117 ) ) PUNCT iajs-3349 60 118 = = SYM iajs-3349 60 119 1 1 NUM iajs-3349 60 120 βˆ’ βˆ’ NUM iajs-3349 60 121 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 60 122 π‘š(π‘₯ π‘š(π‘₯ NOUN iajs-3349 60 123 , , PUNCT iajs-3349 60 124 πœ† πœ† NOUN iajs-3349 60 125 ) ) PUNCT iajs-3349 60 126 = = SYM iajs-3349 60 127 πœ†π‘’βˆ’πœ†π‘₯ πœ†π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 60 128 ( ( PUNCT iajs-3349 60 129 8) 8) NUM iajs-3349 60 130 ( ( PUNCT iajs-3349 60 131 9 9 NUM iajs-3349 60 132 ) ) PUNCT iajs-3349 60 133 now now ADV iajs-3349 60 134 , , PUNCT iajs-3349 60 135 from from ADP iajs-3349 60 136 substituting substitute VERB iajs-3349 60 137 equations equation NOUN iajs-3349 60 138 ( ( PUNCT iajs-3349 60 139 8) 8) NUM iajs-3349 60 140 and and CCONJ iajs-3349 60 141 ( ( PUNCT iajs-3349 60 142 9 9 NUM iajs-3349 60 143 ) ) PUNCT iajs-3349 60 144 in in ADP iajs-3349 60 145 ( ( PUNCT iajs-3349 60 146 6 6 NUM iajs-3349 60 147 ) ) PUNCT iajs-3349 60 148 and and CCONJ iajs-3349 60 149 ( ( PUNCT iajs-3349 60 150 7 7 NUM iajs-3349 60 151 ) ) PUNCT iajs-3349 60 152 , , PUNCT iajs-3349 60 153 we we PRON iajs-3349 60 154 have have AUX iajs-3349 60 155 obtained obtain VERB iajs-3349 60 156 the the DET iajs-3349 60 157 cdf cdf PROPN iajs-3349 60 158 , , PUNCT iajs-3349 60 159 pdf pdf NOUN iajs-3349 60 160 of of ADP iajs-3349 60 161 [ [ X iajs-3349 60 162 0,1 0,1 NUM iajs-3349 60 163 ] ] PUNCT iajs-3349 60 164 truncated truncate VERB iajs-3349 60 165 nadarajah nadarajah PROPN iajs-3349 60 166 - - PUNCT iajs-3349 60 167 haghighi haghighi PROPN iajs-3349 60 168 exponential exponential ADJ iajs-3349 60 169 distribution distribution NOUN iajs-3349 60 170 , , PUNCT iajs-3349 60 171 as as SCONJ iajs-3349 60 172 follows follow VERB iajs-3349 60 173 : : PUNCT iajs-3349 60 174 𝐹(π‘₯)[0,1]𝑇𝑁𝐻𝐸 𝐹(π‘₯)[0,1]𝑇𝑁𝐻𝐸 ADJ iajs-3349 60 175 = = SYM iajs-3349 60 176 1 1 NUM iajs-3349 60 177 βˆ’ βˆ’ NUM iajs-3349 60 178 𝑒1βˆ’[1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 𝑒1βˆ’[1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ NUM iajs-3349 60 179 ) ) PUNCT iajs-3349 60 180 ] ] PUNCT iajs-3349 61 1 π‘Ž π‘Ž PRON iajs-3349 61 2 1 1 NUM iajs-3349 61 3 βˆ’ βˆ’ NOUN iajs-3349 61 4 𝑒1βˆ’[1+𝑏]π‘Ž 𝑒1βˆ’[1+𝑏]π‘Ž NOUN iajs-3349 61 5 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 NOUN iajs-3349 61 6 = = SYM iajs-3349 61 7 π‘Žπ‘πœ†[1 π‘Žπ‘πœ†[1 X iajs-3349 61 8 + + CCONJ iajs-3349 61 9 𝑏(1 𝑏(1 PROPN iajs-3349 61 10 βˆ’ βˆ’ PROPN iajs-3349 61 11 π‘’βˆ’πœ†π‘₯)]π‘Žβˆ’1 π‘’βˆ’πœ†π‘₯)]π‘Žβˆ’1 PROPN iajs-3349 61 12 𝑒1βˆ’[1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯)]π‘Ž 𝑒1βˆ’[1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯)]π‘Ž NOUN iajs-3349 61 13 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 61 14 1 1 NUM iajs-3349 61 15 βˆ’ βˆ’ NOUN iajs-3349 61 16 𝑒1βˆ’[1+𝑏]π‘Ž 𝑒1βˆ’[1+𝑏]π‘Ž NOUN iajs-3349 61 17 ( ( PUNCT iajs-3349 61 18 10 10 NUM iajs-3349 61 19 ) ) PUNCT iajs-3349 61 20 ( ( PUNCT iajs-3349 61 21 11 11 NUM iajs-3349 61 22 ) ) PUNCT iajs-3349 61 23 then then ADV iajs-3349 61 24 , , PUNCT iajs-3349 61 25 equations equation NOUN iajs-3349 61 26 ( ( PUNCT iajs-3349 61 27 10 10 NUM iajs-3349 61 28 ) ) PUNCT iajs-3349 61 29 , , PUNCT iajs-3349 61 30 ( ( PUNCT iajs-3349 61 31 11 11 X iajs-3349 61 32 ) ) PUNCT iajs-3349 61 33 respectivelly respectivelly ADV iajs-3349 61 34 represent represent VERB iajs-3349 61 35 the the DET iajs-3349 61 36 cdf cdf PROPN iajs-3349 61 37 and and CCONJ iajs-3349 61 38 pdf pdf NOUN iajs-3349 61 39 of of ADP iajs-3349 61 40 [ [ X iajs-3349 61 41 0,1 0,1 NUM iajs-3349 61 42 ] ] PUNCT iajs-3349 61 43 truncated truncate VERB iajs-3349 61 44 nadarajahhaghighi nadarajahhaghighi PROPN iajs-3349 61 45 exponentail exponentail NOUN iajs-3349 61 46 distribution distribution NOUN iajs-3349 61 47 . . PUNCT iajs-3349 62 1 according accord VERB iajs-3349 62 2 to to ADP iajs-3349 62 3 equations equation NOUN iajs-3349 62 4 ( ( PUNCT iajs-3349 62 5 10 10 NUM iajs-3349 62 6 ) ) PUNCT iajs-3349 62 7 , , PUNCT iajs-3349 62 8 ( ( PUNCT iajs-3349 62 9 11 11 NUM iajs-3349 62 10 ) ) PUNCT iajs-3349 62 11 , , PUNCT iajs-3349 62 12 we we PRON iajs-3349 62 13 can can AUX iajs-3349 62 14 obtain obtain VERB iajs-3349 62 15 the the DET iajs-3349 62 16 survival survival NOUN iajs-3349 62 17 and and CCONJ iajs-3349 62 18 hazard hazard NOUN iajs-3349 62 19 functions function NOUN iajs-3349 62 20 of of ADP iajs-3349 62 21 [ [ X iajs-3349 62 22 0,1 0,1 NUM iajs-3349 62 23 ] ] X iajs-3349 62 24 tnhedistribuion tnhedistribuion NOUN iajs-3349 62 25 , , PUNCT iajs-3349 62 26 as as SCONJ iajs-3349 62 27 follows follow VERB iajs-3349 62 28 : : PUNCT iajs-3349 62 29 the the DET iajs-3349 62 30 survival survival NOUN iajs-3349 62 31 function function NOUN iajs-3349 62 32 of of ADP iajs-3349 62 33 [ [ X iajs-3349 62 34 0,1 0,1 NUM iajs-3349 62 35 ] ] PUNCT iajs-3349 62 36 tnhe tnhe VERB iajs-3349 62 37 distribution distribution NOUN iajs-3349 62 38 : : PUNCT iajs-3349 62 39 𝑆(π‘₯ 𝑆(π‘₯ NUM iajs-3349 62 40 , , PUNCT iajs-3349 62 41 π‘Ž π‘Ž NOUN iajs-3349 62 42 , , PUNCT iajs-3349 62 43 𝑏 𝑏 NOUN iajs-3349 62 44 , , PUNCT iajs-3349 62 45 πœ†)[0,1]𝑇𝑁𝐻𝐸 πœ†)[0,1]𝑇𝑁𝐻𝐸 NOUN iajs-3349 62 46 = = SYM iajs-3349 62 47 𝑒1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 𝑒1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ NUM iajs-3349 62 48 ) ) PUNCT iajs-3349 62 49 ) ) PUNCT iajs-3349 63 1 π‘Ž π‘Ž DET iajs-3349 63 2 βˆ’ βˆ’ NOUN iajs-3349 63 3 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 63 4 1 1 NUM iajs-3349 63 5 βˆ’ βˆ’ NOUN iajs-3349 63 6 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž NOUN iajs-3349 63 7 ( ( PUNCT iajs-3349 63 8 12 12 NUM iajs-3349 63 9 ) ) PUNCT iajs-3349 63 10 hazard hazard NOUN iajs-3349 63 11 rate rate NOUN iajs-3349 63 12 function function NOUN iajs-3349 63 13 of of ADP iajs-3349 63 14 [ [ X iajs-3349 63 15 0 0 NUM iajs-3349 63 16 , , PUNCT iajs-3349 63 17 1 1 NUM iajs-3349 63 18 ] ] PUNCT iajs-3349 63 19 tnhe tnhe NOUN iajs-3349 63 20 distribution distribution NOUN iajs-3349 63 21 is be AUX iajs-3349 63 22 [ [ X iajs-3349 63 23 22 22 NUM iajs-3349 63 24 ] ] PUNCT iajs-3349 63 25 β„Ž(π‘₯ β„Ž(π‘₯ NUM iajs-3349 63 26 , , PUNCT iajs-3349 63 27 π‘Ž π‘Ž X iajs-3349 63 28 , , PUNCT iajs-3349 63 29 𝑏 𝑏 NOUN iajs-3349 63 30 , , PUNCT iajs-3349 63 31 πœ†)[0,1]𝑇𝑁𝐻𝐸 πœ†)[0,1]𝑇𝑁𝐻𝐸 NOUN iajs-3349 63 32 = = SYM iajs-3349 63 33 π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ ADJ iajs-3349 63 34 ) ) PUNCT iajs-3349 63 35 ) ) PUNCT iajs-3349 64 1 π‘Žβˆ’1 π‘Žβˆ’1 PROPN iajs-3349 64 2 𝑒 𝑒 PROPN iajs-3349 64 3 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 64 4 ) ) PUNCT iajs-3349 64 5 ) ) PUNCT iajs-3349 65 1 π‘Ž π‘Ž DET iajs-3349 65 2 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 65 3 𝑒 𝑒 PROPN iajs-3349 65 4 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 65 5 ) ) PUNCT iajs-3349 65 6 ) ) PUNCT iajs-3349 66 1 π‘Ž π‘Ž X iajs-3349 66 2 βˆ’π‘’1βˆ’(1+𝑏)π‘Ž βˆ’π‘’1βˆ’(1+𝑏)π‘Ž NOUN iajs-3349 66 3 ( ( PUNCT iajs-3349 66 4 13 13 NUM iajs-3349 66 5 ) ) SYM iajs-3349 66 6 4 4 NUM iajs-3349 66 7 . . PUNCT iajs-3349 66 8 statistical statistical ADJ iajs-3349 66 9 properties property NOUN iajs-3349 66 10 of of ADP iajs-3349 66 11 [ [ X iajs-3349 66 12 0,1 0,1 NUM iajs-3349 66 13 ] ] PUNCT iajs-3349 66 14 truncated truncate VERB iajs-3349 66 15 nadarajah nadarajah PROPN iajs-3349 66 16 - - PUNCT iajs-3349 66 17 haghighi haghighi PROPN iajs-3349 66 18 exponential exponential ADJ iajs-3349 66 19 distribution distribution NOUN iajs-3349 66 20 in in ADP iajs-3349 66 21 this this DET iajs-3349 66 22 section section NOUN iajs-3349 66 23 , , PUNCT iajs-3349 66 24 we we PRON iajs-3349 66 25 introduce introduce VERB iajs-3349 66 26 some some PRON iajs-3349 66 27 of of ADP iajs-3349 66 28 important important ADJ iajs-3349 66 29 statistical statistical ADJ iajs-3349 66 30 properties property NOUN iajs-3349 66 31 of of ADP iajs-3349 66 32 [ [ X iajs-3349 66 33 0,1 0,1 NUM iajs-3349 66 34 ] ] PUNCT iajs-3349 66 35 truncated truncated ADJ iajs-3349 66 36 nadarajahhaghighi nadarajahhaghighi PROPN iajs-3349 66 37 exponential exponential ADJ iajs-3349 66 38 distribution distribution NOUN iajs-3349 66 39 like like ADP iajs-3349 66 40 mixture mixture NOUN iajs-3349 66 41 representation representation NOUN iajs-3349 66 42 , , PUNCT iajs-3349 66 43 quantile quantile ADJ iajs-3349 66 44 function function NOUN iajs-3349 66 45 , , PUNCT iajs-3349 66 46 moments moment NOUN iajs-3349 66 47 , , PUNCT iajs-3349 66 48 incomplete incomplete ADJ iajs-3349 66 49 moments moment NOUN iajs-3349 66 50 , , PUNCT iajs-3349 66 51 moment moment NOUN iajs-3349 66 52 generated generate VERB iajs-3349 66 53 function function NOUN iajs-3349 66 54 , , PUNCT iajs-3349 66 55 rΓ©nyi rΓ©nyi PROPN iajs-3349 66 56 entropy entropy PROPN iajs-3349 66 57 , , PUNCT iajs-3349 66 58 shannon shannon PROPN iajs-3349 66 59 entropy entropy PROPN iajs-3349 66 60 , , PUNCT iajs-3349 66 61 𝑐 𝑐 PROPN iajs-3349 66 62 βˆ’ βˆ’ PROPN iajs-3349 66 63 entropy entropy NOUN iajs-3349 66 64 , , PUNCT iajs-3349 66 65 and and CCONJ iajs-3349 66 66 order order NOUN iajs-3349 66 67 statistics statistic NOUN iajs-3349 66 68 . . PUNCT iajs-3349 67 1 4.1 4.1 NUM iajs-3349 67 2 mixture mixture NOUN iajs-3349 67 3 representation representation NOUN iajs-3349 67 4 the the DET iajs-3349 67 5 mixture mixture NOUN iajs-3349 67 6 representation representation NOUN iajs-3349 67 7 of of ADP iajs-3349 67 8 the the DET iajs-3349 67 9 pdf pdf NOUN iajs-3349 67 10 is be AUX iajs-3349 67 11 essential essential ADJ iajs-3349 67 12 in in ADP iajs-3349 67 13 the the DET iajs-3349 67 14 derivation derivation NOUN iajs-3349 67 15 of of ADP iajs-3349 67 16 the the DET iajs-3349 67 17 statistical statistical ADJ iajs-3349 67 18 properties property NOUN iajs-3349 67 19 of of ADP iajs-3349 67 20 [ [ X iajs-3349 67 21 0,1 0,1 NUM iajs-3349 67 22 ] ] PUNCT iajs-3349 67 23 truncated truncate VERB iajs-3349 67 24 nadarajah nadarajah PROPN iajs-3349 67 25 - - PUNCT iajs-3349 67 26 haghighi haghighi PROPN iajs-3349 67 27 exponential exponential ADJ iajs-3349 67 28 distribution distribution NOUN iajs-3349 67 29 . . PUNCT iajs-3349 68 1 the the DET iajs-3349 68 2 mixture mixture NOUN iajs-3349 68 3 representation representation NOUN iajs-3349 68 4 on on ADP iajs-3349 68 5 the the DET iajs-3349 68 6 [ [ NOUN iajs-3349 68 7 0,1 0,1 NUM iajs-3349 68 8 ] ] PUNCT iajs-3349 68 9 truncated truncate VERB iajs-3349 68 10 nadarajah nadarajah PROPN iajs-3349 68 11 - - PUNCT iajs-3349 68 12 haghighi haghighi PROPN iajs-3349 68 13 exponential exponential ADJ iajs-3349 68 14 distribution distribution NOUN iajs-3349 68 15 pdf pdf NOUN iajs-3349 68 16 can can AUX iajs-3349 68 17 be be AUX iajs-3349 68 18 written write VERB iajs-3349 68 19 as as SCONJ iajs-3349 68 20 follows follow VERB iajs-3349 68 21 : : PUNCT iajs-3349 68 22 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 PROPN iajs-3349 68 23 = = PUNCT iajs-3349 68 24 π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 68 25 ) ) PUNCT iajs-3349 68 26 ) ) PUNCT iajs-3349 69 1 π‘Žβˆ’1 π‘Žβˆ’1 PROPN iajs-3349 69 2 𝑒 𝑒 PROPN iajs-3349 69 3 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 69 4 ) ) PUNCT iajs-3349 69 5 ) ) PUNCT iajs-3349 70 1 π‘Ž π‘Ž DET iajs-3349 70 2 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 70 3 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 70 4 using use VERB iajs-3349 70 5 the the DET iajs-3349 70 6 expansion expansion NOUN iajs-3349 70 7 of of ADP iajs-3349 70 8 exponential exponential NOUN iajs-3349 70 9 , , PUNCT iajs-3349 70 10 we we PRON iajs-3349 70 11 get get VERB iajs-3349 70 12 that that DET iajs-3349 70 13 ihjpas ihjpa NOUN iajs-3349 70 14 . . PUNCT iajs-3349 71 1 37 37 NUM iajs-3349 71 2 ( ( PUNCT iajs-3349 71 3 2 2 NUM iajs-3349 71 4 ) ) PUNCT iajs-3349 71 5 2024 2024 NUM iajs-3349 71 6 366 366 NUM iajs-3349 71 7 𝑒1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 𝑒1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ NUM iajs-3349 71 8 ) ) PUNCT iajs-3349 71 9 ) ) PUNCT iajs-3349 72 1 π‘Ž π‘Ž X iajs-3349 72 2 = = SYM iajs-3349 72 3 βˆ‘ βˆ‘ PROPN iajs-3349 72 4 1 1 NUM iajs-3349 72 5 𝑠 𝑠 NOUN iajs-3349 72 6 ! ! PUNCT iajs-3349 73 1 ( ( PUNCT iajs-3349 73 2 1 1 NUM iajs-3349 73 3 βˆ’ βˆ’ PROPN iajs-3349 73 4 ( ( PUNCT iajs-3349 73 5 1 1 NUM iajs-3349 73 6 + + CCONJ iajs-3349 73 7 𝑏(1 𝑏(1 PROPN iajs-3349 73 8 βˆ’ βˆ’ NOUN iajs-3349 73 9 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 73 10 ) ) PUNCT iajs-3349 73 11 ) ) PUNCT iajs-3349 74 1 π‘Ž π‘Ž X iajs-3349 74 2 ) ) PUNCT iajs-3349 74 3 𝑠 𝑠 NOUN iajs-3349 74 4 ∞ ∞ PROPN iajs-3349 74 5 𝑠=0 𝑠=0 PUNCT iajs-3349 74 6 then then ADV iajs-3349 74 7 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 PROPN iajs-3349 74 8 = = PUNCT iajs-3349 74 9 π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 74 10 ) ) PUNCT iajs-3349 74 11 ) ) PUNCT iajs-3349 75 1 π‘Žβˆ’1 π‘Žβˆ’1 PROPN iajs-3349 75 2 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 75 3 1βˆ’π‘’1βˆ’[1+𝑏]π‘Ž 1βˆ’π‘’1βˆ’[1+𝑏]π‘Ž PROPN iajs-3349 75 4 βˆ‘ βˆ‘ PROPN iajs-3349 75 5 1 1 NUM iajs-3349 75 6 𝑠 𝑠 NOUN iajs-3349 75 7 ! ! PUNCT iajs-3349 76 1 ( ( PUNCT iajs-3349 76 2 1 1 NUM iajs-3349 76 3 βˆ’ βˆ’ PROPN iajs-3349 76 4 ( ( PUNCT iajs-3349 76 5 1 1 NUM iajs-3349 76 6 + + CCONJ iajs-3349 76 7 𝑏(1 𝑏(1 PROPN iajs-3349 76 8 βˆ’ βˆ’ NOUN iajs-3349 76 9 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 76 10 ) ) PUNCT iajs-3349 76 11 ) ) PUNCT iajs-3349 77 1 π‘Ž π‘Ž X iajs-3349 77 2 ) ) PUNCT iajs-3349 77 3 𝑠 𝑠 NOUN iajs-3349 77 4 ∞ ∞ PROPN iajs-3349 77 5 𝑠=0 𝑠=0 PUNCT iajs-3349 77 6 now now ADV iajs-3349 77 7 , , PUNCT iajs-3349 77 8 by by ADP iajs-3349 77 9 the the DET iajs-3349 77 10 generalized generalize VERB iajs-3349 77 11 binomial binomial NOUN iajs-3349 77 12 theorem theorem NOUN iajs-3349 77 13 , , PUNCT iajs-3349 77 14 we we PRON iajs-3349 77 15 get get VERB iajs-3349 77 16 that that PRON iajs-3349 77 17 ( ( PUNCT iajs-3349 77 18 1 1 NUM iajs-3349 77 19 βˆ’ βˆ’ PROPN iajs-3349 77 20 ( ( PUNCT iajs-3349 77 21 1 1 NUM iajs-3349 77 22 + + CCONJ iajs-3349 77 23 𝑏(1 𝑏(1 PROPN iajs-3349 77 24 βˆ’ βˆ’ NOUN iajs-3349 77 25 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 77 26 ) ) PUNCT iajs-3349 77 27 ) ) PUNCT iajs-3349 78 1 π‘Ž π‘Ž X iajs-3349 78 2 ) ) PUNCT iajs-3349 78 3 𝑠 𝑠 PROPN iajs-3349 79 1 = = PUNCT iajs-3349 79 2 βˆ‘ βˆ‘ PROPN iajs-3349 79 3 ( ( PUNCT iajs-3349 79 4 𝑠 𝑠 NOUN iajs-3349 79 5 𝑛 𝑛 PROPN iajs-3349 79 6 ) ) PUNCT iajs-3349 79 7 ∞ ∞ NUM iajs-3349 79 8 𝑛=0 𝑛=0 NOUN iajs-3349 79 9 ( ( PUNCT iajs-3349 79 10 βˆ’1)𝑛 βˆ’1)𝑛 X iajs-3349 79 11 ( ( PUNCT iajs-3349 79 12 1 1 NUM iajs-3349 79 13 + + CCONJ iajs-3349 79 14 𝑏(1 𝑏(1 PROPN iajs-3349 79 15 βˆ’ βˆ’ NOUN iajs-3349 79 16 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 79 17 ) ) PUNCT iajs-3349 79 18 ) ) PUNCT iajs-3349 79 19 π‘Žπ‘› π‘Žπ‘› PRON iajs-3349 79 20 so so SCONJ iajs-3349 79 21 that that SCONJ iajs-3349 79 22 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 PROPN iajs-3349 79 23 = = SYM iajs-3349 79 24 π‘Žπ‘πœ† π‘Žπ‘πœ† NOUN iajs-3349 79 25 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 79 26 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 79 27 βˆ‘ βˆ‘ PUNCT iajs-3349 79 28 βˆ‘ βˆ‘ PROPN iajs-3349 79 29 1 1 NUM iajs-3349 79 30 𝑠 𝑠 NOUN iajs-3349 79 31 ! ! PUNCT iajs-3349 80 1 ( ( PUNCT iajs-3349 80 2 𝑠 𝑠 NOUN iajs-3349 80 3 𝑛 𝑛 PROPN iajs-3349 80 4 ) ) PUNCT iajs-3349 80 5 ∞ ∞ NUM iajs-3349 81 1 𝑛=0 𝑛=0 NOUN iajs-3349 81 2 ( ( PUNCT iajs-3349 81 3 βˆ’1)𝑛 βˆ’1)𝑛 X iajs-3349 81 4 ( ( PUNCT iajs-3349 81 5 1 1 NUM iajs-3349 81 6 + + CCONJ iajs-3349 81 7 𝑏(1 𝑏(1 PROPN iajs-3349 81 8 βˆ’ βˆ’ NOUN iajs-3349 81 9 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 81 10 ) ) PUNCT iajs-3349 81 11 ) ) PUNCT iajs-3349 82 1 π‘Žπ‘›+π‘Žβˆ’1∞ π‘Žπ‘›+π‘Žβˆ’1∞ NUM iajs-3349 82 2 𝑠=0 𝑠=0 PUNCT iajs-3349 82 3 again again ADV iajs-3349 82 4 , , PUNCT iajs-3349 82 5 by by ADP iajs-3349 82 6 the the DET iajs-3349 82 7 generalized generalize VERB iajs-3349 82 8 binomial binomial ADJ iajs-3349 82 9 theorem theorem NOUN iajs-3349 82 10 ( ( PUNCT iajs-3349 82 11 1 1 NUM iajs-3349 82 12 + + CCONJ iajs-3349 82 13 𝑏(1 𝑏(1 PROPN iajs-3349 82 14 βˆ’ βˆ’ NOUN iajs-3349 82 15 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 82 16 ) ) PUNCT iajs-3349 82 17 ) ) PUNCT iajs-3349 82 18 π‘Žπ‘›+π‘Žβˆ’1 π‘Žπ‘›+π‘Žβˆ’1 NOUN iajs-3349 82 19 = = PUNCT iajs-3349 82 20 βˆ‘ βˆ‘ PUNCT iajs-3349 82 21 ( ( PUNCT iajs-3349 82 22 π‘Žπ‘›+π‘Žβˆ’1 π‘Žπ‘›+π‘Žβˆ’1 NOUN iajs-3349 82 23 𝑑 𝑑 PROPN iajs-3349 82 24 ) ) PUNCT iajs-3349 82 25 ∞ ∞ NUM iajs-3349 82 26 𝑑=0 𝑑=0 X iajs-3349 82 27 𝑏𝑑 𝑏𝑑 X iajs-3349 82 28 ( ( PUNCT iajs-3349 82 29 ( ( PUNCT iajs-3349 82 30 1 1 NUM iajs-3349 82 31 βˆ’ βˆ’ NOUN iajs-3349 82 32 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 82 33 ) ) PUNCT iajs-3349 82 34 ) ) PUNCT iajs-3349 83 1 𝑑 𝑑 VERB iajs-3349 83 2 then then ADV iajs-3349 83 3 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 PROPN iajs-3349 83 4 = = SYM iajs-3349 83 5 π‘Žπ‘πœ† π‘Žπ‘πœ† NOUN iajs-3349 83 6 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 83 7 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 83 8 βˆ‘ βˆ‘ PUNCT iajs-3349 83 9 βˆ‘ βˆ‘ PROPN iajs-3349 83 10 βˆ‘ βˆ‘ PUNCT iajs-3349 83 11 ( ( PUNCT iajs-3349 83 12 βˆ’1)𝑛 βˆ’1)𝑛 X iajs-3349 83 13 𝑠 𝑠 X iajs-3349 83 14 ! ! PUNCT iajs-3349 84 1 ( ( PUNCT iajs-3349 84 2 𝑠 𝑠 NOUN iajs-3349 84 3 𝑛 𝑛 PROPN iajs-3349 84 4 ) ) PUNCT iajs-3349 84 5 ( ( PUNCT iajs-3349 84 6 π‘Žπ‘›+π‘Žβˆ’1 π‘Žπ‘›+π‘Žβˆ’1 NOUN iajs-3349 84 7 𝑑 𝑑 NOUN iajs-3349 84 8 ) ) PUNCT iajs-3349 84 9 ∞ ∞ NUM iajs-3349 84 10 𝑑=0 𝑑=0 NOUN iajs-3349 84 11 ∞ ∞ NUM iajs-3349 84 12 𝑛=0 𝑛=0 PROPN iajs-3349 84 13 𝑏𝑑 𝑏𝑑 PROPN iajs-3349 84 14 ( ( PUNCT iajs-3349 84 15 ( ( PUNCT iajs-3349 84 16 1 1 NUM iajs-3349 84 17 βˆ’ βˆ’ NOUN iajs-3349 84 18 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 84 19 ) ) PUNCT iajs-3349 84 20 ) ) PUNCT iajs-3349 85 1 π‘‘βˆž π‘‘βˆž NOUN iajs-3349 85 2 𝑠=0 𝑠=0 PUNCT iajs-3349 85 3 by by ADP iajs-3349 85 4 the the DET iajs-3349 85 5 same same ADJ iajs-3349 85 6 way way NOUN iajs-3349 85 7 , , PUNCT iajs-3349 85 8 we we PRON iajs-3349 85 9 obtain obtain VERB iajs-3349 85 10 that that PRON iajs-3349 85 11 ( ( PUNCT iajs-3349 85 12 ( ( PUNCT iajs-3349 85 13 1 1 NUM iajs-3349 85 14 βˆ’ βˆ’ NOUN iajs-3349 85 15 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 85 16 ) ) PUNCT iajs-3349 85 17 ) ) PUNCT iajs-3349 86 1 𝑑 𝑑 PROPN iajs-3349 86 2 = = PUNCT iajs-3349 86 3 βˆ‘ βˆ‘ PUNCT iajs-3349 86 4 ( ( PUNCT iajs-3349 86 5 𝑑 𝑑 PROPN iajs-3349 86 6 π‘˜ π‘˜ PROPN iajs-3349 86 7 ) ) PUNCT iajs-3349 86 8 ∞ ∞ PROPN iajs-3349 86 9 π‘˜=0 π‘˜=0 PROPN iajs-3349 86 10 ( ( PUNCT iajs-3349 86 11 βˆ’1)π‘˜π‘’βˆ’πœ†π‘˜π‘₯ βˆ’1)π‘˜π‘’βˆ’πœ†π‘˜π‘₯ NOUN iajs-3349 86 12 then then ADV iajs-3349 86 13 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 PROPN iajs-3349 87 1 = = NOUN iajs-3349 88 1 π‘Žπ‘πœ† π‘Žπ‘πœ† NOUN iajs-3349 89 1 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 89 2 βˆ‘ βˆ‘ PROPN iajs-3349 89 3 βˆ‘ βˆ‘ PROPN iajs-3349 89 4 βˆ‘ βˆ‘ PUNCT iajs-3349 89 5 ( ( PUNCT iajs-3349 89 6 𝑑 𝑑 PROPN iajs-3349 89 7 π‘˜ π‘˜ PROPN iajs-3349 89 8 ) ) PUNCT iajs-3349 89 9 ∞ ∞ PROPN iajs-3349 89 10 π‘˜=0 π‘˜=0 PROPN iajs-3349 89 11 1 1 NUM iajs-3349 89 12 𝑠 𝑠 NOUN iajs-3349 89 13 ! ! PUNCT iajs-3349 90 1 ( ( PUNCT iajs-3349 90 2 𝑠 𝑠 NOUN iajs-3349 90 3 𝑛 𝑛 PROPN iajs-3349 90 4 ) ) PUNCT iajs-3349 90 5 ∞ ∞ NUM iajs-3349 91 1 𝑛=0 𝑛=0 PROPN iajs-3349 91 2 ( ( PUNCT iajs-3349 91 3 βˆ’1)𝑛+π‘˜ βˆ’1)𝑛+π‘˜ X iajs-3349 91 4 ∞ ∞ PROPN iajs-3349 91 5 𝑠=0 𝑠=0 SYM iajs-3349 91 6 π‘’βˆ’πœ†π‘₯(π‘˜+1 π‘’βˆ’πœ†π‘₯(π‘˜+1 PROPN iajs-3349 91 7 ) ) PUNCT iajs-3349 91 8 moreover moreover ADV iajs-3349 91 9 , , PUNCT iajs-3349 91 10 this this DET iajs-3349 91 11 equation equation NOUN iajs-3349 91 12 can can AUX iajs-3349 91 13 be be AUX iajs-3349 91 14 rewritten rewrite VERB iajs-3349 91 15 as as ADP iajs-3349 91 16 follow follow VERB iajs-3349 91 17 : : PUNCT iajs-3349 91 18 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 NOUN iajs-3349 91 19 = = SYM iajs-3349 91 20 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 PROPN iajs-3349 91 21 𝑒 𝑒 PROPN iajs-3349 91 22 βˆ’πœ†π‘₯(π‘˜+1 βˆ’πœ†π‘₯(π‘˜+1 PROPN iajs-3349 91 23 ) ) PUNCT iajs-3349 91 24 ( ( PUNCT iajs-3349 91 25 14 14 NUM iajs-3349 91 26 ) ) PUNCT iajs-3349 91 27 where where SCONJ iajs-3349 91 28 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 NOUN iajs-3349 91 29 = = SYM iajs-3349 91 30 π‘Žπ‘πœ† π‘Žπ‘πœ† NOUN iajs-3349 91 31 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 91 32 βˆ‘ βˆ‘ PROPN iajs-3349 91 33 βˆ‘ βˆ‘ PROPN iajs-3349 91 34 βˆ‘ βˆ‘ PUNCT iajs-3349 91 35 ( ( PUNCT iajs-3349 91 36 𝑑 𝑑 PROPN iajs-3349 91 37 π‘˜ π‘˜ PROPN iajs-3349 91 38 ) ) PUNCT iajs-3349 91 39 ∞ ∞ PROPN iajs-3349 91 40 π‘˜=0 π‘˜=0 PROPN iajs-3349 91 41 1 1 NUM iajs-3349 91 42 𝑠 𝑠 NOUN iajs-3349 91 43 ! ! PUNCT iajs-3349 92 1 ( ( PUNCT iajs-3349 92 2 𝑠 𝑠 NOUN iajs-3349 92 3 𝑛 𝑛 PROPN iajs-3349 92 4 ) ) PUNCT iajs-3349 92 5 ∞ ∞ NUM iajs-3349 93 1 𝑛=0 𝑛=0 PROPN iajs-3349 93 2 ( ( PUNCT iajs-3349 93 3 βˆ’1)𝑛+π‘˜ βˆ’1)𝑛+π‘˜ X iajs-3349 93 4 ∞ ∞ PROPN iajs-3349 93 5 𝑠=0 𝑠=0 SYM iajs-3349 93 6 4.2 4.2 NUM iajs-3349 93 7 quantile quantile NOUN iajs-3349 93 8 function function VERB iajs-3349 93 9 the the DET iajs-3349 93 10 [ [ X iajs-3349 93 11 0,1 0,1 NUM iajs-3349 93 12 ] ] PUNCT iajs-3349 93 13 truncated truncate VERB iajs-3349 93 14 nadarajah nadarajah PROPN iajs-3349 93 15 - - PUNCT iajs-3349 93 16 haghighi haghighi PROPN iajs-3349 93 17 exponential exponential ADJ iajs-3349 93 18 distribution distribution NOUN iajs-3349 93 19 quantile quantile NOUN iajs-3349 93 20 function function NOUN iajs-3349 93 21 can can AUX iajs-3349 93 22 be be AUX iajs-3349 93 23 obtained obtain VERB iajs-3349 93 24 by by ADP iajs-3349 93 25 inverting invert VERB iajs-3349 93 26 the the DET iajs-3349 93 27 cdf cdf NOUN iajs-3349 93 28 which which PRON iajs-3349 93 29 is be AUX iajs-3349 93 30 defined define VERB iajs-3349 93 31 in in ADP iajs-3349 93 32 ( ( PUNCT iajs-3349 93 33 10 10 NUM iajs-3349 93 34 ) ) PUNCT iajs-3349 93 35 , , PUNCT iajs-3349 93 36 as as SCONJ iajs-3349 93 37 follow follow VERB iajs-3349 93 38 [ [ X iajs-3349 93 39 23 23 NUM iajs-3349 93 40 ] ] SYM iajs-3349 93 41 : : PUNCT iajs-3349 93 42 𝑄(𝑒 𝑄(𝑒 X iajs-3349 93 43 ) ) PUNCT iajs-3349 93 44 = = PUNCT iajs-3349 94 1 πΉβˆ’1 πΉβˆ’1 X iajs-3349 95 1 [ [ X iajs-3349 95 2 0,1]𝑇𝑁𝐻𝐸π‘₯π‘π‘œ 0,1]𝑇𝑁𝐻𝐸π‘₯π‘π‘œ NUM iajs-3349 95 3 ( ( PUNCT iajs-3349 95 4 π‘₯ π‘₯ NOUN iajs-3349 95 5 ) ) PUNCT iajs-3349 95 6 𝑄(𝑒 𝑄(𝑒 NUM iajs-3349 95 7 ) ) PUNCT iajs-3349 95 8 = = SYM iajs-3349 95 9 πΊβˆ’1 πΊβˆ’1 NOUN iajs-3349 95 10 ( ( PUNCT iajs-3349 95 11 βˆ’1 βˆ’1 NOUN iajs-3349 95 12 πœ† πœ† PRON iajs-3349 95 13 ( ( PUNCT iajs-3349 95 14 ln ln X iajs-3349 95 15 { { PUNCT iajs-3349 95 16 1 1 NUM iajs-3349 95 17 βˆ’ βˆ’ PROPN iajs-3349 95 18 1 1 NUM iajs-3349 95 19 𝑏 𝑏 PROPN iajs-3349 95 20 ( ( PUNCT iajs-3349 95 21 ( ( PUNCT iajs-3349 95 22 1 1 NUM iajs-3349 95 23 βˆ’ βˆ’ NOUN iajs-3349 95 24 ln{1 ln{1 NOUN iajs-3349 95 25 βˆ’ βˆ’ PROPN iajs-3349 95 26 𝑒(1 𝑒(1 PROPN iajs-3349 95 27 βˆ’ βˆ’ NOUN iajs-3349 95 28 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž NOUN iajs-3349 95 29 ) ) PUNCT iajs-3349 95 30 } } PUNCT iajs-3349 95 31 ) ) PUNCT iajs-3349 95 32 1 1 NUM iajs-3349 96 1 π‘Ž π‘Ž PRON iajs-3349 96 2 βˆ’ βˆ’ NOUN iajs-3349 96 3 1 1 NUM iajs-3349 96 4 ) ) PUNCT iajs-3349 96 5 } } PUNCT iajs-3349 96 6 ) ) PUNCT iajs-3349 96 7 ) ) PUNCT iajs-3349 96 8 ( ( PUNCT iajs-3349 96 9 15 15 X iajs-3349 96 10 ) ) PUNCT iajs-3349 96 11 now now ADV iajs-3349 96 12 , , PUNCT iajs-3349 96 13 according accord VERB iajs-3349 96 14 to to ADP iajs-3349 96 15 eq eq PROPN iajs-3349 96 16 . . PUNCT iajs-3349 97 1 ( ( PUNCT iajs-3349 97 2 15 15 NUM iajs-3349 97 3 ) ) PUNCT iajs-3349 97 4 , , PUNCT iajs-3349 97 5 the the DET iajs-3349 97 6 median median PROPN iajs-3349 97 7 ( ( PUNCT iajs-3349 97 8 𝑀 𝑀 PROPN iajs-3349 97 9 ) ) PUNCT iajs-3349 97 10 of of ADP iajs-3349 97 11 [ [ X iajs-3349 97 12 0.1 0.1 NUM iajs-3349 97 13 ] ] PUNCT iajs-3349 97 14 tnhe tnhe NOUN iajs-3349 97 15 distribution distribution NOUN iajs-3349 97 16 can can AUX iajs-3349 97 17 be be AUX iajs-3349 97 18 obtained obtain VERB iajs-3349 97 19 by by ADP iajs-3349 97 20 instead instead ADV iajs-3349 97 21 u u NOUN iajs-3349 97 22 = = PROPN iajs-3349 97 23 0.5 0.5 NUM iajs-3349 97 24 . . PUNCT iajs-3349 98 1 4.3 4.3 NUM iajs-3349 98 2 moments moment NOUN iajs-3349 98 3 the the DET iajs-3349 98 4 π‘Ÿπ‘‘β„Ž π‘Ÿπ‘‘β„Ž NOUN iajs-3349 98 5 moments moment NOUN iajs-3349 98 6 of of ADP iajs-3349 98 7 [ [ X iajs-3349 98 8 0,1 0,1 NUM iajs-3349 98 9 ] ] PUNCT iajs-3349 98 10 truncated truncate VERB iajs-3349 98 11 nadarajah nadarajah PROPN iajs-3349 98 12 - - PUNCT iajs-3349 98 13 haghighi haghighi PROPN iajs-3349 98 14 exponential exponential ADJ iajs-3349 98 15 distribution distribution NOUN iajs-3349 98 16 is be AUX iajs-3349 98 17 [ [ X iajs-3349 98 18 24 24 NUM iajs-3349 98 19 ] ] PUNCT iajs-3349 98 20 , , PUNCT iajs-3349 98 21 [ [ X iajs-3349 98 22 25 25 NUM iajs-3349 98 23 ] ] PUNCT iajs-3349 98 24 . . PUNCT iajs-3349 99 1 πœ‡π‘Ÿ πœ‡π‘Ÿ NOUN iajs-3349 99 2 = = SYM iajs-3349 99 3 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 NOUN iajs-3349 99 4 ( ( PUNCT iajs-3349 99 5 1 1 NUM iajs-3349 99 6 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 99 7 ) ) PUNCT iajs-3349 99 8 ) ) PUNCT iajs-3349 100 1 π‘Ÿ+1 π‘Ÿ+1 NUM iajs-3349 101 1 Ξ³(π‘Ÿ Ξ³(π‘Ÿ NOUN iajs-3349 101 2 + + NOUN iajs-3349 101 3 1 1 X iajs-3349 101 4 ) ) PUNCT iajs-3349 101 5 we we PRON iajs-3349 101 6 can can AUX iajs-3349 101 7 obtain obtain VERB iajs-3349 101 8 π‘Ÿπ‘‘β„Ž π‘Ÿπ‘‘β„Ž NOUN iajs-3349 101 9 moment moment NOUN iajs-3349 101 10 of of ADP iajs-3349 101 11 a a DET iajs-3349 101 12 random random ADJ iajs-3349 101 13 variable variable NOUN iajs-3349 101 14 x x PUNCT iajs-3349 101 15 given give VERB iajs-3349 101 16 by by ADP iajs-3349 101 17 the the DET iajs-3349 101 18 following follow VERB iajs-3349 101 19 relation relation NOUN iajs-3349 101 20 : : PUNCT iajs-3349 101 21 πœ‡π‘Ÿ πœ‡π‘Ÿ PROPN iajs-3349 101 22 = = SYM iajs-3349 101 23 ∫ ∫ PROPN iajs-3349 101 24 π‘₯π‘Ÿπ‘“(π‘₯)𝑑π‘₯ π‘₯π‘Ÿπ‘“(π‘₯)𝑑π‘₯ NUM iajs-3349 101 25 ∞ ∞ PROPN iajs-3349 101 26 βˆ’βˆž βˆ’βˆž ADP iajs-3349 101 27 ihjpas ihjpa NOUN iajs-3349 101 28 . . PUNCT iajs-3349 102 1 37 37 NUM iajs-3349 102 2 ( ( PUNCT iajs-3349 102 3 2 2 NUM iajs-3349 102 4 ) ) PUNCT iajs-3349 102 5 2024 2024 NUM iajs-3349 102 6 367 367 NUM iajs-3349 102 7 where where SCONJ iajs-3349 102 8 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3349 102 9 ) ) PUNCT iajs-3349 102 10 is be AUX iajs-3349 102 11 given give VERB iajs-3349 102 12 in in ADP iajs-3349 102 13 ( ( PUNCT iajs-3349 102 14 14 14 NUM iajs-3349 102 15 ) ) PUNCT iajs-3349 102 16 , , PUNCT iajs-3349 102 17 so so SCONJ iajs-3349 102 18 that that SCONJ iajs-3349 102 19 πœ‡π‘Ÿ πœ‡π‘Ÿ NOUN iajs-3349 102 20 = = SYM iajs-3349 102 21 ∫ ∫ PROPN iajs-3349 102 22 π‘₯π‘ŸΟ‰π‘ ,𝑛,π‘˜,𝑑 π‘₯π‘ŸΟ‰π‘ ,𝑛,π‘˜,𝑑 PROPN iajs-3349 102 23 𝑒 𝑒 X iajs-3349 102 24 βˆ’πœ†π‘₯(π‘˜+1)𝑑π‘₯ βˆ’πœ†π‘₯(π‘˜+1)𝑑π‘₯ NOUN iajs-3349 102 25 ∞ ∞ PROPN iajs-3349 102 26 0 0 PUNCT iajs-3349 103 1 let let VERB iajs-3349 103 2 𝑦 𝑦 NOUN iajs-3349 103 3 = = NOUN iajs-3349 103 4 πœ†π‘₯(π‘˜ πœ†π‘₯(π‘˜ X iajs-3349 103 5 + + NOUN iajs-3349 103 6 1 1 NUM iajs-3349 103 7 ) ) PUNCT iajs-3349 103 8 ⟹ ⟹ PUNCT iajs-3349 104 1 π‘₯ π‘₯ NOUN iajs-3349 104 2 = = SYM iajs-3349 104 3 𝑦 𝑦 SYM iajs-3349 104 4 1 1 NUM iajs-3349 104 5 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 104 6 ) ) PUNCT iajs-3349 104 7 ⟹ ⟹ VERB iajs-3349 105 1 𝑑π‘₯ 𝑑π‘₯ NOUN iajs-3349 105 2 = = SYM iajs-3349 105 3 1 1 NUM iajs-3349 105 4 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 105 5 ) ) PUNCT iajs-3349 105 6 𝑑𝑦 𝑑𝑦 ADP iajs-3349 105 7 that that ADV iajs-3349 105 8 is is ADV iajs-3349 105 9 , , PUNCT iajs-3349 105 10 πœ‡π‘Ÿ πœ‡π‘Ÿ NOUN iajs-3349 105 11 = = SYM iajs-3349 105 12 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 NOUN iajs-3349 105 13 ∫ ∫ X iajs-3349 105 14 ( ( PUNCT iajs-3349 105 15 𝑦 𝑦 NOUN iajs-3349 105 16 1 1 NUM iajs-3349 105 17 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 105 18 ) ) PUNCT iajs-3349 105 19 ) ) PUNCT iajs-3349 106 1 π‘Ÿ π‘Ÿ X iajs-3349 106 2 π‘’βˆ’ π‘’βˆ’ VERB iajs-3349 106 3 𝑦 𝑦 NOUN iajs-3349 106 4 1 1 NUM iajs-3349 106 5 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 106 6 ) ) PUNCT iajs-3349 106 7 𝑑𝑦 𝑑𝑦 ADP iajs-3349 106 8 ∞ ∞ NOUN iajs-3349 106 9 0 0 X iajs-3349 107 1 = = SYM iajs-3349 107 2 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 NOUN iajs-3349 107 3 ( ( PUNCT iajs-3349 107 4 1 1 NUM iajs-3349 107 5 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 107 6 ) ) PUNCT iajs-3349 107 7 ) ) PUNCT iajs-3349 108 1 π‘Ÿ+1 π‘Ÿ+1 NUM iajs-3349 108 2 ∫ ∫ NOUN iajs-3349 108 3 ( ( PUNCT iajs-3349 108 4 𝑦)π‘Ÿ 𝑦)π‘Ÿ PUNCT iajs-3349 108 5 π‘’βˆ’ π‘’βˆ’ X iajs-3349 108 6 𝑦 𝑦 NOUN iajs-3349 108 7 𝑑𝑦 𝑑𝑦 ADJ iajs-3349 108 8 ∞ ∞ PROPN iajs-3349 108 9 0 0 NUM iajs-3349 109 1 ( ( PUNCT iajs-3349 109 2 16 16 NUM iajs-3349 109 3 ) ) PUNCT iajs-3349 109 4 the the DET iajs-3349 109 5 mean mean NOUN iajs-3349 109 6 and and CCONJ iajs-3349 109 7 variance variance NOUN iajs-3349 109 8 of of ADP iajs-3349 109 9 [ [ X iajs-3349 109 10 0,1 0,1 NUM iajs-3349 109 11 ] ] PUNCT iajs-3349 109 12 truncated truncate VERB iajs-3349 109 13 nadarajah nadarajah PROPN iajs-3349 109 14 - - PUNCT iajs-3349 109 15 haghighi haghighi PROPN iajs-3349 109 16 exponential exponential ADJ iajs-3349 109 17 distribution distribution NOUN iajs-3349 109 18 can can AUX iajs-3349 109 19 be be AUX iajs-3349 109 20 obtained obtain VERB iajs-3349 109 21 in in ADP iajs-3349 109 22 equation equation NOUN iajs-3349 109 23 ( ( PUNCT iajs-3349 109 24 15 15 NUM iajs-3349 109 25 ) ) PUNCT iajs-3349 109 26 , , PUNCT iajs-3349 109 27 as as ADP iajs-3349 109 28 follow follow VERB iajs-3349 109 29 : : PUNCT iajs-3349 109 30 πœ‡1 πœ‡1 NOUN iajs-3349 109 31 = = SYM iajs-3349 109 32 𝐸(𝑋 𝐸(𝑋 PROPN iajs-3349 109 33 ) ) PUNCT iajs-3349 110 1 [ [ X iajs-3349 110 2 0,1]𝑇𝑁𝐻𝐸 0,1]𝑇𝑁𝐻𝐸 NUM iajs-3349 110 3 = = SYM iajs-3349 110 4 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 NOUN iajs-3349 110 5 ( ( PUNCT iajs-3349 110 6 βˆ’1 βˆ’1 NOUN iajs-3349 110 7 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 110 8 ) ) PUNCT iajs-3349 110 9 ) ) PUNCT iajs-3349 110 10 2 2 NUM iajs-3349 110 11 πœ‡2 πœ‡2 NOUN iajs-3349 110 12 = = PUNCT iajs-3349 110 13 𝐸(𝑋2 𝐸(𝑋2 NOUN iajs-3349 110 14 ) ) PUNCT iajs-3349 111 1 [ [ X iajs-3349 111 2 0,1]𝑇𝑁𝐻𝐸 0,1]𝑇𝑁𝐻𝐸 NUM iajs-3349 111 3 = = SYM iajs-3349 111 4 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 NOUN iajs-3349 111 5 βˆ’2 βˆ’2 NOUN iajs-3349 111 6 ( ( PUNCT iajs-3349 111 7 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 111 8 ) ) PUNCT iajs-3349 111 9 ) ) PUNCT iajs-3349 111 10 3 3 NUM iajs-3349 111 11 πœ‡3 πœ‡3 NOUN iajs-3349 111 12 = = SYM iajs-3349 111 13 𝐸(𝑋3 𝐸(𝑋3 NOUN iajs-3349 111 14 ) ) PUNCT iajs-3349 112 1 [ [ X iajs-3349 112 2 0,1]𝑇𝑁𝐻𝐸 0,1]𝑇𝑁𝐻𝐸 NUM iajs-3349 112 3 = = SYM iajs-3349 112 4 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 NOUN iajs-3349 112 5 6 6 NUM iajs-3349 112 6 ( ( PUNCT iajs-3349 112 7 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 112 8 ) ) PUNCT iajs-3349 112 9 ) ) PUNCT iajs-3349 112 10 4 4 NUM iajs-3349 112 11 πœ‡4 πœ‡4 NOUN iajs-3349 112 12 = = SYM iajs-3349 112 13 𝐸(𝑋4 𝐸(𝑋4 NOUN iajs-3349 112 14 ) ) PUNCT iajs-3349 113 1 [ [ X iajs-3349 113 2 0,1]𝑇𝑁𝐻𝐸 0,1]𝑇𝑁𝐻𝐸 NUM iajs-3349 113 3 = = VERB iajs-3349 113 4 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 NOUN iajs-3349 113 5 βˆ’24 βˆ’24 ADV iajs-3349 113 6 ( ( PUNCT iajs-3349 113 7 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 113 8 ) ) PUNCT iajs-3349 113 9 ) ) PUNCT iajs-3349 113 10 5 5 NUM iajs-3349 113 11 ( ( PUNCT iajs-3349 113 12 17 17 NUM iajs-3349 113 13 ) ) PUNCT iajs-3349 113 14 ( ( PUNCT iajs-3349 113 15 18 18 NUM iajs-3349 113 16 ) ) PUNCT iajs-3349 113 17 ( ( PUNCT iajs-3349 113 18 19 19 NUM iajs-3349 113 19 ) ) PUNCT iajs-3349 113 20 ( ( PUNCT iajs-3349 113 21 20 20 NUM iajs-3349 113 22 ) ) PUNCT iajs-3349 113 23 moreover moreover ADV iajs-3349 113 24 , , PUNCT iajs-3349 113 25 the the DET iajs-3349 113 26 variance variance NOUN iajs-3349 113 27 can can AUX iajs-3349 113 28 be be AUX iajs-3349 113 29 found find VERB iajs-3349 113 30 using use VERB iajs-3349 113 31 the the DET iajs-3349 113 32 following follow VERB iajs-3349 113 33 form form NOUN iajs-3349 113 34 : : PUNCT iajs-3349 113 35 π‘‰π‘Žπ‘Ÿ(𝑋 π‘‰π‘Žπ‘Ÿ(𝑋 NUM iajs-3349 113 36 ) ) PUNCT iajs-3349 113 37 = = PUNCT iajs-3349 113 38 𝐸(𝑋2 𝐸(𝑋2 NOUN iajs-3349 113 39 ) ) PUNCT iajs-3349 114 1 βˆ’ βˆ’ PROPN iajs-3349 114 2 ( ( PUNCT iajs-3349 114 3 𝐸(𝑋 𝐸(𝑋 NOUN iajs-3349 114 4 ) ) PUNCT iajs-3349 114 5 ) ) PUNCT iajs-3349 114 6 2 2 NUM iajs-3349 114 7 ( ( PUNCT iajs-3349 114 8 21 21 NUM iajs-3349 114 9 ) ) PUNCT iajs-3349 115 1 so so SCONJ iajs-3349 115 2 that that SCONJ iajs-3349 115 3 π‘‰π‘Žπ‘Ÿ(𝑋)[0,1]𝑇𝑁𝐻𝐸 π‘‰π‘Žπ‘Ÿ(𝑋)[0,1]𝑇𝑁𝐻𝐸 PROPN iajs-3349 116 1 = = PUNCT iajs-3349 116 2 𝐷𝑗 𝐷𝑗 PROPN iajs-3349 116 3 , , PUNCT iajs-3349 116 4 π‘˜,π‘š π‘˜,π‘š PROPN iajs-3349 116 5 βˆ’2 βˆ’2 NOUN iajs-3349 116 6 ( ( PUNCT iajs-3349 116 7 πœ†(π‘˜+1 πœ†(π‘˜+1 NOUN iajs-3349 116 8 ) ) PUNCT iajs-3349 116 9 ) ) PUNCT iajs-3349 117 1 3 3 NUM iajs-3349 117 2 βˆ’ βˆ’ PROPN iajs-3349 117 3 𝐷𝑗 𝐷𝑗 PROPN iajs-3349 117 4 , , PUNCT iajs-3349 117 5 π‘˜,π‘š π‘˜,π‘š PROPN iajs-3349 117 6 1 1 NUM iajs-3349 117 7 ( ( PUNCT iajs-3349 117 8 πœ†(π‘˜+1 πœ†(π‘˜+1 NUM iajs-3349 117 9 ) ) PUNCT iajs-3349 117 10 ) ) PUNCT iajs-3349 117 11 4 4 NUM iajs-3349 117 12 ( ( PUNCT iajs-3349 117 13 22 22 NUM iajs-3349 117 14 ) ) PUNCT iajs-3349 117 15 in in ADP iajs-3349 117 16 addition addition NOUN iajs-3349 117 17 , , PUNCT iajs-3349 117 18 measures measure NOUN iajs-3349 117 19 of of ADP iajs-3349 117 20 skewness skewness NOUN iajs-3349 117 21 and and CCONJ iajs-3349 117 22 kurtosis kurtosis NOUN iajs-3349 117 23 of of ADP iajs-3349 117 24 [ [ X iajs-3349 117 25 0,1 0,1 NUM iajs-3349 117 26 ] ] PUNCT iajs-3349 117 27 tnhe tnhe VERB iajs-3349 117 28 distribution distribution NOUN iajs-3349 117 29 based base VERB iajs-3349 117 30 on on ADP iajs-3349 117 31 the the DET iajs-3349 117 32 above above ADJ iajs-3349 117 33 equations equation NOUN iajs-3349 117 34 can can AUX iajs-3349 117 35 be be AUX iajs-3349 117 36 obtained obtain VERB iajs-3349 117 37 according accord VERB iajs-3349 117 38 to to ADP iajs-3349 117 39 the the DET iajs-3349 117 40 following follow VERB iajs-3349 117 41 relations relation NOUN iajs-3349 117 42 : : PUNCT iajs-3349 117 43 π‘†π‘˜ π‘†π‘˜ PROPN iajs-3349 117 44 = = SYM iajs-3349 117 45 πœ‡3βˆ’3πœ‡2πœ‡1 πœ‡3βˆ’3πœ‡2πœ‡1 PROPN iajs-3349 117 46 + + NOUN iajs-3349 117 47 2πœ‡1 2πœ‡1 NUM iajs-3349 117 48 3 3 NUM iajs-3349 117 49 ( ( PUNCT iajs-3349 117 50 πœ‡2βˆ’πœ‡1 πœ‡2βˆ’πœ‡1 PROPN iajs-3349 117 51 2 2 NUM iajs-3349 117 52 ) ) PUNCT iajs-3349 117 53 3/2 3/2 NUM iajs-3349 117 54 , , PUNCT iajs-3349 117 55 𝐾𝑒 𝐾𝑒 PROPN iajs-3349 117 56 = = PUNCT iajs-3349 117 57 πœ‡4βˆ’4πœ‡3πœ‡1 πœ‡4βˆ’4πœ‡3πœ‡1 NOUN iajs-3349 117 58 + + PROPN iajs-3349 117 59 6πœ‡2πœ‡1 6πœ‡2πœ‡1 NUM iajs-3349 117 60 2βˆ’3πœ‡1 2βˆ’3πœ‡1 NUM iajs-3349 117 61 4 4 NUM iajs-3349 117 62 ( ( PUNCT iajs-3349 117 63 πœ‡2βˆ’πœ‡1 πœ‡2βˆ’πœ‡1 PROPN iajs-3349 117 64 2 2 NUM iajs-3349 117 65 ) ) PUNCT iajs-3349 117 66 2 2 NUM iajs-3349 117 67 4.4 4.4 NUM iajs-3349 117 68 incomplete incomplete ADJ iajs-3349 117 69 moments moment NOUN iajs-3349 117 70 the the DET iajs-3349 117 71 incomplete incomplete ADJ iajs-3349 117 72 moments moment NOUN iajs-3349 117 73 of of ADP iajs-3349 117 74 [ [ X iajs-3349 117 75 0,1 0,1 NUM iajs-3349 117 76 ] ] PUNCT iajs-3349 117 77 truncated truncate VERB iajs-3349 117 78 nadarajah nadarajah PROPN iajs-3349 117 79 - - PUNCT iajs-3349 117 80 haghighi haghighi PROPN iajs-3349 117 81 exponential exponential ADJ iajs-3349 117 82 distribution distribution NOUN iajs-3349 117 83 can can AUX iajs-3349 117 84 be be AUX iajs-3349 117 85 obtained obtain VERB iajs-3349 117 86 in in ADP iajs-3349 117 87 the the DET iajs-3349 117 88 same same ADJ iajs-3349 117 89 way way NOUN iajs-3349 117 90 of of ADP iajs-3349 117 91 ( ( PUNCT iajs-3349 117 92 4.3 4.3 NUM iajs-3349 117 93 ) ) PUNCT iajs-3349 117 94 , , PUNCT iajs-3349 117 95 as as SCONJ iajs-3349 117 96 follows follow VERB iajs-3349 117 97 : : PUNCT iajs-3349 117 98 π‘€π‘Ÿ(𝑦 π‘€π‘Ÿ(𝑦 ADJ iajs-3349 117 99 ) ) PUNCT iajs-3349 117 100 = = SYM iajs-3349 117 101 ∫ ∫ PROPN iajs-3349 117 102 π‘₯π‘Ÿπ‘“(π‘₯)[0,1]𝑇𝑁𝐻𝐸 π‘₯π‘Ÿπ‘“(π‘₯)[0,1]𝑇𝑁𝐻𝐸 PROPN iajs-3349 117 103 𝑑π‘₯ 𝑑π‘₯ VERB iajs-3349 117 104 𝑦 𝑦 PRON iajs-3349 117 105 βˆ’βˆž βˆ’βˆž NOUN iajs-3349 117 106 = = SYM iajs-3349 117 107 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 NOUN iajs-3349 117 108 1 1 NUM iajs-3349 117 109 ( ( PUNCT iajs-3349 117 110 πœ†(π‘˜+1 πœ†(π‘˜+1 NUM iajs-3349 117 111 ) ) PUNCT iajs-3349 117 112 ) ) PUNCT iajs-3349 118 1 π‘Ÿ+1 π‘Ÿ+1 PROPN iajs-3349 118 2 𝛾(π‘Ÿ 𝛾(π‘Ÿ PROPN iajs-3349 118 3 + + CCONJ iajs-3349 118 4 2 2 NUM iajs-3349 118 5 , , PUNCT iajs-3349 118 6 πœ†π‘¦(π‘š πœ†π‘¦(π‘š PUNCT iajs-3349 118 7 + + X iajs-3349 118 8 1 1 NUM iajs-3349 118 9 ) ) PUNCT iajs-3349 118 10 ) ) PUNCT iajs-3349 118 11 ( ( PUNCT iajs-3349 118 12 23 23 NUM iajs-3349 118 13 ) ) PUNCT iajs-3349 118 14 4.5 4.5 NUM iajs-3349 118 15 moment moment NOUN iajs-3349 118 16 generating generating NOUN iajs-3349 118 17 functions function NOUN iajs-3349 118 18 the the DET iajs-3349 118 19 moment moment NOUN iajs-3349 118 20 generating generating NOUN iajs-3349 118 21 functions function NOUN iajs-3349 118 22 of of ADP iajs-3349 118 23 [ [ X iajs-3349 118 24 0,1 0,1 NUM iajs-3349 118 25 ] ] PUNCT iajs-3349 118 26 truncated truncate VERB iajs-3349 118 27 nadarajah nadarajah PROPN iajs-3349 118 28 - - PUNCT iajs-3349 118 29 haghighi haghighi PROPN iajs-3349 118 30 exponential exponential ADJ iajs-3349 118 31 distribution distribution NOUN iajs-3349 118 32 can can AUX iajs-3349 118 33 be be AUX iajs-3349 118 34 obtained obtain VERB iajs-3349 118 35 in in ADP iajs-3349 118 36 the the DET iajs-3349 118 37 following following NOUN iajs-3349 118 38 : : PUNCT iajs-3349 118 39 ihjpas ihjpas PROPN iajs-3349 118 40 . . PUNCT iajs-3349 119 1 37 37 NUM iajs-3349 119 2 ( ( PUNCT iajs-3349 119 3 2 2 NUM iajs-3349 119 4 ) ) PUNCT iajs-3349 119 5 2024 2024 NUM iajs-3349 119 6 368 368 NUM iajs-3349 119 7 𝑀𝑋(𝑑 𝑀𝑋(𝑑 NUM iajs-3349 119 8 ) ) PUNCT iajs-3349 120 1 = = PUNCT iajs-3349 120 2 βˆ‘ βˆ‘ PUNCT iajs-3349 120 3 π‘™π‘Ÿ π‘™π‘Ÿ ADP iajs-3349 120 4 𝑙 𝑙 PROPN iajs-3349 120 5 ! ! PUNCT iajs-3349 121 1 ∞ ∞ NUM iajs-3349 122 1 𝑙=0 𝑙=0 PUNCT iajs-3349 122 2 ∫ ∫ PROPN iajs-3349 123 1 π‘₯𝑙𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 π‘₯𝑙𝑓(π‘₯)[0,1]𝑇𝑁𝐻𝐸 PROPN iajs-3349 123 2 𝑑π‘₯ 𝑑π‘₯ VERB iajs-3349 123 3 ∞ ∞ PROPN iajs-3349 123 4 βˆ’βˆž βˆ’βˆž ADP iajs-3349 123 5 = = NOUN iajs-3349 123 6 βˆ‘ βˆ‘ PROPN iajs-3349 123 7 π‘™π‘Ÿ π‘™π‘Ÿ ADP iajs-3349 123 8 𝑙 𝑙 PROPN iajs-3349 123 9 ! ! PUNCT iajs-3349 124 1 ∞ ∞ NUM iajs-3349 125 1 𝑙=0 𝑙=0 PUNCT iajs-3349 126 1 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 NOUN iajs-3349 126 2 ∫ ∫ PROPN iajs-3349 126 3 π‘₯π‘™π‘’βˆ’πœ†(π‘˜+1)π‘₯ π‘₯π‘™π‘’βˆ’πœ†(π‘˜+1)π‘₯ X iajs-3349 126 4 𝑑π‘₯ 𝑑π‘₯ VERB iajs-3349 126 5 ∞ ∞ PROPN iajs-3349 126 6 0 0 NUM iajs-3349 126 7 ( ( PUNCT iajs-3349 126 8 24 24 NUM iajs-3349 126 9 ) ) PUNCT iajs-3349 126 10 4.6 4.6 NUM iajs-3349 126 11 entropy entropy NOUN iajs-3349 126 12 in in ADP iajs-3349 126 13 this this DET iajs-3349 126 14 sub sub NOUN iajs-3349 126 15 - - NOUN iajs-3349 126 16 section section NOUN iajs-3349 126 17 , , PUNCT iajs-3349 126 18 we we PRON iajs-3349 126 19 will will AUX iajs-3349 126 20 find find VERB iajs-3349 126 21 three three NUM iajs-3349 126 22 common common ADJ iajs-3349 126 23 measures measure NOUN iajs-3349 126 24 of of ADP iajs-3349 126 25 entropy entropy NOUN iajs-3349 126 26 for for ADP iajs-3349 126 27 a a DET iajs-3349 126 28 random random ADJ iajs-3349 126 29 variable variable NOUN iajs-3349 126 30 x. x. NOUN iajs-3349 127 1 these these PRON iajs-3349 127 2 are be AUX iajs-3349 127 3 rΓ©nyi rΓ©nyi NOUN iajs-3349 127 4 entropy entropy PROPN iajs-3349 127 5 , , PUNCT iajs-3349 127 6 shannon shannon PROPN iajs-3349 127 7 entropy entropy PROPN iajs-3349 127 8 and and CCONJ iajs-3349 127 9 delta delta PROPN iajs-3349 127 10 entropy entropy PROPN iajs-3349 127 11 . . PUNCT iajs-3349 128 1 shannon shannon PROPN iajs-3349 128 2 entropy entropy PROPN iajs-3349 128 3 is be AUX iajs-3349 128 4 a a DET iajs-3349 128 5 special special ADJ iajs-3349 128 6 case case NOUN iajs-3349 128 7 of of ADP iajs-3349 128 8 rΓ©nyi rΓ©nyi PROPN iajs-3349 128 9 entropy entropy NOUN iajs-3349 128 10 . . PUNCT iajs-3349 129 1 4.6.1 4.6.1 NUM iajs-3349 129 2 rΓ©nyi rΓ©nyi PROPN iajs-3349 129 3 entropy entropy PROPN iajs-3349 129 4 rΓ©nyi rΓ©nyi PROPN iajs-3349 129 5 entropy entropy PROPN iajs-3349 129 6 of of ADP iajs-3349 129 7 [ [ X iajs-3349 129 8 0,1 0,1 NUM iajs-3349 129 9 ] ] PUNCT iajs-3349 129 10 truncated truncate VERB iajs-3349 129 11 nadarajah nadarajah PROPN iajs-3349 129 12 - - PUNCT iajs-3349 129 13 haghighi haghighi PROPN iajs-3349 129 14 exponential exponential ADJ iajs-3349 129 15 distribution distribution NOUN iajs-3349 129 16 is be AUX iajs-3349 129 17 defined define VERB iajs-3349 129 18 as as SCONJ iajs-3349 129 19 follows follow VERB iajs-3349 129 20 𝐼𝑅(𝑐)[0,1]𝑇𝑁𝐻𝐸 𝐼𝑅(𝑐)[0,1]𝑇𝑁𝐻𝐸 PROPN iajs-3349 129 21 = = SYM iajs-3349 129 22 1 1 NUM iajs-3349 129 23 1βˆ’π‘ 1βˆ’π‘ NUM iajs-3349 129 24 log{∫ log{∫ NOUN iajs-3349 129 25 𝑓𝑐(π‘₯)𝑑π‘₯ 𝑓𝑐(π‘₯)𝑑π‘₯ PUNCT iajs-3349 129 26 ∞ ∞ PROPN iajs-3349 129 27 βˆ’βˆž βˆ’βˆž ADP iajs-3349 129 28 } } PUNCT iajs-3349 129 29 , , PUNCT iajs-3349 129 30 𝑐 𝑐 PROPN iajs-3349 129 31 β‰  β‰  PROPN iajs-3349 129 32 1 1 NUM iajs-3349 129 33 , , PUNCT iajs-3349 129 34 𝑐 𝑐 NOUN iajs-3349 129 35 > > X iajs-3349 129 36 0 0 NUM iajs-3349 129 37 the the DET iajs-3349 129 38 rΓ©nyi rΓ©nyi NOUN iajs-3349 129 39 entropy entropy NOUN iajs-3349 129 40 for for ADP iajs-3349 129 41 the the DET iajs-3349 129 42 random random ADJ iajs-3349 129 43 variable variable NOUN iajs-3349 129 44 x x PUNCT iajs-3349 129 45 is be AUX iajs-3349 129 46 defined define VERB iajs-3349 129 47 by by ADP iajs-3349 129 48 𝐼𝑅(𝑐)[0,1]𝑇𝑁𝐻𝐸 𝐼𝑅(𝑐)[0,1]𝑇𝑁𝐻𝐸 PROPN iajs-3349 129 49 = = SYM iajs-3349 129 50 1 1 NUM iajs-3349 129 51 1βˆ’π‘ 1βˆ’π‘ NUM iajs-3349 129 52 log{∫ log{∫ NOUN iajs-3349 129 53 𝑓𝑐(π‘₯)𝑑π‘₯ 𝑓𝑐(π‘₯)𝑑π‘₯ PUNCT iajs-3349 129 54 ∞ ∞ PROPN iajs-3349 129 55 βˆ’βˆž βˆ’βˆž ADP iajs-3349 129 56 } } PUNCT iajs-3349 129 57 , , PUNCT iajs-3349 129 58 𝑐 𝑐 PROPN iajs-3349 129 59 β‰  β‰  PROPN iajs-3349 129 60 1 1 NUM iajs-3349 129 61 , , PUNCT iajs-3349 129 62 𝑐 𝑐 NOUN iajs-3349 129 63 > > X iajs-3349 129 64 0 0 PUNCT iajs-3349 129 65 now now ADV iajs-3349 129 66 𝑓𝑐(π‘₯ 𝑓𝑐(π‘₯ NUM iajs-3349 129 67 ) ) PUNCT iajs-3349 129 68 = = PRON iajs-3349 129 69 ( ( PUNCT iajs-3349 129 70 π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ ADJ iajs-3349 129 71 ) ) PUNCT iajs-3349 129 72 ) ) PUNCT iajs-3349 130 1 π‘Žβˆ’1 π‘Žβˆ’1 PROPN iajs-3349 130 2 𝑒 𝑒 PROPN iajs-3349 130 3 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 130 4 ) ) PUNCT iajs-3349 130 5 ) ) PUNCT iajs-3349 131 1 π‘Ž π‘Ž PRON iajs-3349 131 2 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 131 3 1+𝑏 1+𝑏 NUM iajs-3349 131 4 ) ) PUNCT iajs-3349 131 5 𝑐 𝑐 NOUN iajs-3349 131 6 = = SYM iajs-3349 131 7 ( ( PUNCT iajs-3349 131 8 π‘Žπ‘πœ†)𝑐 π‘Žπ‘πœ†)𝑐 PROPN iajs-3349 131 9 π‘’βˆ’π‘πœ†π‘₯ π‘’βˆ’π‘πœ†π‘₯ NOUN iajs-3349 131 10 ( ( PUNCT iajs-3349 131 11 1+𝑏)𝑐 1+𝑏)𝑐 NOUN iajs-3349 131 12 ( ( PUNCT iajs-3349 131 13 ( ( PUNCT iajs-3349 131 14 1 1 NUM iajs-3349 131 15 + + CCONJ iajs-3349 131 16 𝑏(1 𝑏(1 PROPN iajs-3349 131 17 βˆ’ βˆ’ NOUN iajs-3349 131 18 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 131 19 ) ) PUNCT iajs-3349 131 20 ) ) PUNCT iajs-3349 131 21 𝑐(π‘Žβˆ’1 𝑐(π‘Žβˆ’1 X iajs-3349 131 22 ) ) PUNCT iajs-3349 131 23 𝑒𝑐(1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 𝑒𝑐(1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PUNCT iajs-3349 131 24 ) ) PUNCT iajs-3349 131 25 ) ) PUNCT iajs-3349 132 1 π‘Ž π‘Ž X iajs-3349 132 2 ) ) PUNCT iajs-3349 132 3 ) ) PUNCT iajs-3349 132 4 now now ADV iajs-3349 132 5 , , PUNCT iajs-3349 132 6 by by ADP iajs-3349 132 7 use use VERB iajs-3349 132 8 the the DET iajs-3349 132 9 expansion expansion NOUN iajs-3349 132 10 exponential exponential ADJ iajs-3349 132 11 formula formula NOUN iajs-3349 132 12 , , PUNCT iajs-3349 132 13 we we PRON iajs-3349 132 14 get get VERB iajs-3349 132 15 𝑒𝑐(1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 𝑒𝑐(1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ ADJ iajs-3349 132 16 ) ) PUNCT iajs-3349 132 17 ) ) PUNCT iajs-3349 133 1 π‘Ž π‘Ž X iajs-3349 133 2 ) ) PUNCT iajs-3349 133 3 = = SYM iajs-3349 133 4 βˆ‘ βˆ‘ PROPN iajs-3349 133 5 1 1 NUM iajs-3349 133 6 𝑗 𝑗 NOUN iajs-3349 133 7 ! ! PUNCT iajs-3349 134 1 𝑐𝑗(1 𝑐𝑗(1 PROPN iajs-3349 134 2 βˆ’ βˆ’ PROPN iajs-3349 134 3 ( ( PUNCT iajs-3349 134 4 1 1 NUM iajs-3349 134 5 + + CCONJ iajs-3349 134 6 𝑏(1 𝑏(1 PROPN iajs-3349 134 7 βˆ’ βˆ’ NOUN iajs-3349 134 8 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 134 9 ) ) PUNCT iajs-3349 134 10 ) ) PUNCT iajs-3349 135 1 π‘Ž π‘Ž X iajs-3349 135 2 ) ) PUNCT iajs-3349 135 3 𝑗 𝑗 NOUN iajs-3349 135 4 ∞ ∞ NUM iajs-3349 135 5 𝑗=0 𝑗=0 PUNCT iajs-3349 136 1 so so SCONJ iajs-3349 136 2 that that SCONJ iajs-3349 136 3 𝑓𝑐(π‘₯ 𝑓𝑐(π‘₯ X iajs-3349 136 4 ) ) PUNCT iajs-3349 136 5 = = SYM iajs-3349 136 6 ( ( PUNCT iajs-3349 136 7 π‘Žπ‘πœ†)𝑐 π‘Žπ‘πœ†)𝑐 PROPN iajs-3349 136 8 π‘’βˆ’π‘πœ†π‘₯ π‘’βˆ’π‘πœ†π‘₯ NOUN iajs-3349 136 9 ( ( PUNCT iajs-3349 136 10 1+𝑏)𝑐 1+𝑏)𝑐 NOUN iajs-3349 136 11 ( ( PUNCT iajs-3349 136 12 ( ( PUNCT iajs-3349 136 13 1 1 NUM iajs-3349 136 14 + + CCONJ iajs-3349 136 15 𝑏(1 𝑏(1 PROPN iajs-3349 136 16 βˆ’ βˆ’ NOUN iajs-3349 136 17 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 136 18 ) ) PUNCT iajs-3349 136 19 ) ) PUNCT iajs-3349 136 20 𝑐(π‘Žβˆ’1 𝑐(π‘Žβˆ’1 NUM iajs-3349 136 21 ) ) PUNCT iajs-3349 136 22 βˆ‘ βˆ‘ ADV iajs-3349 136 23 1 1 NUM iajs-3349 136 24 𝑗 𝑗 NOUN iajs-3349 136 25 ! ! PUNCT iajs-3349 137 1 𝑐𝑗(1 𝑐𝑗(1 PROPN iajs-3349 137 2 βˆ’ βˆ’ PROPN iajs-3349 137 3 ( ( PUNCT iajs-3349 137 4 1 1 NUM iajs-3349 137 5 + + CCONJ iajs-3349 137 6 𝑏(1 𝑏(1 PROPN iajs-3349 137 7 βˆ’ βˆ’ NOUN iajs-3349 137 8 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 137 9 ) ) PUNCT iajs-3349 137 10 ) ) PUNCT iajs-3349 138 1 π‘Ž π‘Ž X iajs-3349 138 2 ) ) PUNCT iajs-3349 138 3 𝑗 𝑗 NOUN iajs-3349 138 4 ∞ ∞ PROPN iajs-3349 138 5 𝑗=0 𝑗=0 PUNCT iajs-3349 138 6 ) ) PUNCT iajs-3349 138 7 and and CCONJ iajs-3349 138 8 by by ADP iajs-3349 138 9 the the DET iajs-3349 138 10 generalized generalized ADJ iajs-3349 138 11 binomial binomial ADJ iajs-3349 138 12 theorem theorem NOUN iajs-3349 138 13 ( ( PUNCT iajs-3349 138 14 1 1 NUM iajs-3349 138 15 βˆ’ βˆ’ PROPN iajs-3349 138 16 ( ( PUNCT iajs-3349 138 17 1 1 NUM iajs-3349 138 18 + + CCONJ iajs-3349 138 19 𝑏(1 𝑏(1 PROPN iajs-3349 138 20 βˆ’ βˆ’ NOUN iajs-3349 138 21 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 138 22 ) ) PUNCT iajs-3349 138 23 ) ) PUNCT iajs-3349 139 1 π‘Ž π‘Ž X iajs-3349 139 2 ) ) PUNCT iajs-3349 139 3 𝑗 𝑗 NOUN iajs-3349 139 4 = = PUNCT iajs-3349 139 5 βˆ‘ βˆ‘ PUNCT iajs-3349 139 6 ( ( PUNCT iajs-3349 139 7 𝑗 𝑗 PROPN iajs-3349 139 8 π‘˜ π‘˜ NOUN iajs-3349 139 9 ) ) PUNCT iajs-3349 139 10 ( ( PUNCT iajs-3349 139 11 βˆ’1)π‘˜(1 βˆ’1)π‘˜(1 CCONJ iajs-3349 139 12 + + CCONJ iajs-3349 139 13 𝑏(1 𝑏(1 PROPN iajs-3349 139 14 βˆ’ βˆ’ PROPN iajs-3349 139 15 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 139 16 ) ) PUNCT iajs-3349 139 17 ) ) PUNCT iajs-3349 140 1 π‘Žπ‘—βˆž π‘Žπ‘—βˆž PROPN iajs-3349 140 2 π‘˜=0 π‘˜=0 VERB iajs-3349 140 3 hence hence ADV iajs-3349 140 4 𝑓𝑐(π‘₯ 𝑓𝑐(π‘₯ NUM iajs-3349 140 5 ) ) PUNCT iajs-3349 140 6 = = SYM iajs-3349 140 7 ( ( PUNCT iajs-3349 140 8 π‘Žπ‘πœ†)𝑐 π‘Žπ‘πœ†)𝑐 PROPN iajs-3349 140 9 π‘’βˆ’π‘πœ†π‘₯ π‘’βˆ’π‘πœ†π‘₯ NOUN iajs-3349 140 10 ( ( PUNCT iajs-3349 140 11 1+𝑏)𝑐 1+𝑏)𝑐 NOUN iajs-3349 140 12 βˆ‘ βˆ‘ PROPN iajs-3349 140 13 βˆ‘ βˆ‘ PUNCT iajs-3349 140 14 ( ( PUNCT iajs-3349 140 15 𝑗 𝑗 INTJ iajs-3349 140 16 π‘˜ π‘˜ NOUN iajs-3349 140 17 ) ) PUNCT iajs-3349 140 18 𝑐𝑗 𝑐𝑗 NOUN iajs-3349 140 19 𝑗 𝑗 NOUN iajs-3349 140 20 ! ! PUNCT iajs-3349 141 1 ( ( PUNCT iajs-3349 141 2 βˆ’1)π‘˜(1 βˆ’1)π‘˜(1 PUNCT iajs-3349 141 3 + + CCONJ iajs-3349 141 4 𝑏(1 𝑏(1 PROPN iajs-3349 141 5 βˆ’ βˆ’ PROPN iajs-3349 141 6 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 141 7 ) ) PUNCT iajs-3349 141 8 ) ) PUNCT iajs-3349 142 1 π‘Žπ‘—+π‘π‘Žβˆ’π‘βˆž π‘Žπ‘—+π‘π‘Žβˆ’π‘βˆž PROPN iajs-3349 142 2 π‘˜=0 π‘˜=0 SYM iajs-3349 142 3 ∞ ∞ NUM iajs-3349 142 4 𝑗=0 𝑗=0 PROPN iajs-3349 143 1 in in ADP iajs-3349 143 2 the the DET iajs-3349 143 3 same same ADJ iajs-3349 143 4 way way NOUN iajs-3349 143 5 ( ( PUNCT iajs-3349 143 6 1 1 NUM iajs-3349 143 7 + + CCONJ iajs-3349 143 8 𝑏(1 𝑏(1 PROPN iajs-3349 143 9 βˆ’ βˆ’ NOUN iajs-3349 143 10 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ NOUN iajs-3349 143 11 ) ) PUNCT iajs-3349 143 12 ) ) PUNCT iajs-3349 143 13 π‘Žπ‘—+π‘π‘Žβˆ’π‘ π‘Žπ‘—+π‘π‘Žβˆ’π‘ PUNCT iajs-3349 144 1 = = PUNCT iajs-3349 144 2 βˆ‘ βˆ‘ PUNCT iajs-3349 144 3 ( ( PUNCT iajs-3349 144 4 π‘Žπ‘—+π‘π‘Žβˆ’π‘ π‘Žπ‘—+π‘π‘Žβˆ’π‘ NOUN iajs-3349 144 5 π‘Ÿ π‘Ÿ NOUN iajs-3349 144 6 ) ) PUNCT iajs-3349 144 7 π‘π‘Ÿ π‘π‘Ÿ NOUN iajs-3349 144 8 ( ( PUNCT iajs-3349 144 9 1 1 NUM iajs-3349 144 10 βˆ’ βˆ’ NUM iajs-3349 144 11 π‘’βˆ’πœ†π‘₯)π‘Ÿβˆž π‘’βˆ’πœ†π‘₯)π‘Ÿβˆž PRON iajs-3349 144 12 π‘Ÿ=0 π‘Ÿ=0 NUM iajs-3349 144 13 then then ADV iajs-3349 144 14 𝑓𝑐(π‘₯ 𝑓𝑐(π‘₯ X iajs-3349 144 15 ) ) PUNCT iajs-3349 144 16 = = SYM iajs-3349 144 17 ( ( PUNCT iajs-3349 144 18 π‘Žπ‘πœ†)𝑐 π‘Žπ‘πœ†)𝑐 PROPN iajs-3349 144 19 π‘’βˆ’π‘πœ†π‘₯ π‘’βˆ’π‘πœ†π‘₯ NOUN iajs-3349 144 20 ( ( PUNCT iajs-3349 144 21 1+𝑏)𝑐 1+𝑏)𝑐 NOUN iajs-3349 144 22 βˆ‘ βˆ‘ PROPN iajs-3349 144 23 βˆ‘ βˆ‘ PROPN iajs-3349 144 24 βˆ‘ βˆ‘ PUNCT iajs-3349 144 25 ( ( PUNCT iajs-3349 144 26 𝑗 𝑗 INTJ iajs-3349 144 27 π‘˜ π‘˜ NOUN iajs-3349 144 28 ) ) PUNCT iajs-3349 144 29 𝑐𝑗 𝑐𝑗 NOUN iajs-3349 144 30 𝑗 𝑗 NOUN iajs-3349 144 31 ! ! PUNCT iajs-3349 145 1 ( ( PUNCT iajs-3349 145 2 βˆ’1)π‘˜(π‘Žπ‘—+π‘π‘Žβˆ’π‘ βˆ’1)π‘˜(π‘Žπ‘—+π‘π‘Žβˆ’π‘ NOUN iajs-3349 145 3 π‘Ÿ π‘Ÿ NOUN iajs-3349 145 4 ) ) PUNCT iajs-3349 145 5 π‘π‘Ÿ π‘π‘Ÿ NOUN iajs-3349 145 6 ( ( PUNCT iajs-3349 145 7 1 1 NUM iajs-3349 145 8 βˆ’ βˆ’ NUM iajs-3349 145 9 π‘’βˆ’πœ†π‘₯)π‘Ÿβˆž π‘’βˆ’πœ†π‘₯)π‘Ÿβˆž PRON iajs-3349 145 10 π‘Ÿ=0 π‘Ÿ=0 PUNCT iajs-3349 145 11 ∞ ∞ NUM iajs-3349 145 12 π‘˜=0 π‘˜=0 PROPN iajs-3349 145 13 ∞ ∞ PROPN iajs-3349 145 14 𝑗=0 𝑗=0 PUNCT iajs-3349 146 1 also also ADV iajs-3349 146 2 ( ( PUNCT iajs-3349 146 3 1 1 NUM iajs-3349 146 4 βˆ’ βˆ’ PROPN iajs-3349 146 5 π‘’βˆ’πœ†π‘₯)π‘Ÿ π‘’βˆ’πœ†π‘₯)π‘Ÿ PROPN iajs-3349 146 6 = = SYM iajs-3349 146 7 βˆ‘ βˆ‘ PUNCT iajs-3349 146 8 ( ( PUNCT iajs-3349 146 9 π‘Ÿ π‘Ÿ NOUN iajs-3349 146 10 𝑠 𝑠 NOUN iajs-3349 146 11 ) ) PUNCT iajs-3349 146 12 ( ( PUNCT iajs-3349 146 13 βˆ’1)π‘ π‘’βˆ’π‘ πœ†π‘₯∞ βˆ’1)π‘ π‘’βˆ’π‘ πœ†π‘₯∞ AUX iajs-3349 146 14 𝑠=0 𝑠=0 PUNCT iajs-3349 146 15 ihjpas ihjpas PROPN iajs-3349 146 16 . . PUNCT iajs-3349 147 1 37 37 NUM iajs-3349 147 2 ( ( PUNCT iajs-3349 147 3 2 2 NUM iajs-3349 147 4 ) ) PUNCT iajs-3349 147 5 2024 2024 NUM iajs-3349 147 6 369 369 NUM iajs-3349 147 7 finally finally ADV iajs-3349 147 8 𝑓𝑐(π‘₯ 𝑓𝑐(π‘₯ NUM iajs-3349 147 9 ) ) PUNCT iajs-3349 147 10 = = PUNCT iajs-3349 148 1 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 PROPN iajs-3349 148 2 𝑒 𝑒 PROPN iajs-3349 148 3 βˆ’πœ†π‘₯(𝑠+𝑐 βˆ’πœ†π‘₯(𝑠+𝑐 NUM iajs-3349 148 4 ) ) PUNCT iajs-3349 148 5 where where SCONJ iajs-3349 148 6 Οˆπ‘—,π‘˜,π‘Ÿ,𝑠 Οˆπ‘—,π‘˜,π‘Ÿ,𝑠 PROPN iajs-3349 148 7 = = SYM iajs-3349 148 8 ( ( PUNCT iajs-3349 148 9 π‘Žπ‘πœ†)𝑐 π‘Žπ‘πœ†)𝑐 PROPN iajs-3349 148 10 ( ( PUNCT iajs-3349 148 11 1+𝑏)𝑐 1+𝑏)𝑐 NOUN iajs-3349 148 12 βˆ‘ βˆ‘ PROPN iajs-3349 148 13 βˆ‘ βˆ‘ PROPN iajs-3349 148 14 βˆ‘ βˆ‘ PROPN iajs-3349 148 15 βˆ‘ βˆ‘ PUNCT iajs-3349 148 16 ( ( PUNCT iajs-3349 148 17 π‘Ÿ π‘Ÿ NOUN iajs-3349 148 18 𝑠 𝑠 NOUN iajs-3349 148 19 ) ) PUNCT iajs-3349 148 20 ( ( PUNCT iajs-3349 148 21 𝑗 𝑗 PROPN iajs-3349 148 22 π‘˜ π‘˜ NOUN iajs-3349 148 23 ) ) PUNCT iajs-3349 148 24 ( ( PUNCT iajs-3349 148 25 π‘Žπ‘—+π‘π‘Žβˆ’π‘ π‘Žπ‘—+π‘π‘Žβˆ’π‘ ADP iajs-3349 148 26 π‘Ÿ π‘Ÿ NOUN iajs-3349 148 27 ) ) PUNCT iajs-3349 149 1 𝑐𝑗 𝑐𝑗 X iajs-3349 149 2 𝑗 𝑗 NOUN iajs-3349 149 3 ! ! PUNCT iajs-3349 150 1 ( ( PUNCT iajs-3349 150 2 βˆ’1)π‘˜+𝑠 βˆ’1)π‘˜+𝑠 NUM iajs-3349 150 3 π‘π‘Ÿ π‘π‘Ÿ ADP iajs-3349 150 4 ∞ ∞ PROPN iajs-3349 150 5 𝑠=0 𝑠=0 PROPN iajs-3349 150 6 ∞ ∞ PROPN iajs-3349 150 7 π‘Ÿ=0 π‘Ÿ=0 PUNCT iajs-3349 151 1 ∞ ∞ NUM iajs-3349 151 2 π‘˜=0 π‘˜=0 PROPN iajs-3349 151 3 ∞ ∞ PROPN iajs-3349 151 4 𝑗=0 𝑗=0 PUNCT iajs-3349 152 1 now now ADV iajs-3349 152 2 𝐼𝑅(𝑐)[0,1]π‘‡π‘π»βˆ’πΈπ‘₯π‘π‘œ 𝐼𝑅(𝑐)[0,1]π‘‡π‘π»βˆ’πΈπ‘₯π‘π‘œ NOUN iajs-3349 152 3 = = SYM iajs-3349 152 4 1 1 NUM iajs-3349 152 5 1βˆ’π‘ 1βˆ’π‘ NUM iajs-3349 152 6 log{∫ log{∫ NOUN iajs-3349 152 7 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 PROPN iajs-3349 152 8 𝑒 𝑒 PROPN iajs-3349 152 9 βˆ’πœ†π‘₯(𝑠+𝑐 βˆ’πœ†π‘₯(𝑠+𝑐 NUM iajs-3349 152 10 ) ) PUNCT iajs-3349 152 11 𝑑π‘₯ 𝑑π‘₯ VERB iajs-3349 152 12 ∞ ∞ NOUN iajs-3349 152 13 0 0 NUM iajs-3349 152 14 } } PUNCT iajs-3349 152 15 , , PUNCT iajs-3349 152 16 𝑐 𝑐 PROPN iajs-3349 152 17 β‰  β‰  PROPN iajs-3349 152 18 1 1 NUM iajs-3349 152 19 , , PUNCT iajs-3349 152 20 𝑐 𝑐 NOUN iajs-3349 152 21 > > X iajs-3349 152 22 0 0 PUNCT iajs-3349 152 23 to to PART iajs-3349 152 24 find find VERB iajs-3349 152 25 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 PROPN iajs-3349 152 26 ∫ ∫ PROPN iajs-3349 152 27 𝑒 𝑒 PROPN iajs-3349 152 28 βˆ’πœ†π‘₯(𝑠+𝑐 βˆ’πœ†π‘₯(𝑠+𝑐 PROPN iajs-3349 152 29 ) ) PUNCT iajs-3349 152 30 𝑑π‘₯ 𝑑π‘₯ VERB iajs-3349 152 31 ∞ ∞ PROPN iajs-3349 152 32 0 0 NUM iajs-3349 153 1 = = PUNCT iajs-3349 153 2 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 NUM iajs-3349 153 3 1 1 NUM iajs-3349 153 4 πœ†(𝑠+𝑐 πœ†(𝑠+𝑐 NUM iajs-3349 153 5 ) ) PUNCT iajs-3349 153 6 then then ADV iajs-3349 153 7 the the DET iajs-3349 153 8 final final ADJ iajs-3349 153 9 form form NOUN iajs-3349 153 10 of of ADP iajs-3349 153 11 rΓ©nyi rΓ©nyi NOUN iajs-3349 153 12 entropy entropy NOUN iajs-3349 153 13 for for ADP iajs-3349 153 14 [ [ X iajs-3349 153 15 0,1 0,1 NOUN iajs-3349 153 16 ] ] PUNCT iajs-3349 153 17 tnhe tnhe NOUN iajs-3349 153 18 distribution distribution NOUN iajs-3349 153 19 is be AUX iajs-3349 153 20 𝐼𝑅(𝑐)[0,1]π‘‡π‘π»βˆ’πΈπ‘₯π‘π‘œ 𝐼𝑅(𝑐)[0,1]π‘‡π‘π»βˆ’πΈπ‘₯π‘π‘œ NOUN iajs-3349 153 21 = = SYM iajs-3349 153 22 1 1 NUM iajs-3349 153 23 1βˆ’π‘ 1βˆ’π‘ NUM iajs-3349 153 24 log log NOUN iajs-3349 153 25 { { PUNCT iajs-3349 153 26 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 NUM iajs-3349 153 27 1 1 NUM iajs-3349 153 28 πœ†(𝑠+𝑐 πœ†(𝑠+𝑐 NUM iajs-3349 153 29 ) ) PUNCT iajs-3349 153 30 } } PUNCT iajs-3349 153 31 , , PUNCT iajs-3349 153 32 𝑐 𝑐 PROPN iajs-3349 153 33 β‰  β‰  PROPN iajs-3349 153 34 1 1 NUM iajs-3349 153 35 , , PUNCT iajs-3349 153 36 𝑐 𝑐 NOUN iajs-3349 153 37 > > X iajs-3349 153 38 0 0 PUNCT iajs-3349 154 1 ( ( PUNCT iajs-3349 154 2 25 25 NUM iajs-3349 154 3 ) ) PUNCT iajs-3349 154 4 4.6.2 4.6.2 PROPN iajs-3349 154 5 shannon shannon PROPN iajs-3349 154 6 entropy entropy VERB iajs-3349 154 7 the the DET iajs-3349 154 8 shannon shannon PROPN iajs-3349 154 9 entropy entropy PROPN iajs-3349 154 10 of of ADP iajs-3349 154 11 the the DET iajs-3349 154 12 new new ADJ iajs-3349 154 13 distribution distribution NOUN iajs-3349 154 14 is be AUX iajs-3349 154 15 given give VERB iajs-3349 154 16 by by ADP iajs-3349 154 17 πœ‚π‘₯ πœ‚π‘₯ PROPN iajs-3349 154 18 = = SYM iajs-3349 154 19 𝐸(βˆ’ 𝐸(βˆ’ NOUN iajs-3349 154 20 log log NOUN iajs-3349 154 21 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 PROPN iajs-3349 154 22 𝑒 𝑒 PROPN iajs-3349 154 23 βˆ’πœ†π‘₯(π‘˜+1 βˆ’πœ†π‘₯(π‘˜+1 PROPN iajs-3349 154 24 ) ) PUNCT iajs-3349 154 25 ) ) PUNCT iajs-3349 155 1 ( ( PUNCT iajs-3349 155 2 26 26 NUM iajs-3349 155 3 ) ) PUNCT iajs-3349 155 4 shannon shannon PROPN iajs-3349 155 5 entropy entropy PROPN iajs-3349 155 6 , , PUNCT iajs-3349 155 7 defined define VERB iajs-3349 155 8 as as ADP iajs-3349 155 9 an an DET iajs-3349 155 10 a a DET iajs-3349 155 11 random random ADJ iajs-3349 155 12 variable variable NOUN iajs-3349 155 13 x x PUNCT iajs-3349 155 14 with with ADP iajs-3349 155 15 a a DET iajs-3349 155 16 pdf pdf NOUN iajs-3349 155 17 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3349 155 18 ) ) PUNCT iajs-3349 155 19 , , PUNCT iajs-3349 155 20 is be AUX iajs-3349 155 21 a a DET iajs-3349 155 22 special special ADJ iajs-3349 155 23 case case NOUN iajs-3349 155 24 of of ADP iajs-3349 155 25 the the DET iajs-3349 155 26 renyi renyi PROPN iajs-3349 155 27 entropy entropy PROPN iajs-3349 155 28 when when SCONJ iajs-3349 155 29 c↑ c↑ PROPN iajs-3349 155 30 1 1 NUM iajs-3349 155 31 and and CCONJ iajs-3349 155 32 is be AUX iajs-3349 155 33 defined define VERB iajs-3349 155 34 as as ADP iajs-3349 155 35 follow follow NOUN iajs-3349 155 36 πœ‚π‘₯ πœ‚π‘₯ PROPN iajs-3349 155 37 = = SYM iajs-3349 155 38 𝐸(βˆ’ 𝐸(βˆ’ NOUN iajs-3349 155 39 log log NOUN iajs-3349 155 40 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3349 155 41 ) ) PUNCT iajs-3349 155 42 ) ) PUNCT iajs-3349 156 1 so so ADV iajs-3349 156 2 , , PUNCT iajs-3349 156 3 the the DET iajs-3349 156 4 [ [ X iajs-3349 156 5 0,1]tnhe 0,1]tnhe ADJ iajs-3349 156 6 distribution distribution NOUN iajs-3349 156 7 random random ADJ iajs-3349 156 8 variable variable NOUN iajs-3349 156 9 is be AUX iajs-3349 156 10 given give VERB iajs-3349 156 11 by by ADP iajs-3349 156 12 πœ‚π‘₯ πœ‚π‘₯ PROPN iajs-3349 156 13 = = SYM iajs-3349 156 14 𝐸(βˆ’ 𝐸(βˆ’ NOUN iajs-3349 156 15 log log NOUN iajs-3349 156 16 ω𝑠,𝑛,π‘˜,𝑑 ω𝑠,𝑛,π‘˜,𝑑 PROPN iajs-3349 156 17 𝑒 𝑒 PROPN iajs-3349 156 18 βˆ’πœ†π‘₯(π‘˜+1 βˆ’πœ†π‘₯(π‘˜+1 PROPN iajs-3349 156 19 ) ) PUNCT iajs-3349 156 20 ) ) PUNCT iajs-3349 157 1 4.6.3 4.6.3 NUM iajs-3349 157 2 delta delta NOUN iajs-3349 157 3 entropy entropy VERB iajs-3349 157 4 the the DET iajs-3349 157 5 𝑐 𝑐 PROPN iajs-3349 157 6 βˆ’ βˆ’ PROPN iajs-3349 157 7 entropy entropy NOUN iajs-3349 157 8 of of ADP iajs-3349 157 9 a a DET iajs-3349 157 10 random random ADJ iajs-3349 157 11 variable variable NOUN iajs-3349 157 12 x x PUNCT iajs-3349 157 13 is be AUX iajs-3349 157 14 given give VERB iajs-3349 157 15 by by ADP iajs-3349 157 16 𝐻(𝑐 𝐻(𝑐 NOUN iajs-3349 157 17 ) ) PUNCT iajs-3349 157 18 = = SYM iajs-3349 157 19 1 1 NUM iajs-3349 157 20 1βˆ’π›Ώ 1βˆ’π›Ώ NUM iajs-3349 157 21 log{1 log{1 NUM iajs-3349 157 22 βˆ’ βˆ’ PROPN iajs-3349 157 23 ∫ ∫ PROPN iajs-3349 157 24 𝑓𝑐(π‘₯)𝑑π‘₯ 𝑓𝑐(π‘₯)𝑑π‘₯ NUM iajs-3349 157 25 ∞ ∞ PROPN iajs-3349 157 26 βˆ’βˆž βˆ’βˆž ADP iajs-3349 157 27 } } PUNCT iajs-3349 157 28 hence hence ADV iajs-3349 157 29 , , PUNCT iajs-3349 157 30 [ [ X iajs-3349 157 31 0,1 0,1 NOUN iajs-3349 157 32 ] ] PUNCT iajs-3349 157 33 truncated truncate VERB iajs-3349 157 34 nadarajah nadarajah PROPN iajs-3349 157 35 - - PUNCT iajs-3349 157 36 haghighi haghighi PROPN iajs-3349 157 37 exponential exponential ADJ iajs-3349 157 38 distribution distribution NOUN iajs-3349 157 39 is be AUX iajs-3349 157 40 given give VERB iajs-3349 157 41 by by ADP iajs-3349 157 42 : : PUNCT iajs-3349 157 43 𝐻(𝑐 𝐻(𝑐 NUM iajs-3349 157 44 ) ) PUNCT iajs-3349 157 45 = = SYM iajs-3349 157 46 1 1 NUM iajs-3349 157 47 1βˆ’π‘ 1βˆ’π‘ NUM iajs-3349 157 48 log{1 log{1 ADJ iajs-3349 157 49 βˆ’ βˆ’ PROPN iajs-3349 158 1 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 NUM iajs-3349 158 2 ∫ ∫ PROPN iajs-3349 158 3 𝑒 𝑒 PROPN iajs-3349 158 4 βˆ’πœ†π‘₯(𝑠+𝑐)𝑑π‘₯ βˆ’πœ†π‘₯(𝑠+𝑐)𝑑π‘₯ NUM iajs-3349 158 5 ∞ ∞ PROPN iajs-3349 158 6 0 0 NUM iajs-3349 158 7 } } PUNCT iajs-3349 158 8 𝐻(𝛿 𝐻(𝛿 PROPN iajs-3349 158 9 ) ) PUNCT iajs-3349 158 10 = = SYM iajs-3349 158 11 1 1 NUM iajs-3349 158 12 1βˆ’π›Ώ 1βˆ’π›Ώ NUM iajs-3349 158 13 log log NOUN iajs-3349 158 14 { { PUNCT iajs-3349 158 15 1 1 NUM iajs-3349 158 16 βˆ’ βˆ’ PROPN iajs-3349 158 17 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 𝛹𝑗,π‘˜,π‘Ÿ,𝑠 NUM iajs-3349 158 18 1 1 NUM iajs-3349 158 19 πœ†(𝑠+𝑐 πœ†(𝑠+𝑐 NUM iajs-3349 158 20 ) ) PUNCT iajs-3349 158 21 } } PUNCT iajs-3349 158 22 4.7 4.7 NUM iajs-3349 158 23 order order NOUN iajs-3349 158 24 statistic statistic NOUN iajs-3349 158 25 let let VERB iajs-3349 158 26 𝑋1 𝑋1 PROPN iajs-3349 158 27 , , PUNCT iajs-3349 158 28 𝑋2 𝑋2 VERB iajs-3349 158 29 , , PUNCT iajs-3349 158 30 𝑋3 𝑋3 NOUN iajs-3349 158 31 , , PUNCT iajs-3349 158 32 … … PUNCT iajs-3349 158 33 , , PUNCT iajs-3349 158 34 𝑋𝑛 𝑋𝑛 NOUN iajs-3349 158 35 have have VERB iajs-3349 158 36 [ [ X iajs-3349 158 37 0,1 0,1 NUM iajs-3349 158 38 ] ] PUNCT iajs-3349 158 39 truncated truncate VERB iajs-3349 158 40 nadarajah nadarajah PROPN iajs-3349 158 41 - - PUNCT iajs-3349 158 42 haghighi haghighi PROPN iajs-3349 158 43 exponential exponential ADJ iajs-3349 158 44 distribution distribution NOUN iajs-3349 158 45 with with ADP iajs-3349 158 46 cdf cdf PROPN iajs-3349 158 47 , , PUNCT iajs-3349 158 48 pdf pdf NOUN iajs-3349 158 49 defined define VERB iajs-3349 158 50 in in ADP iajs-3349 158 51 ( ( PUNCT iajs-3349 158 52 10 10 NUM iajs-3349 158 53 ) ) PUNCT iajs-3349 158 54 , , PUNCT iajs-3349 158 55 ( ( PUNCT iajs-3349 158 56 11 11 NUM iajs-3349 158 57 ) ) PUNCT iajs-3349 158 58 , , PUNCT iajs-3349 158 59 respectively respectively ADV iajs-3349 158 60 and and CCONJ iajs-3349 158 61 let let VERB iajs-3349 158 62 𝑋1:𝑛 𝑋1:𝑛 ADJ iajs-3349 158 63 , , PUNCT iajs-3349 158 64 𝑋2:𝑛 𝑋2:𝑛 INTJ iajs-3349 158 65 , , PUNCT iajs-3349 158 66 𝑋3:𝑛 𝑋3:𝑛 VERB iajs-3349 158 67 , , PUNCT iajs-3349 158 68 … … PUNCT iajs-3349 158 69 , , PUNCT iajs-3349 158 70 𝑋𝑛:𝑛 𝑋𝑛:𝑛 ADV iajs-3349 158 71 be be AUX iajs-3349 158 72 the the DET iajs-3349 158 73 order order NOUN iajs-3349 158 74 statistic statistic NOUN iajs-3349 158 75 obtained obtain VERB iajs-3349 158 76 from from ADP iajs-3349 158 77 this this DET iajs-3349 158 78 sample sample NOUN iajs-3349 158 79 . . PUNCT iajs-3349 159 1 then then ADV iajs-3349 159 2 , , PUNCT iajs-3349 159 3 the the DET iajs-3349 159 4 probability probability NOUN iajs-3349 159 5 density density NOUN iajs-3349 159 6 function function NOUN iajs-3349 159 7 of of ADP iajs-3349 159 8 π‘π‘‘β„Ž π‘π‘‘β„Ž NOUN iajs-3349 159 9 order order NOUN iajs-3349 159 10 statistic statistic NOUN iajs-3349 159 11 from from ADP iajs-3349 159 12 [ [ PUNCT iajs-3349 159 13 0,1]tnhe 0,1]tnhe ADJ iajs-3349 159 14 distribution distribution NOUN iajs-3349 159 15 is be AUX iajs-3349 159 16 obtained obtain VERB iajs-3349 159 17 as as SCONJ iajs-3349 159 18 follows follow VERB iajs-3349 159 19 [ [ X iajs-3349 159 20 26 26 NUM iajs-3349 159 21 ] ] X iajs-3349 159 22 : : PUNCT iajs-3349 159 23 the the DET iajs-3349 159 24 pdf pdf NOUN iajs-3349 159 25 of of ADP iajs-3349 159 26 order order NOUN iajs-3349 159 27 statistic statistic NOUN iajs-3349 159 28 with with ADP iajs-3349 159 29 order order NOUN iajs-3349 159 30 𝑝 𝑝 NOUN iajs-3349 159 31 , , PUNCT iajs-3349 159 32 𝑋𝑝;𝑛 𝑋𝑝;𝑛 PROPN iajs-3349 159 33 is be AUX iajs-3349 159 34 given give VERB iajs-3349 159 35 by by ADP iajs-3349 159 36 the the DET iajs-3349 159 37 form form NOUN iajs-3349 159 38 : : PUNCT iajs-3349 159 39 𝑓𝑝:𝑛(π‘₯ 𝑓𝑝:𝑛(π‘₯ PROPN iajs-3349 159 40 ) ) PUNCT iajs-3349 160 1 = = SYM iajs-3349 160 2 𝑛 𝑛 PROPN iajs-3349 160 3 ! ! PUNCT iajs-3349 160 4 ( ( PUNCT iajs-3349 160 5 pβˆ’1)!(π‘›βˆ’π‘ pβˆ’1)!(π‘›βˆ’π‘ PROPN iajs-3349 160 6 ) ) PUNCT iajs-3349 160 7 ! ! PUNCT iajs-3349 161 1 [ [ X iajs-3349 161 2 𝐹(π‘₯)]π‘βˆ’1[1 𝐹(π‘₯)]π‘βˆ’1[1 X iajs-3349 161 3 βˆ’ βˆ’ PROPN iajs-3349 161 4 𝐹(π‘₯)]π‘›βˆ’π‘ 𝐹(π‘₯)]π‘›βˆ’π‘ SYM iajs-3349 161 5 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3349 161 6 ) ) PUNCT iajs-3349 161 7 = = PUNCT iajs-3349 162 1 βˆ‘ βˆ‘ PUNCT iajs-3349 162 2 𝑑(βˆ’1)𝑠(π‘›βˆ’π‘ 𝑑(βˆ’1)𝑠(π‘›βˆ’π‘ ADP iajs-3349 162 3 𝑠 𝑠 NUM iajs-3349 162 4 ) ) PUNCT iajs-3349 163 1 [ [ X iajs-3349 163 2 𝐹(π‘₯)]𝑝+π‘ βˆ’1π‘›βˆ’π‘ 𝐹(π‘₯)]𝑝+π‘ βˆ’1π‘›βˆ’π‘ PUNCT iajs-3349 163 3 𝑠=0 𝑠=0 PUNCT iajs-3349 163 4 𝑓(π‘₯ 𝑓(π‘₯ NOUN iajs-3349 163 5 ) ) PUNCT iajs-3349 163 6 ( ( PUNCT iajs-3349 163 7 27 27 NUM iajs-3349 163 8 ) ) PUNCT iajs-3349 163 9 where where SCONJ iajs-3349 163 10 𝑑 𝑑 PROPN iajs-3349 163 11 = = SYM iajs-3349 163 12 𝑛 𝑛 PROPN iajs-3349 163 13 ! ! PUNCT iajs-3349 163 14 ( ( PUNCT iajs-3349 163 15 pβˆ’1)!(π‘›βˆ’π‘ pβˆ’1)!(π‘›βˆ’π‘ PROPN iajs-3349 163 16 ) ) PUNCT iajs-3349 163 17 ! ! PUNCT iajs-3349 164 1 now now ADV iajs-3349 164 2 , , PUNCT iajs-3349 164 3 substituting substitute VERB iajs-3349 164 4 ( ( PUNCT iajs-3349 164 5 10 10 NUM iajs-3349 164 6 ) ) PUNCT iajs-3349 164 7 , , PUNCT iajs-3349 164 8 ( ( PUNCT iajs-3349 164 9 11 11 NUM iajs-3349 164 10 ) ) PUNCT iajs-3349 164 11 in in ADP iajs-3349 164 12 ( ( PUNCT iajs-3349 164 13 27 27 NUM iajs-3349 164 14 ) ) PUNCT iajs-3349 164 15 , , PUNCT iajs-3349 164 16 we we PRON iajs-3349 164 17 have have AUX iajs-3349 164 18 get get VERB iajs-3349 164 19 ihjpas ihjpa NOUN iajs-3349 164 20 . . PUNCT iajs-3349 165 1 37 37 NUM iajs-3349 165 2 ( ( PUNCT iajs-3349 165 3 2 2 NUM iajs-3349 165 4 ) ) PUNCT iajs-3349 165 5 2024 2024 NUM iajs-3349 165 6 370 370 NUM iajs-3349 165 7 𝑓𝑝:𝑛(π‘₯ 𝑓𝑝:𝑛(π‘₯ NOUN iajs-3349 165 8 ) ) PUNCT iajs-3349 165 9 = = PUNCT iajs-3349 166 1 βˆ‘ βˆ‘ PROPN iajs-3349 166 2 𝑑 𝑑 PROPN iajs-3349 166 3 ( ( PUNCT iajs-3349 166 4 βˆ’1)𝑠 βˆ’1)𝑠 X iajs-3349 166 5 ( ( PUNCT iajs-3349 166 6 𝑛 𝑛 PROPN iajs-3349 166 7 βˆ’ βˆ’ PROPN iajs-3349 166 8 𝑝 𝑝 PROPN iajs-3349 166 9 𝑠 𝑠 PROPN iajs-3349 166 10 ) ) PUNCT iajs-3349 166 11 ( ( PUNCT iajs-3349 166 12 1 1 NUM iajs-3349 166 13 βˆ’ βˆ’ NOUN iajs-3349 166 14 𝑒1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 𝑒1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ NUM iajs-3349 166 15 ) ) PUNCT iajs-3349 166 16 ) ) PUNCT iajs-3349 166 17 1 1 NUM iajs-3349 166 18 βˆ’ βˆ’ NOUN iajs-3349 166 19 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž ADJ iajs-3349 166 20 ) ) PUNCT iajs-3349 166 21 𝑝+π‘ βˆ’1π‘›βˆ’π‘ 𝑝+π‘ βˆ’1π‘›βˆ’π‘ NOUN iajs-3349 166 22 𝑠=0 𝑠=0 PROPN iajs-3349 166 23 Γ— Γ— PROPN iajs-3349 166 24 ( ( PUNCT iajs-3349 166 25 π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 166 26 ) ) PUNCT iajs-3349 166 27 ) ) PUNCT iajs-3349 167 1 π‘Žβˆ’1 π‘Žβˆ’1 PROPN iajs-3349 167 2 𝑒 𝑒 PROPN iajs-3349 167 3 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 167 4 ) ) PUNCT iajs-3349 167 5 ) ) PUNCT iajs-3349 168 1 π‘Ž π‘Ž DET iajs-3349 168 2 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 168 3 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 168 4 ) ) PUNCT iajs-3349 168 5 ( ( PUNCT iajs-3349 168 6 28 28 NUM iajs-3349 168 7 ) ) PUNCT iajs-3349 168 8 now now ADV iajs-3349 168 9 , , PUNCT iajs-3349 168 10 for for ADP iajs-3349 168 11 𝑝 𝑝 NOUN iajs-3349 168 12 = = SYM iajs-3349 168 13 1 1 NUM iajs-3349 168 14 , , PUNCT iajs-3349 168 15 we we PRON iajs-3349 168 16 get get VERB iajs-3349 168 17 the the DET iajs-3349 168 18 smallest small ADJ iajs-3349 168 19 order order NOUN iajs-3349 168 20 statistic statistic NOUN iajs-3349 168 21 ( ( PUNCT iajs-3349 168 22 least least ADJ iajs-3349 168 23 value value NOUN iajs-3349 168 24 function function NOUN iajs-3349 168 25 ): ): PUNCT iajs-3349 168 26 𝑓𝑝:𝑛(π‘₯ 𝑓𝑝:𝑛(π‘₯ PROPN iajs-3349 168 27 ) ) PUNCT iajs-3349 168 28 = = PUNCT iajs-3349 169 1 βˆ‘ βˆ‘ PROPN iajs-3349 169 2 𝑑 𝑑 PROPN iajs-3349 169 3 ( ( PUNCT iajs-3349 169 4 βˆ’1)𝑠 βˆ’1)𝑠 X iajs-3349 169 5 ( ( PUNCT iajs-3349 169 6 𝑛 𝑛 DET iajs-3349 169 7 βˆ’ βˆ’ NUM iajs-3349 169 8 1 1 NUM iajs-3349 169 9 𝑠 𝑠 NOUN iajs-3349 169 10 ) ) PUNCT iajs-3349 169 11 ( ( PUNCT iajs-3349 169 12 1 1 NUM iajs-3349 169 13 βˆ’ βˆ’ NOUN iajs-3349 169 14 𝑒1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 𝑒1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ NUM iajs-3349 169 15 ) ) PUNCT iajs-3349 169 16 ) ) PUNCT iajs-3349 169 17 1 1 NUM iajs-3349 169 18 βˆ’ βˆ’ NOUN iajs-3349 169 19 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž ADJ iajs-3349 169 20 ) ) PUNCT iajs-3349 169 21 π‘ π‘›βˆ’1 π‘ π‘›βˆ’1 PROPN iajs-3349 169 22 𝑠=0 𝑠=0 SYM iajs-3349 169 23 Γ— Γ— PROPN iajs-3349 169 24 ( ( PUNCT iajs-3349 169 25 π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 169 26 ) ) PUNCT iajs-3349 169 27 ) ) PUNCT iajs-3349 170 1 π‘Žβˆ’1 π‘Žβˆ’1 PROPN iajs-3349 170 2 𝑒 𝑒 PROPN iajs-3349 170 3 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 170 4 ) ) PUNCT iajs-3349 170 5 ) ) PUNCT iajs-3349 171 1 π‘Ž π‘Ž DET iajs-3349 171 2 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 171 3 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 171 4 ) ) PUNCT iajs-3349 171 5 ( ( PUNCT iajs-3349 171 6 29 29 NUM iajs-3349 171 7 ) ) PUNCT iajs-3349 171 8 and and CCONJ iajs-3349 171 9 for for ADP iajs-3349 171 10 𝑝 𝑝 PROPN iajs-3349 171 11 = = SYM iajs-3349 171 12 𝑛 𝑛 PROPN iajs-3349 171 13 , , PUNCT iajs-3349 171 14 we we PRON iajs-3349 171 15 get get VERB iajs-3349 171 16 the the DET iajs-3349 171 17 largest large ADJ iajs-3349 171 18 order order NOUN iajs-3349 171 19 statistic statistic NOUN iajs-3349 171 20 ( ( PUNCT iajs-3349 171 21 big big ADJ iajs-3349 171 22 value value NOUN iajs-3349 171 23 function function NOUN iajs-3349 171 24 ): ): PUNCT iajs-3349 171 25 𝑓𝑛:𝑛(π‘₯ 𝑓𝑛:𝑛(π‘₯ NUM iajs-3349 171 26 ) ) PUNCT iajs-3349 171 27 = = PUNCT iajs-3349 171 28 ( ( PUNCT iajs-3349 171 29 1βˆ’π‘’ 1βˆ’π‘’ NUM iajs-3349 171 30 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 171 31 ) ) PUNCT iajs-3349 171 32 ) ) PUNCT iajs-3349 171 33 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 171 34 ) ) PUNCT iajs-3349 171 35 𝑛+π‘ βˆ’1 𝑛+π‘ βˆ’1 NOUN iajs-3349 171 36 ( ( PUNCT iajs-3349 171 37 30 30 NUM iajs-3349 171 38 ) ) PUNCT iajs-3349 171 39 Γ— Γ— NOUN iajs-3349 171 40 ( ( PUNCT iajs-3349 171 41 π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ π‘Žπ‘πœ†(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ ADJ iajs-3349 171 42 ) ) PUNCT iajs-3349 171 43 ) ) PUNCT iajs-3349 172 1 π‘Žβˆ’1 π‘Žβˆ’1 PROPN iajs-3349 172 2 𝑒 𝑒 PROPN iajs-3349 172 3 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 172 4 ) ) PUNCT iajs-3349 172 5 ) ) PUNCT iajs-3349 173 1 π‘Ž π‘Ž DET iajs-3349 173 2 π‘’βˆ’πœ†π‘₯ π‘’βˆ’πœ†π‘₯ PROPN iajs-3349 173 3 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž 1βˆ’π‘’1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 173 4 ) ) PUNCT iajs-3349 173 5 5 5 NUM iajs-3349 173 6 . . PUNCT iajs-3349 173 7 maximum maximum ADJ iajs-3349 173 8 likelihood likelihood NOUN iajs-3349 173 9 method method NOUN iajs-3349 173 10 assume assume VERB iajs-3349 173 11 that that SCONJ iajs-3349 173 12 , , PUNCT iajs-3349 173 13 -1 -1 PROPN iajs-3349 173 14 . . PUNCT iajs-3349 173 15 , , PUNCT iajs-3349 173 16 , , PUNCT iajs-3349 173 17 π‘₯-2 π‘₯-2 PROPN iajs-3349 173 18 . . PUNCT iajs-3349 174 1 , , PUNCT iajs-3349 174 2 … … PUNCT iajs-3349 174 3 , , PUNCT iajs-3349 174 4 , , PUNCT iajs-3349 174 5 π‘₯-𝑛. π‘₯-𝑛. PROPN iajs-3349 174 6 is be AUX iajs-3349 174 7 a a DET iajs-3349 174 8 random random ADJ iajs-3349 174 9 sample sample NOUN iajs-3349 174 10 of of ADP iajs-3349 174 11 size size NOUN iajs-3349 174 12 n n CCONJ iajs-3349 174 13 from from ADP iajs-3349 174 14 the the DET iajs-3349 174 15 [ [ X iajs-3349 174 16 0,1 0,1 NUM iajs-3349 174 17 ] ] PUNCT iajs-3349 174 18 truncated truncate VERB iajs-3349 174 19 nadarajah nadarajah PROPN iajs-3349 174 20 - - PUNCT iajs-3349 174 21 haghighi haghighi PROPN iajs-3349 174 22 exponential exponential ADJ iajs-3349 174 23 distribution distribution NOUN iajs-3349 174 24 . . PUNCT iajs-3349 175 1 the the DET iajs-3349 175 2 corresponding corresponding ADJ iajs-3349 175 3 log log NOUN iajs-3349 175 4 - - PUNCT iajs-3349 175 5 likelihood likelihood NOUN iajs-3349 175 6 function function NOUN iajs-3349 175 7 is be AUX iajs-3349 175 8 then then ADV iajs-3349 175 9 given give VERB iajs-3349 175 10 by by ADP iajs-3349 175 11 [ [ PUNCT iajs-3349 175 12 27 27 NUM iajs-3349 175 13 ] ] PUNCT iajs-3349 175 14 , , PUNCT iajs-3349 176 1 [ [ X iajs-3349 176 2 28 28 NUM iajs-3349 176 3 ] ] PUNCT iajs-3349 176 4 : : PUNCT iajs-3349 176 5 𝐿(πœ‘\𝑋 𝐿(πœ‘\𝑋 NUM iajs-3349 176 6 ) ) PUNCT iajs-3349 176 7 = = PRON iajs-3349 176 8 ( ( PUNCT iajs-3349 176 9 π‘Žπ‘πœ†)𝑛 π‘Žπ‘πœ†)𝑛 PROPN iajs-3349 176 10 π‘’βˆ’πœ† π‘’βˆ’πœ† PROPN iajs-3349 176 11 βˆ‘ βˆ‘ PUNCT iajs-3349 176 12 π‘₯𝑖 π‘₯𝑖 ADP iajs-3349 176 13 𝑛 𝑛 PRON iajs-3349 176 14 𝑖=1 𝑖=1 PUNCT iajs-3349 176 15 βˆ‘ βˆ‘ PROPN iajs-3349 176 16 ( ( PUNCT iajs-3349 176 17 ( ( PUNCT iajs-3349 176 18 1 1 NUM iajs-3349 176 19 + + CCONJ iajs-3349 176 20 𝑏(1 𝑏(1 PROPN iajs-3349 176 21 βˆ’ βˆ’ PROPN iajs-3349 176 22 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 176 23 ) ) PUNCT iajs-3349 176 24 ) ) PUNCT iajs-3349 176 25 ) ) PUNCT iajs-3349 177 1 π‘Žβˆ’1 π‘Žβˆ’1 PROPN iajs-3349 177 2 𝑛 𝑛 PRON iajs-3349 177 3 𝑖=1 𝑖=1 PROPN iajs-3349 177 4 π‘’βˆ‘ π‘’βˆ‘ PROPN iajs-3349 177 5 ( ( PUNCT iajs-3349 177 6 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯𝑖 1βˆ’(1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯𝑖 NUM iajs-3349 177 7 ) ) PUNCT iajs-3349 177 8 ) ) PUNCT iajs-3349 177 9 π‘Žπ‘› π‘Žπ‘› PROPN iajs-3349 177 10 𝑖=1 𝑖=1 PROPN iajs-3349 177 11 ) ) PUNCT iajs-3349 177 12 ( ( PUNCT iajs-3349 177 13 1 1 NUM iajs-3349 177 14 βˆ’ βˆ’ PROPN iajs-3349 177 15 𝑒1βˆ’(1+𝑏)π‘Ž)𝑛 𝑒1βˆ’(1+𝑏)π‘Ž)𝑛 NOUN iajs-3349 177 16 let let VERB iajs-3349 177 17 𝑙 𝑙 X iajs-3349 177 18 = = SYM iajs-3349 177 19 π‘™π‘œπ‘”πΏ(πœ‘\𝑋 π‘™π‘œπ‘”πΏ(πœ‘\𝑋 NOUN iajs-3349 177 20 ) ) PUNCT iajs-3349 177 21 be be VERB iajs-3349 177 22 the the DET iajs-3349 177 23 natural natural ADJ iajs-3349 177 24 logarithm logarithm NOUN iajs-3349 177 25 probability probability NOUN iajs-3349 177 26 function function NOUN iajs-3349 177 27 . . PUNCT iajs-3349 178 1 𝑙 𝑙 X iajs-3349 178 2 = = SYM iajs-3349 178 3 π‘›π‘™π‘œπ‘”(π‘Žπ‘πœ† π‘›π‘™π‘œπ‘”(π‘Žπ‘πœ† PROPN iajs-3349 178 4 ) ) PUNCT iajs-3349 179 1 βˆ’ βˆ’ PROPN iajs-3349 180 1 πœ† πœ† ADP iajs-3349 180 2 βˆ‘ βˆ‘ PROPN iajs-3349 180 3 π‘₯𝑖 π‘₯𝑖 PROPN iajs-3349 180 4 𝑛 𝑛 PRON iajs-3349 180 5 𝑖=1 𝑖=1 PROPN iajs-3349 181 1 + + CCONJ iajs-3349 181 2 ( ( PUNCT iajs-3349 181 3 π‘Ž π‘Ž X iajs-3349 181 4 βˆ’ βˆ’ NOUN iajs-3349 181 5 1 1 NUM iajs-3349 181 6 ) ) PUNCT iajs-3349 181 7 βˆ‘ βˆ‘ PUNCT iajs-3349 181 8 π‘™π‘œπ‘”{(1 π‘™π‘œπ‘”{(1 NOUN iajs-3349 181 9 + + CCONJ iajs-3349 181 10 𝑏(1 𝑏(1 PROPN iajs-3349 181 11 βˆ’ βˆ’ PROPN iajs-3349 181 12 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 181 13 ) ) PUNCT iajs-3349 181 14 ) ) PUNCT iajs-3349 181 15 } } PUNCT iajs-3349 182 1 𝑛 𝑛 PRON iajs-3349 182 2 𝑖=1 𝑖=1 PUNCT iajs-3349 183 1 + + CCONJ iajs-3349 183 2 βˆ‘(1 βˆ‘(1 NUM iajs-3349 183 3 βˆ’ βˆ’ PROPN iajs-3349 183 4 ( ( PUNCT iajs-3349 183 5 1 1 NUM iajs-3349 183 6 + + CCONJ iajs-3349 183 7 𝑏(1 𝑏(1 PROPN iajs-3349 183 8 βˆ’ βˆ’ PROPN iajs-3349 183 9 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 183 10 ) ) PUNCT iajs-3349 183 11 ) ) PUNCT iajs-3349 184 1 π‘Ž π‘Ž X iajs-3349 184 2 𝑛 𝑛 PROPN iajs-3349 184 3 𝑖=1 𝑖=1 PROPN iajs-3349 184 4 ) ) PUNCT iajs-3349 184 5 βˆ’ βˆ’ NOUN iajs-3349 184 6 π‘›π‘™π‘œπ‘”{1 π‘›π‘™π‘œπ‘”{1 VERB iajs-3349 184 7 βˆ’ βˆ’ NOUN iajs-3349 184 8 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž ADJ iajs-3349 184 9 } } PUNCT iajs-3349 184 10 𝑙 𝑙 NOUN iajs-3349 184 11 = = PUNCT iajs-3349 184 12 π‘›π‘™π‘œπ‘”(π‘Ž π‘›π‘™π‘œπ‘”(π‘Ž NOUN iajs-3349 184 13 ) ) PUNCT iajs-3349 184 14 + + NUM iajs-3349 184 15 π‘›π‘™π‘œπ‘”(𝑏 π‘›π‘™π‘œπ‘”(𝑏 X iajs-3349 184 16 ) ) PUNCT iajs-3349 184 17 + + NUM iajs-3349 184 18 π‘›π‘™π‘œπ‘”(πœ† π‘›π‘™π‘œπ‘”(πœ† NOUN iajs-3349 184 19 ) ) PUNCT iajs-3349 184 20 βˆ’ βˆ’ PROPN iajs-3349 185 1 πœ† πœ† ADP iajs-3349 185 2 βˆ‘ βˆ‘ PROPN iajs-3349 185 3 π‘₯𝑖 π‘₯𝑖 PROPN iajs-3349 185 4 𝑛 𝑛 PRON iajs-3349 185 5 𝑖=1 𝑖=1 PROPN iajs-3349 186 1 + + CCONJ iajs-3349 186 2 ( ( PUNCT iajs-3349 186 3 π‘Ž π‘Ž X iajs-3349 186 4 βˆ‘ βˆ‘ VERB iajs-3349 186 5 π‘™π‘œπ‘” π‘™π‘œπ‘” X iajs-3349 186 6 { { PUNCT iajs-3349 186 7 ( ( PUNCT iajs-3349 186 8 1 1 NUM iajs-3349 186 9 + + CCONJ iajs-3349 186 10 𝑏(1 𝑏(1 PROPN iajs-3349 186 11 βˆ’ βˆ’ PROPN iajs-3349 186 12 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 186 13 ) ) PUNCT iajs-3349 186 14 ) ) PUNCT iajs-3349 186 15 } } PUNCT iajs-3349 186 16 𝑛 𝑛 DET iajs-3349 186 17 𝑖=1 𝑖=1 PUNCT iajs-3349 186 18 βˆ’ βˆ’ PRON iajs-3349 186 19 βˆ‘ βˆ‘ PUNCT iajs-3349 186 20 π‘™π‘œπ‘” π‘™π‘œπ‘” X iajs-3349 186 21 { { PUNCT iajs-3349 186 22 ( ( PUNCT iajs-3349 186 23 1 1 NUM iajs-3349 186 24 + + CCONJ iajs-3349 186 25 𝑏(1 𝑏(1 PROPN iajs-3349 186 26 βˆ’ βˆ’ PROPN iajs-3349 186 27 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 186 28 ) ) PUNCT iajs-3349 186 29 ) ) PUNCT iajs-3349 186 30 } } PUNCT iajs-3349 186 31 𝑛 𝑛 PRON iajs-3349 186 32 𝑖=1 𝑖=1 PUNCT iajs-3349 186 33 ) ) PUNCT iajs-3349 187 1 + + CCONJ iajs-3349 187 2 βˆ‘ βˆ‘ PROPN iajs-3349 187 3 1 1 NUM iajs-3349 187 4 𝑛 𝑛 DET iajs-3349 187 5 𝑖=1 𝑖=1 PROPN iajs-3349 187 6 βˆ’ βˆ’ PROPN iajs-3349 187 7 βˆ‘(1 βˆ‘(1 PROPN iajs-3349 187 8 + + CCONJ iajs-3349 187 9 𝑏(1 𝑏(1 PROPN iajs-3349 187 10 βˆ’ βˆ’ PROPN iajs-3349 187 11 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 187 12 ) ) PUNCT iajs-3349 187 13 ) ) PUNCT iajs-3349 188 1 π‘Ž π‘Ž X iajs-3349 188 2 𝑛 𝑛 PROPN iajs-3349 188 3 𝑖=1 𝑖=1 PROPN iajs-3349 188 4 ) ) PUNCT iajs-3349 188 5 βˆ’ βˆ’ PROPN iajs-3349 188 6 π‘›π‘™π‘œπ‘”(1 π‘›π‘™π‘œπ‘”(1 NOUN iajs-3349 188 7 βˆ’ βˆ’ NOUN iajs-3349 188 8 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž NOUN iajs-3349 188 9 ) ) PUNCT iajs-3349 188 10 ihjpas ihjpa NOUN iajs-3349 188 11 . . PUNCT iajs-3349 189 1 37 37 NUM iajs-3349 189 2 ( ( PUNCT iajs-3349 189 3 2 2 NUM iajs-3349 189 4 ) ) PUNCT iajs-3349 189 5 2024 2024 NUM iajs-3349 189 6 371 371 NUM iajs-3349 189 7 𝑙 𝑙 NOUN iajs-3349 189 8 = = PUNCT iajs-3349 189 9 π‘›π‘™π‘œπ‘”(π‘Ž π‘›π‘™π‘œπ‘”(π‘Ž NOUN iajs-3349 189 10 ) ) PUNCT iajs-3349 189 11 + + NUM iajs-3349 189 12 π‘›π‘™π‘œπ‘”(𝑏 π‘›π‘™π‘œπ‘”(𝑏 X iajs-3349 189 13 ) ) PUNCT iajs-3349 189 14 + + NUM iajs-3349 189 15 π‘›π‘™π‘œπ‘”(πœ† π‘›π‘™π‘œπ‘”(πœ† NOUN iajs-3349 189 16 ) ) PUNCT iajs-3349 189 17 βˆ’ βˆ’ PROPN iajs-3349 189 18 πœ† πœ† ADP iajs-3349 189 19 βˆ‘ βˆ‘ PROPN iajs-3349 189 20 π‘₯𝑖 π‘₯𝑖 PROPN iajs-3349 189 21 𝑛 𝑛 PRON iajs-3349 189 22 𝑖=1 𝑖=1 PROPN iajs-3349 190 1 + + CCONJ iajs-3349 190 2 ( ( PUNCT iajs-3349 190 3 π‘Ž π‘Ž X iajs-3349 190 4 βˆ‘ βˆ‘ PUNCT iajs-3349 190 5 π‘™π‘œπ‘”{(1 π‘™π‘œπ‘”{(1 NOUN iajs-3349 190 6 + + CCONJ iajs-3349 190 7 𝑏(1 𝑏(1 PROPN iajs-3349 190 8 βˆ’ βˆ’ PROPN iajs-3349 190 9 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 190 10 ) ) PUNCT iajs-3349 190 11 ) ) PUNCT iajs-3349 190 12 } } PUNCT iajs-3349 191 1 𝑛 𝑛 PRON iajs-3349 191 2 𝑖=1 𝑖=1 PUNCT iajs-3349 191 3 βˆ’ βˆ’ NOUN iajs-3349 191 4 βˆ‘ βˆ‘ PUNCT iajs-3349 191 5 π‘™π‘œπ‘”{(1 π‘™π‘œπ‘”{(1 PROPN iajs-3349 191 6 + + CCONJ iajs-3349 191 7 𝑏(1 𝑏(1 PROPN iajs-3349 191 8 βˆ’ βˆ’ PROPN iajs-3349 191 9 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 191 10 ) ) PUNCT iajs-3349 191 11 ) ) PUNCT iajs-3349 191 12 } } PUNCT iajs-3349 191 13 𝑛 𝑛 PRON iajs-3349 191 14 𝑖=1 𝑖=1 PUNCT iajs-3349 191 15 ) ) PUNCT iajs-3349 192 1 + + CCONJ iajs-3349 192 2 𝑛 𝑛 DET iajs-3349 192 3 βˆ’ βˆ’ PROPN iajs-3349 192 4 βˆ‘(1 βˆ‘(1 PROPN iajs-3349 192 5 + + CCONJ iajs-3349 192 6 𝑏(1 𝑏(1 PROPN iajs-3349 192 7 βˆ’ βˆ’ PROPN iajs-3349 192 8 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 192 9 ) ) PUNCT iajs-3349 192 10 ) ) PUNCT iajs-3349 192 11 π‘Ž π‘Ž X iajs-3349 192 12 𝑛 𝑛 PROPN iajs-3349 192 13 𝑖=1 𝑖=1 PROPN iajs-3349 192 14 ) ) PUNCT iajs-3349 192 15 βˆ’ βˆ’ NOUN iajs-3349 192 16 π‘›π‘™π‘œπ‘”{1 π‘›π‘™π‘œπ‘”{1 VERB iajs-3349 192 17 βˆ’ βˆ’ NOUN iajs-3349 192 18 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž NOUN iajs-3349 192 19 } } PUNCT iajs-3349 192 20 now now ADV iajs-3349 192 21 , , PUNCT iajs-3349 192 22 by by ADP iajs-3349 192 23 taking take VERB iajs-3349 192 24 the the DET iajs-3349 192 25 first first ADJ iajs-3349 192 26 partial partial ADJ iajs-3349 192 27 derivative derivative NOUN iajs-3349 192 28 of of ADP iajs-3349 192 29 the the DET iajs-3349 192 30 log log NOUN iajs-3349 192 31 likelihood likelihood NOUN iajs-3349 192 32 function function NOUN iajs-3349 192 33 with with ADP iajs-3349 192 34 respect respect NOUN iajs-3349 192 35 to to ADP iajs-3349 192 36 the the DET iajs-3349 192 37 parameters parameter NOUN iajs-3349 192 38 ( ( PUNCT iajs-3349 192 39 π‘Ž π‘Ž X iajs-3349 192 40 , , PUNCT iajs-3349 192 41 𝑏 𝑏 NOUN iajs-3349 192 42 , , PUNCT iajs-3349 192 43 πœ† πœ† NOUN iajs-3349 192 44 ) ) PUNCT iajs-3349 192 45 , , PUNCT iajs-3349 192 46 we we PRON iajs-3349 192 47 get get VERB iajs-3349 192 48 : : PUNCT iajs-3349 192 49 πœ•(𝑙 πœ•(𝑙 NOUN iajs-3349 192 50 ) ) PUNCT iajs-3349 192 51 πœ•π‘Ž πœ•π‘Ž NOUN iajs-3349 193 1 = = PUNCT iajs-3349 193 2 { { PUNCT iajs-3349 193 3 𝑛 𝑛 ADP iajs-3349 193 4 π‘Ž π‘Ž PROPN iajs-3349 193 5 + + X iajs-3349 193 6 βˆ‘ βˆ‘ PUNCT iajs-3349 193 7 π‘™π‘œπ‘”{(1 π‘™π‘œπ‘”{(1 NOUN iajs-3349 193 8 + + CCONJ iajs-3349 193 9 𝑏(1 𝑏(1 PROPN iajs-3349 193 10 βˆ’ βˆ’ PROPN iajs-3349 193 11 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 193 12 ) ) PUNCT iajs-3349 193 13 ) ) PUNCT iajs-3349 193 14 } } PUNCT iajs-3349 194 1 𝑛 𝑛 PRON iajs-3349 194 2 𝑖=1 𝑖=1 PUNCT iajs-3349 194 3 + + CCONJ iajs-3349 194 4 ( ( PUNCT iajs-3349 194 5 1 1 NUM iajs-3349 194 6 + + CCONJ iajs-3349 194 7 𝑏(1 𝑏(1 PROPN iajs-3349 194 8 βˆ’ βˆ’ PROPN iajs-3349 194 9 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 194 10 ) ) PUNCT iajs-3349 194 11 ) ) PUNCT iajs-3349 195 1 π‘Ž π‘Ž X iajs-3349 195 2 π‘™π‘œπ‘”{(1 π‘™π‘œπ‘”{(1 NOUN iajs-3349 195 3 + + CCONJ iajs-3349 195 4 𝑏(1 𝑏(1 PROPN iajs-3349 195 5 βˆ’ βˆ’ PROPN iajs-3349 195 6 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 195 7 ) ) PUNCT iajs-3349 195 8 ) ) PUNCT iajs-3349 196 1 π‘Ž π‘Ž X iajs-3349 196 2 } } PUNCT iajs-3349 196 3 βˆ’ βˆ’ PROPN iajs-3349 196 4 𝑛 𝑛 PROPN iajs-3349 196 5 ( ( PUNCT iajs-3349 196 6 1 1 NUM iajs-3349 196 7 + + CCONJ iajs-3349 196 8 𝑏)π‘Ž 𝑏)π‘Ž ADJ iajs-3349 196 9 π‘™π‘œπ‘” π‘™π‘œπ‘” X iajs-3349 196 10 { { PUNCT iajs-3349 196 11 1 1 NUM iajs-3349 196 12 + + CCONJ iajs-3349 196 13 𝑏}𝑒1βˆ’(1+𝑏)π‘Ž 𝑏}𝑒1βˆ’(1+𝑏)π‘Ž ADJ iajs-3349 196 14 1 1 NUM iajs-3349 196 15 βˆ’ βˆ’ NOUN iajs-3349 196 16 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž NOUN iajs-3349 196 17 } } PUNCT iajs-3349 196 18 ( ( PUNCT iajs-3349 196 19 31 31 NUM iajs-3349 196 20 ) ) PUNCT iajs-3349 196 21 πœ•(𝑙 πœ•(𝑙 NOUN iajs-3349 196 22 ) ) PUNCT iajs-3349 196 23 πœ•π‘ πœ•π‘ PROPN iajs-3349 196 24 = = SYM iajs-3349 196 25 𝑛 𝑛 PRON iajs-3349 196 26 𝑏 𝑏 NOUN iajs-3349 196 27 + + CCONJ iajs-3349 196 28 ( ( PUNCT iajs-3349 196 29 π‘Ž π‘Ž PRON iajs-3349 196 30 βˆ’ βˆ’ PROPN iajs-3349 196 31 1 1 NUM iajs-3349 196 32 ) ) PUNCT iajs-3349 196 33 βˆ‘ βˆ‘ ADP iajs-3349 196 34 1 1 NUM iajs-3349 196 35 βˆ’ βˆ’ NOUN iajs-3349 196 36 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 VERB iajs-3349 196 37 1 1 NUM iajs-3349 196 38 + + CCONJ iajs-3349 196 39 𝑏(1 𝑏(1 PROPN iajs-3349 196 40 βˆ’ βˆ’ PROPN iajs-3349 196 41 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 196 42 ) ) PUNCT iajs-3349 196 43 𝑛 𝑛 PRON iajs-3349 196 44 𝑖=1 𝑖=1 PUNCT iajs-3349 197 1 βˆ’ βˆ’ NOUN iajs-3349 197 2 βˆ‘ βˆ‘ PUNCT iajs-3349 197 3 ( ( PUNCT iajs-3349 197 4 π‘Ž(1 π‘Ž(1 PROPN iajs-3349 197 5 βˆ’ βˆ’ NOUN iajs-3349 197 6 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 197 7 ) ) PUNCT iajs-3349 197 8 ( ( PUNCT iajs-3349 197 9 1 1 NUM iajs-3349 197 10 + + CCONJ iajs-3349 197 11 𝑏(1 𝑏(1 PROPN iajs-3349 197 12 βˆ’ βˆ’ PROPN iajs-3349 197 13 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 197 14 ) ) PUNCT iajs-3349 197 15 ) ) PUNCT iajs-3349 198 1 π‘Žβˆ’1 π‘Žβˆ’1 PROPN iajs-3349 198 2 ) ) PUNCT iajs-3349 198 3 𝑛 𝑛 PRON iajs-3349 199 1 𝑖=1 𝑖=1 PUNCT iajs-3349 199 2 βˆ’ βˆ’ NOUN iajs-3349 199 3 π‘›π‘Ž(1 π‘›π‘Ž(1 NOUN iajs-3349 199 4 + + CCONJ iajs-3349 199 5 𝑏)π‘Žβˆ’1 𝑏)π‘Žβˆ’1 PRON iajs-3349 199 6 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž NUM iajs-3349 199 7 1 1 NUM iajs-3349 199 8 βˆ’ βˆ’ NOUN iajs-3349 199 9 𝑒1βˆ’(1+𝑏)π‘Ž 𝑒1βˆ’(1+𝑏)π‘Ž NOUN iajs-3349 199 10 ( ( PUNCT iajs-3349 199 11 32 32 NUM iajs-3349 199 12 ) ) PUNCT iajs-3349 199 13 πœ•(𝑙 πœ•(𝑙 NOUN iajs-3349 199 14 ) ) PUNCT iajs-3349 199 15 πœ•πœ† πœ•πœ† NOUN iajs-3349 199 16 = = PUNCT iajs-3349 199 17 𝑛 𝑛 PROPN iajs-3349 199 18 πœ† πœ† NOUN iajs-3349 199 19 βˆ’ βˆ’ PROPN iajs-3349 199 20 βˆ‘ βˆ‘ PROPN iajs-3349 199 21 π‘₯𝑖 π‘₯𝑖 PROPN iajs-3349 199 22 𝑛 𝑛 PRON iajs-3349 199 23 𝑖=1 𝑖=1 PROPN iajs-3349 200 1 + + CCONJ iajs-3349 200 2 ( ( PUNCT iajs-3349 200 3 π‘Ž π‘Ž X iajs-3349 200 4 βˆ’ βˆ’ NOUN iajs-3349 200 5 1 1 NUM iajs-3349 200 6 ) ) PUNCT iajs-3349 200 7 βˆ‘ βˆ‘ ADP iajs-3349 200 8 𝑏 𝑏 PROPN iajs-3349 200 9 π‘₯𝑖 π‘₯𝑖 PROPN iajs-3349 200 10 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 VERB iajs-3349 200 11 1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯𝑖 1+𝑏(1βˆ’π‘’βˆ’πœ†π‘₯𝑖 NUM iajs-3349 200 12 ) ) PUNCT iajs-3349 200 13 βˆ’ βˆ’ NOUN iajs-3349 200 14 βˆ‘ βˆ‘ PUNCT iajs-3349 200 15 π‘Žπ‘ π‘Žπ‘ PROPN iajs-3349 200 16 π‘₯𝑖 π‘₯𝑖 PROPN iajs-3349 200 17 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 VERB iajs-3349 200 18 ( ( PUNCT iajs-3349 200 19 1 1 NUM iajs-3349 200 20 + + CCONJ iajs-3349 200 21 𝑏(1 𝑏(1 PROPN iajs-3349 200 22 βˆ’π‘› βˆ’π‘› NOUN iajs-3349 200 23 𝑖=1 𝑖=1 PUNCT iajs-3349 200 24 𝑛 𝑛 DET iajs-3349 200 25 𝑖=1 𝑖=1 PROPN iajs-3349 200 26 π‘’βˆ’πœ†π‘₯𝑖 π‘’βˆ’πœ†π‘₯𝑖 NOUN iajs-3349 200 27 ) ) PUNCT iajs-3349 200 28 ) ) PUNCT iajs-3349 201 1 π‘Žβˆ’1 π‘Žβˆ’1 PROPN iajs-3349 201 2 ( ( PUNCT iajs-3349 201 3 33 33 NUM iajs-3349 201 4 ) ) PUNCT iajs-3349 201 5 by by ADP iajs-3349 201 6 setting set VERB iajs-3349 201 7 the the DET iajs-3349 201 8 above above ADJ iajs-3349 201 9 equations equation NOUN iajs-3349 201 10 ( ( PUNCT iajs-3349 201 11 31 31 NUM iajs-3349 201 12 ) ) PUNCT iajs-3349 201 13 , , PUNCT iajs-3349 201 14 ( ( PUNCT iajs-3349 201 15 32 32 NUM iajs-3349 201 16 ) ) PUNCT iajs-3349 201 17 , , PUNCT iajs-3349 201 18 and and CCONJ iajs-3349 201 19 ( ( PUNCT iajs-3349 201 20 33 33 NUM iajs-3349 201 21 ) ) PUNCT iajs-3349 201 22 to to ADP iajs-3349 201 23 zero zero NUM iajs-3349 201 24 , , PUNCT iajs-3349 201 25 solving solve VERB iajs-3349 201 26 it it PRON iajs-3349 201 27 numerically numerically ADV iajs-3349 201 28 through through ADP iajs-3349 201 29 using use VERB iajs-3349 201 30 iterative iterative NOUN iajs-3349 201 31 methods method NOUN iajs-3349 201 32 , , PUNCT iajs-3349 201 33 such such ADJ iajs-3349 201 34 as as ADP iajs-3349 201 35 newton newton PROPN iajs-3349 201 36 - - PUNCT iajs-3349 201 37 raphson raphson NOUN iajs-3349 201 38 type type NOUN iajs-3349 201 39 algorithms algorithm NOUN iajs-3349 201 40 , , PUNCT iajs-3349 201 41 we we PRON iajs-3349 201 42 can can AUX iajs-3349 201 43 get get VERB iajs-3349 201 44 the the DET iajs-3349 201 45 estimators estimator NOUN iajs-3349 201 46 of of ADP iajs-3349 201 47 the the DET iajs-3349 201 48 parameters parameter NOUN iajs-3349 201 49 . . PUNCT iajs-3349 202 1 6 6 X iajs-3349 202 2 . . X iajs-3349 202 3 simulation simulation NOUN iajs-3349 202 4 study study NOUN iajs-3349 202 5 in in ADP iajs-3349 202 6 this this DET iajs-3349 202 7 section section NOUN iajs-3349 202 8 , , PUNCT iajs-3349 202 9 we we PRON iajs-3349 202 10 have have AUX iajs-3349 202 11 conducted conduct VERB iajs-3349 202 12 simulation simulation NOUN iajs-3349 202 13 study study NOUN iajs-3349 202 14 for for ADP iajs-3349 202 15 [ [ X iajs-3349 202 16 0,1 0,1 NOUN iajs-3349 202 17 ] ] PUNCT iajs-3349 202 18 tnhe tnhe VERB iajs-3349 202 19 distribution distribution NOUN iajs-3349 202 20 . . PUNCT iajs-3349 203 1 we we PRON iajs-3349 203 2 have have AUX iajs-3349 203 3 generated generate VERB iajs-3349 203 4 samples sample NOUN iajs-3349 203 5 of of ADP iajs-3349 203 6 sizes size NOUN iajs-3349 203 7 𝑛 𝑛 VERB iajs-3349 203 8 = = PUNCT iajs-3349 203 9 { { PUNCT iajs-3349 203 10 30 30 NUM iajs-3349 203 11 , , PUNCT iajs-3349 203 12 50 50 NUM iajs-3349 203 13 , , PUNCT iajs-3349 203 14 80 80 NUM iajs-3349 203 15 , , PUNCT iajs-3349 203 16 120 120 NUM iajs-3349 203 17 , , PUNCT iajs-3349 203 18 200 200 NUM iajs-3349 203 19 } } PUNCT iajs-3349 203 20 from from ADP iajs-3349 203 21 the the DET iajs-3349 203 22 proposed propose VERB iajs-3349 203 23 model model NOUN iajs-3349 203 24 and and CCONJ iajs-3349 203 25 parameters parameter NOUN iajs-3349 203 26 estimated estimate VERB iajs-3349 203 27 by by ADP iajs-3349 203 28 mle mle PROPN iajs-3349 203 29 method method NOUN iajs-3349 203 30 , , PUNCT iajs-3349 203 31 the the DET iajs-3349 203 32 simulation simulation NOUN iajs-3349 203 33 study study NOUN iajs-3349 203 34 is be AUX iajs-3349 203 35 in in ADP iajs-3349 203 36 terms term NOUN iajs-3349 203 37 of of ADP iajs-3349 203 38 the the DET iajs-3349 203 39 averages average NOUN iajs-3349 203 40 of of ADP iajs-3349 203 41 the the DET iajs-3349 203 42 three three NUM iajs-3349 203 43 quantities quantity NOUN iajs-3349 203 44 : : PUNCT iajs-3349 203 45 absolute absolute ADJ iajs-3349 203 46 bias bias NOUN iajs-3349 203 47 |bais(𝝉)| |bais(𝝉)| PUNCT iajs-3349 203 48 = = SYM iajs-3349 203 49 1 1 NUM iajs-3349 203 50 n n NUM iajs-3349 203 51 βˆ‘ βˆ‘ ADV iajs-3349 203 52 |n |n PROPN iajs-3349 203 53 i=1 i=1 PROPN iajs-3349 203 54 οΏ½ οΏ½ PROPN iajs-3349 203 55 Μ‚ Μ‚ NUM iajs-3349 203 56 οΏ½ οΏ½ NOUN iajs-3349 203 57 βˆ’ βˆ’ NOUN iajs-3349 203 58 𝝉| 𝝉| PROPN iajs-3349 203 59 , , PUNCT iajs-3349 203 60 mean mean ADJ iajs-3349 203 61 square square ADJ iajs-3349 203 62 error error NOUN iajs-3349 203 63 ( ( PUNCT iajs-3349 203 64 mse mse NOUN iajs-3349 203 65 ) ) PUNCT iajs-3349 203 66 , , PUNCT iajs-3349 203 67 𝑀𝑆𝐸(𝝉 𝑀𝑆𝐸(𝝉 NUM iajs-3349 203 68 ) ) PUNCT iajs-3349 203 69 = = SYM iajs-3349 203 70 1 1 NUM iajs-3349 203 71 n n X iajs-3349 203 72 βˆ‘ βˆ‘ ADP iajs-3349 203 73 ( ( PUNCT iajs-3349 203 74 οΏ½ οΏ½ PROPN iajs-3349 203 75 Μ‚ Μ‚ SYM iajs-3349 203 76 οΏ½ οΏ½ PROPN iajs-3349 203 77 βˆ’ βˆ’ PROPN iajs-3349 203 78 𝝉)2n 𝝉)2n NOUN iajs-3349 203 79 i=1 i=1 X iajs-3349 203 80 , , PUNCT iajs-3349 203 81 and and CCONJ iajs-3349 203 82 the the DET iajs-3349 203 83 mean mean ADJ iajs-3349 203 84 relative relative ADJ iajs-3349 203 85 error error NOUN iajs-3349 203 86 ( ( PUNCT iajs-3349 203 87 mre mre NOUN iajs-3349 203 88 ) ) PUNCT iajs-3349 203 89 , , PUNCT iajs-3349 203 90 𝑀𝑅𝐸(𝝉 𝑀𝑅𝐸(𝝉 NUM iajs-3349 203 91 ) ) PUNCT iajs-3349 203 92 = = SYM iajs-3349 203 93 1 1 NUM iajs-3349 203 94 n n X iajs-3349 203 95 βˆ‘ βˆ‘ ADV iajs-3349 203 96 |n |n PROPN iajs-3349 203 97 i=1 i=1 PROPN iajs-3349 203 98 οΏ½ οΏ½ PROPN iajs-3349 203 99 Μ‚ Μ‚ NUM iajs-3349 203 100 οΏ½ οΏ½ PROPN iajs-3349 203 101 βˆ’ βˆ’ PROPN iajs-3349 203 102 𝝉|/𝝉 𝝉|/𝝉 NUM iajs-3349 203 103 . . PUNCT iajs-3349 204 1 all all DET iajs-3349 204 2 the the DET iajs-3349 204 3 computations computation NOUN iajs-3349 204 4 are be AUX iajs-3349 204 5 made make VERB iajs-3349 204 6 by by ADP iajs-3349 204 7 using use VERB iajs-3349 204 8 r r NOUN iajs-3349 204 9 statistical statistical ADJ iajs-3349 204 10 software software NOUN iajs-3349 204 11 . . PUNCT iajs-3349 205 1 table table NOUN iajs-3349 205 2 1 1 NUM iajs-3349 205 3 shows show VERB iajs-3349 205 4 some some DET iajs-3349 205 5 simulation simulation NOUN iajs-3349 205 6 results result NOUN iajs-3349 205 7 for for ADP iajs-3349 205 8 different different ADJ iajs-3349 205 9 values value NOUN iajs-3349 205 10 of of ADP iajs-3349 205 11 𝝉 𝝉 NOUN iajs-3349 205 12 = = PUNCT iajs-3349 205 13 ( ( PUNCT iajs-3349 205 14 π‘Ž π‘Ž X iajs-3349 205 15 , , PUNCT iajs-3349 205 16 𝑏 𝑏 NOUN iajs-3349 205 17 , , PUNCT iajs-3349 205 18 πœ†)𝑇. πœ†)𝑇. NOUN iajs-3349 205 19 based base VERB iajs-3349 205 20 on on ADP iajs-3349 205 21 the the DET iajs-3349 205 22 simulation simulation NOUN iajs-3349 205 23 criteria criterion NOUN iajs-3349 205 24 , , PUNCT iajs-3349 205 25 table table NOUN iajs-3349 205 26 1 1 NUM iajs-3349 205 27 discovered discover VERB iajs-3349 205 28 that that SCONJ iajs-3349 205 29 the the DET iajs-3349 205 30 maximum maximum ADJ iajs-3349 205 31 likelihood likelihood NOUN iajs-3349 205 32 estimate estimate NOUN iajs-3349 205 33 strategy strategy NOUN iajs-3349 205 34 performs perform VERB iajs-3349 205 35 pretty pretty ADV iajs-3349 205 36 well well ADV iajs-3349 205 37 in in ADP iajs-3349 205 38 estimating estimate VERB iajs-3349 205 39 the the DET iajs-3349 205 40 [ [ X iajs-3349 205 41 0,1 0,1 NUM iajs-3349 205 42 ] ] PUNCT iajs-3349 205 43 tnhe tnhe VERB iajs-3349 205 44 distribution distribution NOUN iajs-3349 205 45 parameters parameter NOUN iajs-3349 205 46 . . PUNCT iajs-3349 206 1 7 7 X iajs-3349 206 2 . . X iajs-3349 206 3 application application NOUN iajs-3349 206 4 in in ADP iajs-3349 206 5 this this DET iajs-3349 206 6 section section NOUN iajs-3349 206 7 , , PUNCT iajs-3349 206 8 we we PRON iajs-3349 206 9 fit fit VERB iajs-3349 206 10 the the DET iajs-3349 206 11 tnhe tnhe NOUN iajs-3349 206 12 distribution distribution NOUN iajs-3349 206 13 to to ADP iajs-3349 206 14 a a DET iajs-3349 206 15 real real ADJ iajs-3349 206 16 data datum NOUN iajs-3349 206 17 set set VERB iajs-3349 206 18 to to PART iajs-3349 206 19 show show VERB iajs-3349 206 20 that that SCONJ iajs-3349 206 21 the the DET iajs-3349 206 22 proposed propose VERB iajs-3349 206 23 distribution distribution NOUN iajs-3349 206 24 fits fit VERB iajs-3349 206 25 well well ADV iajs-3349 206 26 compared compare VERB iajs-3349 206 27 to to ADP iajs-3349 206 28 competing compete VERB iajs-3349 206 29 distributions distribution NOUN iajs-3349 206 30 . . PUNCT iajs-3349 207 1 the the DET iajs-3349 207 2 statistical statistical ADJ iajs-3349 207 3 software software NOUN iajs-3349 207 4 r r NOUN iajs-3349 207 5 is be AUX iajs-3349 207 6 used use VERB iajs-3349 207 7 to to PART iajs-3349 207 8 calculate calculate VERB iajs-3349 207 9 all all DET iajs-3349 207 10 results result NOUN iajs-3349 207 11 . . PUNCT iajs-3349 208 1 to to PART iajs-3349 208 2 obtain obtain VERB iajs-3349 208 3 the the DET iajs-3349 208 4 best good ADJ iajs-3349 208 5 results result NOUN iajs-3349 208 6 , , PUNCT iajs-3349 208 7 we we PRON iajs-3349 208 8 used use VERB iajs-3349 208 9 the the DET iajs-3349 208 10 following following ADJ iajs-3349 208 11 statistical statistical ADJ iajs-3349 208 12 criteria criterion NOUN iajs-3349 208 13 ( ( PUNCT iajs-3349 208 14 -𝑙 -𝑙 PROPN iajs-3349 208 15 , , PUNCT iajs-3349 208 16 aic aic PROPN iajs-3349 208 17 , , PUNCT iajs-3349 208 18 aic aic PROPN iajs-3349 208 19 , , PUNCT iajs-3349 208 20 bic bic PROPN iajs-3349 208 21 , , PUNCT iajs-3349 208 22 hqic hqic NOUN iajs-3349 208 23 ) ) PUNCT iajs-3349 208 24 for for ADP iajs-3349 208 25 the the DET iajs-3349 208 26 proposed propose VERB iajs-3349 208 27 model model NOUN iajs-3349 208 28 compared compare VERB iajs-3349 208 29 to to ADP iajs-3349 208 30 other other ADJ iajs-3349 208 31 models model NOUN iajs-3349 208 32 such such ADJ iajs-3349 208 33 as as ADP iajs-3349 208 34 beta beta NOUN iajs-3349 208 35 - - PUNCT iajs-3349 208 36 exponential exponential NOUN iajs-3349 208 37 ihjpas ihjpa NOUN iajs-3349 208 38 . . PUNCT iajs-3349 209 1 37 37 NUM iajs-3349 209 2 ( ( PUNCT iajs-3349 209 3 2 2 NUM iajs-3349 209 4 ) ) PUNCT iajs-3349 209 5 2024 2024 NUM iajs-3349 209 6 372 372 NUM iajs-3349 209 7 ( ( PUNCT iajs-3349 209 8 beex beex PROPN iajs-3349 209 9 ) ) PUNCT iajs-3349 209 10 , , PUNCT iajs-3349 209 11 kumaraswamy kumaraswamy NOUN iajs-3349 209 12 - - PUNCT iajs-3349 209 13 exponential exponential NOUN iajs-3349 209 14 ( ( PUNCT iajs-3349 209 15 kuex kuex PROPN iajs-3349 209 16 ) ) PUNCT iajs-3349 209 17 , , PUNCT iajs-3349 209 18 exponential exponential ADJ iajs-3349 209 19 generalized generalize VERB iajs-3349 209 20 exponential exponential NOUN iajs-3349 209 21 ( ( PUNCT iajs-3349 209 22 egex egex NOUN iajs-3349 209 23 ) ) PUNCT iajs-3349 209 24 , , PUNCT iajs-3349 209 25 weibull weibull NOUN iajs-3349 209 26 - - PUNCT iajs-3349 209 27 exponential exponential NOUN iajs-3349 209 28 ( ( PUNCT iajs-3349 209 29 weex weex NOUN iajs-3349 209 30 ) ) PUNCT iajs-3349 209 31 , , PUNCT iajs-3349 209 32 gompertez gompertez NOUN iajs-3349 209 33 - - PUNCT iajs-3349 209 34 exponential exponential NOUN iajs-3349 209 35 ( ( PUNCT iajs-3349 209 36 goex goex NOUN iajs-3349 209 37 ) ) PUNCT iajs-3349 209 38 , , PUNCT iajs-3349 209 39 marshal marshal NOUN iajs-3349 209 40 - - PUNCT iajs-3349 209 41 olkin olkin NOUN iajs-3349 209 42 - - PUNCT iajs-3349 209 43 ex ex NOUN iajs-3349 209 44 ( ( PUNCT iajs-3349 209 45 moex moex NOUN iajs-3349 209 46 ) ) PUNCT iajs-3349 209 47 and and CCONJ iajs-3349 209 48 exponential exponential ADJ iajs-3349 209 49 ( ( PUNCT iajs-3349 209 50 ex ex NOUN iajs-3349 209 51 ) ) PUNCT iajs-3349 209 52 . . PUNCT iajs-3349 210 1 this this DET iajs-3349 210 2 data datum NOUN iajs-3349 210 3 set set VERB iajs-3349 210 4 is be AUX iajs-3349 210 5 the the DET iajs-3349 210 6 use use NOUN iajs-3349 210 7 of of ADP iajs-3349 210 8 the the DET iajs-3349 210 9 failure failure NOUN iajs-3349 210 10 rate rate NOUN iajs-3349 210 11 data datum NOUN iajs-3349 210 12 set set VERB iajs-3349 210 13 ( ( PUNCT iajs-3349 210 14 103 103 NUM iajs-3349 210 15 hours hour NOUN iajs-3349 210 16 ) ) PUNCT iajs-3349 210 17 for for ADP iajs-3349 210 18 the the DET iajs-3349 210 19 turbocharger turbocharger NOUN iajs-3349 210 20 for for ADP iajs-3349 210 21 the the DET iajs-3349 210 22 engine engine NOUN iajs-3349 210 23 type type NOUN iajs-3349 210 24 . . PUNCT iajs-3349 211 1 for for ADP iajs-3349 211 2 the the DET iajs-3349 211 3 dataset dataset NOUN iajs-3349 211 4 , , PUNCT iajs-3349 211 5 we we PRON iajs-3349 211 6 consider consider VERB iajs-3349 211 7 the the DET iajs-3349 211 8 large large ADJ iajs-3349 211 9 recorded record VERB iajs-3349 211 10 intensities intensity NOUN iajs-3349 211 11 ( ( PUNCT iajs-3349 211 12 on on ADP iajs-3349 211 13 the the DET iajs-3349 211 14 richter richter NOUN iajs-3349 211 15 scale scale NOUN iajs-3349 211 16 ) ) PUNCT iajs-3349 211 17 of of ADP iajs-3349 211 18 earthquakes earthquake NOUN iajs-3349 211 19 at at ADP iajs-3349 211 20 seismometer seismometer NOUN iajs-3349 211 21 locations location NOUN iajs-3349 211 22 in in ADP iajs-3349 211 23 western western ADJ iajs-3349 211 24 north north PROPN iajs-3349 211 25 america america PROPN iajs-3349 211 26 between between ADP iajs-3349 211 27 1940 1940 NUM iajs-3349 211 28 and and CCONJ iajs-3349 211 29 1980 1980 NUM iajs-3349 211 30 , , PUNCT iajs-3349 211 31 as as ADP iajs-3349 211 32 in in ADP iajs-3349 211 33 [ [ X iajs-3349 211 34 29 29 NUM iajs-3349 211 35 ] ] PUNCT iajs-3349 211 36 , , PUNCT iajs-3349 211 37 [ [ X iajs-3349 211 38 30 30 NUM iajs-3349 211 39 ] ] PUNCT iajs-3349 211 40 , , PUNCT iajs-3349 211 41 [ [ X iajs-3349 211 42 31 31 NUM iajs-3349 211 43 ] ] PUNCT iajs-3349 211 44 . . PUNCT iajs-3349 212 1 7.5,8.8,8.9,9.4,9.7,9.7,10.5,10.5,12,12.2,12.8,14.6,14.9,17.6,23.9,25,2.9,3.2,7.6,17,8,10,10,8,1 7.5,8.8,8.9,9.4,9.7,9.7,10.5,10.5,12,12.2,12.8,14.6,14.9,17.6,23.9,25,2.9,3.2,7.6,17,8,10,10,8,1 PROPN iajs-3349 212 2 9,21,13,22,29,31,5.8,12,12.1,20.5,20.5,25.3,35.9,36.1,36.3,38.5,41.4,43.6,44.4,46.1,47.1,47.7 9,21,13,22,29,31,5.8,12,12.1,20.5,20.5,25.3,35.9,36.1,36.3,38.5,41.4,43.6,44.4,46.1,47.1,47.7 PROPN iajs-3349 212 3 , , PUNCT iajs-3349 212 4 49.2,53.1,4,10.1,11.1,17.7,22.5,26.5,29,30.9,37.8,48.3,62,50,16,62,1.2,1.6,9.1,3.7,5.3,7.4,17.9 49.2,53.1,4,10.1,11.1,17.7,22.5,26.5,29,30.9,37.8,48.3,62,50,16,62,1.2,1.6,9.1,3.7,5.3,7.4,17.9 NOUN iajs-3349 212 5 , , PUNCT iajs-3349 212 6 19.2,23.4,30,38.9,10.8,15.7,16.7,20.8,28.5,33.1,40.3,8,32,30,31,16.1,63.6,6.6,9.3,13,17.3,105 19.2,23.4,30,38.9,10.8,15.7,16.7,20.8,28.5,33.1,40.3,8,32,30,31,16.1,63.6,6.6,9.3,13,17.3,105 NUM iajs-3349 212 7 , , PUNCT iajs-3349 212 8 112,123,5,23.5,26,0.5,0.6,1.3,1.4,2.6,3.8,4,5.1,6.2,6.8,7.5,7.6,8.4,8.5,8.5,10.6,12.6,12.7,12.9,1 112,123,5,23.5,26,0.5,0.6,1.3,1.4,2.6,3.8,4,5.1,6.2,6.8,7.5,7.6,8.4,8.5,8.5,10.6,12.6,12.7,12.9,1 NUM iajs-3349 212 9 4,15,16,17.7,18,22,22,23,23.2,29,32,32.7,36,43.5,49,60,64,105,122,141,200,45,130,147,187,1 4,15,16,17.7,18,22,22,23,23.2,29,32,32.7,36,43.5,49,60,64,105,122,141,200,45,130,147,187,1 PROPN iajs-3349 212 10 97,203,211,17,19.6,20.2,21.1,21.9,66,87,23.4,24.6,25.7,28.6,37.4,46.7,56.9,60.7,61.4,62,64,82 97,203,211,17,19.6,20.2,21.1,21.9,66,87,23.4,24.6,25.7,28.6,37.4,46.7,56.9,60.7,61.4,62,64,82 NUM iajs-3349 212 11 , , PUNCT iajs-3349 212 12 88,91,12,24.2,148,42,85,107,109,156,224,293,359,370,25.4,32.9,92.2,45,145,300 88,91,12,24.2,148,42,85,107,109,156,224,293,359,370,25.4,32.9,92.2,45,145,300 PROPN iajs-3349 212 13 . . PUNCT iajs-3349 213 1 according accord VERB iajs-3349 213 2 to to ADP iajs-3349 213 3 the the DET iajs-3349 213 4 values value NOUN iajs-3349 213 5 shown show VERB iajs-3349 213 6 in in ADP iajs-3349 213 7 tables table NOUN iajs-3349 213 8 2 2 NUM iajs-3349 213 9 and and CCONJ iajs-3349 213 10 3 3 NUM iajs-3349 213 11 , , PUNCT iajs-3349 213 12 it it PRON iajs-3349 213 13 is be AUX iajs-3349 213 14 clear clear ADJ iajs-3349 213 15 that that SCONJ iajs-3349 213 16 the the DET iajs-3349 213 17 tnhe tnhe NOUN iajs-3349 213 18 distribution distribution NOUN iajs-3349 213 19 is be AUX iajs-3349 213 20 superior superior ADJ iajs-3349 213 21 to to ADP iajs-3349 213 22 the the DET iajs-3349 213 23 comparative comparative ADJ iajs-3349 213 24 distributions distribution NOUN iajs-3349 213 25 . . PUNCT iajs-3349 214 1 the the DET iajs-3349 214 2 proposed propose VERB iajs-3349 214 3 expanded expand VERB iajs-3349 214 4 distribution distribution NOUN iajs-3349 214 5 provides provide VERB iajs-3349 214 6 an an DET iajs-3349 214 7 accurate accurate ADJ iajs-3349 214 8 representation representation NOUN iajs-3349 214 9 because because SCONJ iajs-3349 214 10 it it PRON iajs-3349 214 11 has have VERB iajs-3349 214 12 the the DET iajs-3349 214 13 lowest low ADJ iajs-3349 214 14 values value NOUN iajs-3349 214 15 according accord VERB iajs-3349 214 16 to to ADP iajs-3349 214 17 the the DET iajs-3349 214 18 statistical statistical ADJ iajs-3349 214 19 and and CCONJ iajs-3349 214 20 informational informational ADJ iajs-3349 214 21 criteria criterion NOUN iajs-3349 214 22 and and CCONJ iajs-3349 214 23 the the DET iajs-3349 214 24 largest large ADJ iajs-3349 214 25 value value NOUN iajs-3349 214 26 of of ADP iajs-3349 214 27 the𝑝-value the𝑝-value NOUN iajs-3349 214 28 . . PUNCT iajs-3349 215 1 it it PRON iajs-3349 215 2 is be AUX iajs-3349 215 3 from from ADP iajs-3349 215 4 figures figure NOUN iajs-3349 215 5 1 1 NUM iajs-3349 215 6 and and CCONJ iajs-3349 215 7 2 2 NUM iajs-3349 215 8 , , PUNCT iajs-3349 215 9 the the DET iajs-3349 215 10 [ [ X iajs-3349 215 11 0,1 0,1 NUM iajs-3349 215 12 ] ] PUNCT iajs-3349 215 13 tnhe tnhe NOUN iajs-3349 215 14 model model NOUN iajs-3349 215 15 provides provide VERB iajs-3349 215 16 the the DET iajs-3349 215 17 overall overall ADJ iajs-3349 215 18 best well ADV iajs-3349 215 19 fit fit ADJ iajs-3349 215 20 and and CCONJ iajs-3349 215 21 therefore therefore ADV iajs-3349 215 22 could could AUX iajs-3349 215 23 be be AUX iajs-3349 215 24 chosen choose VERB iajs-3349 215 25 as as ADP iajs-3349 215 26 the the DET iajs-3349 215 27 adequate adequate ADJ iajs-3349 215 28 model model NOUN iajs-3349 215 29 for for ADP iajs-3349 215 30 explaining explain VERB iajs-3349 215 31 data datum NOUN iajs-3349 215 32 . . PUNCT iajs-3349 216 1 table table NOUN iajs-3349 216 2 1 1 NUM iajs-3349 216 3 . . PUNCT iajs-3349 216 4 bias bias NOUN iajs-3349 216 5 , , PUNCT iajs-3349 216 6 mse mse PROPN iajs-3349 216 7 and and CCONJ iajs-3349 216 8 mre mre NOUN iajs-3349 216 9 of of ADP iajs-3349 216 10 parameters parameter NOUN iajs-3349 216 11 of of ADP iajs-3349 216 12 [ [ X iajs-3349 216 13 0,1 0,1 NUM iajs-3349 216 14 ] ] PUNCT iajs-3349 216 15 tnhe tnhe VERB iajs-3349 216 16 distribution distribution NOUN iajs-3349 216 17 . . PUNCT iajs-3349 217 1 Ο„ Ο„ X iajs-3349 217 2 = = PUNCT iajs-3349 217 3 ( ( PUNCT iajs-3349 217 4 a a PRON iajs-3349 217 5 = = SYM iajs-3349 217 6 0.75 0.75 NUM iajs-3349 217 7 , , PUNCT iajs-3349 217 8 b b NOUN iajs-3349 217 9 = = SYM iajs-3349 217 10 1.75 1.75 NUM iajs-3349 217 11 , , PUNCT iajs-3349 217 12 Ξ» Ξ» X iajs-3349 217 13 = = PUNCT iajs-3349 217 14 3)t 3)t NUM iajs-3349 217 15 𝑛 𝑛 NOUN iajs-3349 217 16 = = SYM iajs-3349 217 17 200 200 NUM iajs-3349 217 18 𝑛 𝑛 NOUN iajs-3349 217 19 = = SYM iajs-3349 217 20 120 120 NUM iajs-3349 217 21 𝑛 𝑛 NOUN iajs-3349 217 22 = = SYM iajs-3349 217 23 80 80 NUM iajs-3349 217 24 𝑛 𝑛 NOUN iajs-3349 217 25 = = SYM iajs-3349 217 26 50 50 NUM iajs-3349 217 27 𝑛 𝑛 NOUN iajs-3349 217 28 = = SYM iajs-3349 217 29 30 30 NUM iajs-3349 217 30 est est X iajs-3349 217 31 . . PUNCT iajs-3349 218 1 par par NOUN iajs-3349 218 2 . . PUNCT iajs-3349 219 1 est est X iajs-3349 219 2 . . PUNCT iajs-3349 220 1 0.61788 0.61788 NUM iajs-3349 220 2 0.68144 0.68144 NUM iajs-3349 220 3 0.73878 0.73878 NUM iajs-3349 220 4 0.80613 0.80613 NUM iajs-3349 220 5 0.87476 0.87476 NUM iajs-3349 220 6 οΏ½ οΏ½ PROPN iajs-3349 220 7 Μ‚ Μ‚ SYM iajs-3349 220 8 οΏ½ οΏ½ PROPN iajs-3349 220 9 |bias| |bias| NUM iajs-3349 220 10 0.92474 0.92474 NUM iajs-3349 220 11 1.17777 1.17777 NUM iajs-3349 220 12 1.38657 1.38657 NUM iajs-3349 220 13 1.66894 1.66894 NUM iajs-3349 220 14 1.96507 1.96507 NUM iajs-3349 220 15 οΏ½ οΏ½ NOUN iajs-3349 220 16 Μ‚ Μ‚ SYM iajs-3349 220 17 οΏ½ οΏ½ NOUN iajs-3349 220 18 0.55274 0.55274 NUM iajs-3349 220 19 0.67574 0.67574 NUM iajs-3349 220 20 0.78722 0.78722 NUM iajs-3349 220 21 0.90548 0.90548 NUM iajs-3349 220 22 1.08240 1.08240 NUM iajs-3349 220 23 πœ† πœ† NOUN iajs-3349 220 24 Μ‚ Μ‚ PUNCT iajs-3349 220 25 0.56899 0.56899 NUM iajs-3349 220 26 0.71129 0.71129 NUM iajs-3349 221 1 0.87907 0.87907 NUM iajs-3349 221 2 1.02999 1.02999 NUM iajs-3349 221 3 1.23050 1.23050 NUM iajs-3349 221 4 οΏ½ οΏ½ SYM iajs-3349 221 5 Μ‚ Μ‚ SYM iajs-3349 221 6 οΏ½ οΏ½ PROPN iajs-3349 221 7 mse mse NOUN iajs-3349 221 8 1.48122 1.48122 NUM iajs-3349 221 9 2.61345 2.61345 NUM iajs-3349 221 10 3.67168 3.67168 NUM iajs-3349 221 11 5.59836 5.59836 NUM iajs-3349 221 12 7.99706 7.99706 NUM iajs-3349 221 13 οΏ½ οΏ½ SYM iajs-3349 221 14 Μ‚ Μ‚ NOUN iajs-3349 221 15 οΏ½ οΏ½ NOUN iajs-3349 221 16 0.47828 0.47828 NUM iajs-3349 221 17 0.72033 0.72033 NUM iajs-3349 221 18 0.98659 0.98659 NUM iajs-3349 221 19 1.34383 1.34383 NUM iajs-3349 221 20 1.94217 1.94217 NUM iajs-3349 221 21 πœ† πœ† NOUN iajs-3349 221 22 Μ‚ Μ‚ SYM iajs-3349 221 23 0.82383 0.82383 NUM iajs-3349 221 24 0.90858 0.90858 NUM iajs-3349 221 25 0.98504 0.98504 NUM iajs-3349 221 26 1.07483 1.07483 NUM iajs-3349 221 27 1.16634 1.16634 NUM iajs-3349 221 28 οΏ½ οΏ½ PROPN iajs-3349 221 29 Μ‚ Μ‚ SYM iajs-3349 221 30 οΏ½ οΏ½ PROPN iajs-3349 221 31 mre mre NOUN iajs-3349 221 32 0.52842 0.52842 NUM iajs-3349 221 33 0.67301 0.67301 NUM iajs-3349 221 34 0.79233 0.79233 NUM iajs-3349 221 35 0.95368 0.95368 NUM iajs-3349 221 36 1.12290 1.12290 NUM iajs-3349 221 37 οΏ½ οΏ½ SYM iajs-3349 221 38 Μ‚ Μ‚ SYM iajs-3349 221 39 οΏ½ οΏ½ PROPN iajs-3349 221 40 0.18425 0.18425 NUM iajs-3349 221 41 0.22525 0.22525 NUM iajs-3349 221 42 0.26241 0.26241 NUM iajs-3349 221 43 0.30183 0.30183 NUM iajs-3349 221 44 0.36080 0.36080 NUM iajs-3349 221 45 πœ† πœ† ADP iajs-3349 221 46 Μ‚ Μ‚ PUNCT iajs-3349 221 47 𝜏 𝜏 X iajs-3349 221 48 = = PUNCT iajs-3349 221 49 ( ( PUNCT iajs-3349 221 50 a a PRON iajs-3349 221 51 = = SYM iajs-3349 221 52 1.5 1.5 NUM iajs-3349 221 53 , , PUNCT iajs-3349 221 54 b b NOUN iajs-3349 221 55 = = SYM iajs-3349 221 56 0.5 0.5 NUM iajs-3349 221 57 , , PUNCT iajs-3349 221 58 Ξ» Ξ» X iajs-3349 221 59 = = PUNCT iajs-3349 221 60 0.4)t 0.4)t X iajs-3349 221 61 𝑛 𝑛 NOUN iajs-3349 221 62 = = NUM iajs-3349 221 63 200 200 NUM iajs-3349 221 64 𝑛 𝑛 NOUN iajs-3349 221 65 = = SYM iajs-3349 221 66 120 120 NUM iajs-3349 221 67 𝑛 𝑛 NOUN iajs-3349 221 68 = = SYM iajs-3349 221 69 80 80 NUM iajs-3349 221 70 𝑛 𝑛 NOUN iajs-3349 221 71 = = SYM iajs-3349 221 72 50 50 NUM iajs-3349 221 73 𝑛 𝑛 NOUN iajs-3349 221 74 = = SYM iajs-3349 221 75 30 30 NUM iajs-3349 221 76 est est X iajs-3349 221 77 . . PUNCT iajs-3349 222 1 par par NOUN iajs-3349 222 2 . . PUNCT iajs-3349 223 1 est est X iajs-3349 223 2 . . PUNCT iajs-3349 224 1 0.60516 0.60516 NUM iajs-3349 224 2 0.62884 0.62884 NUM iajs-3349 224 3 0.64801 0.64801 NUM iajs-3349 224 4 0.66331 0.66331 NUM iajs-3349 224 5 0.69942 0.69942 NUM iajs-3349 224 6 οΏ½ οΏ½ SYM iajs-3349 224 7 Μ‚ Μ‚ SYM iajs-3349 224 8 οΏ½ οΏ½ PROPN iajs-3349 224 9 |bias| |bias| NUM iajs-3349 224 10 0.50648 0.50648 NUM iajs-3349 224 11 0.61350 0.61350 NUM iajs-3349 224 12 0.71024 0.71024 NUM iajs-3349 224 13 0.81820 0.81820 NUM iajs-3349 224 14 0.91634 0.91634 NUM iajs-3349 224 15 οΏ½ οΏ½ SYM iajs-3349 224 16 Μ‚ Μ‚ VERB iajs-3349 224 17 οΏ½ οΏ½ NOUN iajs-3349 224 18 0.06481 0.06481 NUM iajs-3349 224 19 0.07838 0.07838 NUM iajs-3349 224 20 0.08630 0.08630 NUM iajs-3349 224 21 0.09739 0.09739 NUM iajs-3349 224 22 0.11670 0.11670 NUM iajs-3349 224 23 πœ† πœ† X iajs-3349 224 24 Μ‚ Μ‚ PUNCT iajs-3349 224 25 0.74607 0.74607 NUM iajs-3349 224 26 0.79344 0.79344 NUM iajs-3349 224 27 0.82684 0.82684 NUM iajs-3349 224 28 0.8509 0.8509 NUM iajs-3349 224 29 0.93774 0.93774 NUM iajs-3349 224 30 οΏ½ οΏ½ NOUN iajs-3349 224 31 Μ‚ Μ‚ SYM iajs-3349 224 32 οΏ½ οΏ½ PROPN iajs-3349 224 33 mse mse NOUN iajs-3349 224 34 0.44504 0.44504 NUM iajs-3349 224 35 0.63727 0.63727 NUM iajs-3349 224 36 0.81895 0.81895 NUM iajs-3349 224 37 1.07283 1.07283 NUM iajs-3349 224 38 1.34474 1.34474 NUM iajs-3349 224 39 οΏ½ οΏ½ PROPN iajs-3349 224 40 Μ‚ Μ‚ SYM iajs-3349 224 41 οΏ½ οΏ½ NOUN iajs-3349 224 42 0.00685 0.00685 NUM iajs-3349 224 43 0.00962 0.00962 NUM iajs-3349 224 44 0.01156 0.01156 NUM iajs-3349 224 45 0.01483 0.01483 NUM iajs-3349 224 46 0.02236 0.02236 NUM iajs-3349 224 47 πœ† πœ† X iajs-3349 224 48 Μ‚ Μ‚ SYM iajs-3349 224 49 0.40344 0.40344 NUM iajs-3349 224 50 0.41923 0.41923 NUM iajs-3349 224 51 0.43201 0.43201 NUM iajs-3349 224 52 0.44220 0.44220 NUM iajs-3349 224 53 0.46628 0.46628 NUM iajs-3349 224 54 οΏ½ οΏ½ SYM iajs-3349 224 55 Μ‚ Μ‚ SYM iajs-3349 224 56 οΏ½ οΏ½ PROPN iajs-3349 224 57 mre mre PROPN iajs-3349 224 58 1.01296 1.01296 NUM iajs-3349 224 59 1.22700 1.22700 NUM iajs-3349 224 60 1.42047 1.42047 NUM iajs-3349 224 61 1.63640 1.63640 NUM iajs-3349 224 62 1.83267 1.83267 NUM iajs-3349 224 63 οΏ½ οΏ½ SYM iajs-3349 224 64 Μ‚ Μ‚ SYM iajs-3349 224 65 οΏ½ οΏ½ NOUN iajs-3349 224 66 0.16203 0.16203 NUM iajs-3349 224 67 0.19595 0.19595 NUM iajs-3349 224 68 0.21574 0.21574 NUM iajs-3349 224 69 0.24347 0.24347 NUM iajs-3349 224 70 0.29175 0.29175 NUM iajs-3349 224 71 πœ† πœ† X iajs-3349 224 72 Μ‚ Μ‚ PUNCT iajs-3349 224 73 𝜏 𝜏 X iajs-3349 224 74 = = PUNCT iajs-3349 224 75 ( ( PUNCT iajs-3349 224 76 a a DET iajs-3349 224 77 = = SYM iajs-3349 224 78 3 3 NUM iajs-3349 224 79 , , PUNCT iajs-3349 224 80 b b NOUN iajs-3349 224 81 = = SYM iajs-3349 224 82 3 3 NUM iajs-3349 224 83 , , PUNCT iajs-3349 224 84 Ξ» Ξ» X iajs-3349 224 85 = = NOUN iajs-3349 224 86 1.6)𝑇 1.6)𝑇 NUM iajs-3349 224 87 𝑛 𝑛 NOUN iajs-3349 225 1 = = SYM iajs-3349 225 2 200 200 NUM iajs-3349 225 3 𝑛 𝑛 NOUN iajs-3349 225 4 = = SYM iajs-3349 225 5 120 120 NUM iajs-3349 225 6 𝑛 𝑛 NOUN iajs-3349 225 7 = = SYM iajs-3349 225 8 80 80 NUM iajs-3349 225 9 𝑛 𝑛 NOUN iajs-3349 225 10 = = SYM iajs-3349 225 11 50 50 NUM iajs-3349 225 12 𝑛 𝑛 NOUN iajs-3349 225 13 = = SYM iajs-3349 225 14 30 30 NUM iajs-3349 225 15 est est X iajs-3349 225 16 . . PUNCT iajs-3349 226 1 par par NOUN iajs-3349 226 2 . . PUNCT iajs-3349 227 1 est est X iajs-3349 227 2 . . PUNCT iajs-3349 228 1 0.74159 0.74159 NUM iajs-3349 228 2 0.94522 0.94522 NUM iajs-3349 228 3 1.11533 1.11533 NUM iajs-3349 228 4 1.31950 1.31950 NUM iajs-3349 228 5 1.62784 1.62784 NUM iajs-3349 228 6 οΏ½ οΏ½ NOUN iajs-3349 228 7 Μ‚ Μ‚ SYM iajs-3349 228 8 οΏ½ οΏ½ PROPN iajs-3349 228 9 |bias| |bias| NUM iajs-3349 228 10 0.43437 0.43437 NUM iajs-3349 228 11 0.48626 0.48626 NUM iajs-3349 228 12 0.54433 0.54433 NUM iajs-3349 228 13 0.60525 0.60525 NUM iajs-3349 228 14 0.67291 0.67291 NUM iajs-3349 228 15 οΏ½ οΏ½ SYM iajs-3349 228 16 Μ‚ Μ‚ SYM iajs-3349 228 17 οΏ½ οΏ½ NOUN iajs-3349 228 18 0.38669 0.38669 NUM iajs-3349 228 19 0.46325 0.46325 NUM iajs-3349 228 20 0.53197 0.53197 NUM iajs-3349 228 21 0.62328 0.62328 NUM iajs-3349 228 22 0.74430 0.74430 NUM iajs-3349 228 23 πœ† πœ† X iajs-3349 228 24 Μ‚ Μ‚ PUNCT iajs-3349 228 25 0.93697 0.93697 NUM iajs-3349 228 26 1.51047 1.51047 NUM iajs-3349 228 27 2.05816 2.05816 NUM iajs-3349 228 28 2.85928 2.85928 NUM iajs-3349 228 29 4.13035 4.13035 NUM iajs-3349 228 30 οΏ½ οΏ½ SYM iajs-3349 228 31 Μ‚ Μ‚ SYM iajs-3349 228 32 οΏ½ οΏ½ NOUN iajs-3349 228 33 mse mse NOUN iajs-3349 229 1 0.31474 0.31474 NUM iajs-3349 229 2 0.40259 0.40259 NUM iajs-3349 229 3 0.50889 0.50889 NUM iajs-3349 229 4 0.64676 0.64676 NUM iajs-3349 229 5 0.80849 0.80849 NUM iajs-3349 229 6 οΏ½ οΏ½ SYM iajs-3349 229 7 Μ‚ Μ‚ SYM iajs-3349 229 8 οΏ½ οΏ½ NOUN iajs-3349 229 9 0.22417 0.22417 NUM iajs-3349 229 10 0.32020 0.32020 NUM iajs-3349 229 11 0.42608 0.42608 NUM iajs-3349 229 12 0.63751 0.63751 NUM iajs-3349 229 13 1.0537 1.0537 NUM iajs-3349 229 14 πœ† πœ† PRON iajs-3349 229 15 Μ‚ Μ‚ VERB iajs-3349 229 16 ihjpas ihjpa NOUN iajs-3349 229 17 . . PUNCT iajs-3349 230 1 37 37 NUM iajs-3349 230 2 ( ( PUNCT iajs-3349 230 3 2 2 NUM iajs-3349 230 4 ) ) PUNCT iajs-3349 230 5 2024 2024 NUM iajs-3349 230 6 373 373 NUM iajs-3349 230 7 table table NOUN iajs-3349 230 8 2 2 NUM iajs-3349 230 9 . . PUNCT iajs-3349 231 1 the the DET iajs-3349 231 2 k k PROPN iajs-3349 231 3 - - PUNCT iajs-3349 231 4 s s NOUN iajs-3349 231 5 value value NOUN iajs-3349 231 6 with with ADP iajs-3349 231 7 its its PRON iajs-3349 231 8 corresponding correspond VERB iajs-3349 231 9 𝑝-value 𝑝-value NOUN iajs-3349 231 10 and and CCONJ iajs-3349 231 11 w w NOUN iajs-3349 231 12 value value NOUN iajs-3349 231 13 of of ADP iajs-3349 231 14 the the DET iajs-3349 231 15 data datum NOUN iajs-3349 231 16 set set VERB iajs-3349 231 17 value value NOUN iajs-3349 231 18 with with ADP iajs-3349 231 19 icorrue icorrue NOUN iajs-3349 231 20 of of ADP iajs-3349 231 21 the the DET iajs-3349 231 22 table table NOUN iajs-3349 231 23 3 3 NUM iajs-3349 231 24 . . PUNCT iajs-3349 231 25 represented represent VERB iajs-3349 231 26 the the DET iajs-3349 231 27 values value NOUN iajs-3349 231 28 of of ADP iajs-3349 231 29 statistically statistically ADJ iajs-3349 231 30 criteria criterion NOUN iajs-3349 231 31 ( ( PUNCT iajs-3349 231 32 -ll -ll ADP iajs-3349 231 33 , , PUNCT iajs-3349 231 34 aic aic PROPN iajs-3349 231 35 , , PUNCT iajs-3349 231 36 caic caic PROPN iajs-3349 231 37 , , PUNCT iajs-3349 231 38 bic bic PROPN iajs-3349 231 39 , , PUNCT iajs-3349 231 40 hqic hqic NOUN iajs-3349 231 41 ) ) PUNCT iajs-3349 231 42 . . PUNCT iajs-3349 232 1 0.24720 0.24720 NUM iajs-3349 232 2 0.31507 0.31507 NUM iajs-3349 233 1 0.37178 0.37178 NUM iajs-3349 233 2 0.43983 0.43983 NUM iajs-3349 233 3 0.54261 0.54261 NUM iajs-3349 233 4 οΏ½ οΏ½ SYM iajs-3349 233 5 Μ‚ Μ‚ SYM iajs-3349 233 6 οΏ½ οΏ½ NOUN iajs-3349 233 7 mre mre NOUN iajs-3349 233 8 0.14479 0.14479 NUM iajs-3349 234 1 0.16209 0.16209 NUM iajs-3349 234 2 0.18144 0.18144 NUM iajs-3349 234 3 0.20175 0.20175 NUM iajs-3349 234 4 0.22430 0.22430 NUM iajs-3349 234 5 οΏ½ οΏ½ SYM iajs-3349 234 6 Μ‚ Μ‚ SYM iajs-3349 234 7 οΏ½ οΏ½ NOUN iajs-3349 234 8 0.24168 0.24168 NUM iajs-3349 234 9 0.28953 0.28953 NUM iajs-3349 234 10 0.33248 0.33248 NUM iajs-3349 234 11 0.38955 0.38955 NUM iajs-3349 234 12 0.46519 0.46519 NUM iajs-3349 234 13 πœ† πœ† DET iajs-3349 234 14 Μ‚ Μ‚ PUNCT iajs-3349 234 15 model model NOUN iajs-3349 234 16 w w ADP iajs-3349 234 17 a a DET iajs-3349 234 18 k k PROPN iajs-3349 234 19 - - PUNCT iajs-3349 234 20 s s X iajs-3349 234 21 𝒑-value 𝒑-value NOUN iajs-3349 234 22 [ [ X iajs-3349 234 23 0,1]tnh 0,1]tnh X iajs-3349 234 24 - - ADJ iajs-3349 234 25 expo expo ADJ iajs-3349 234 26 0.2156 0.2156 NUM iajs-3349 234 27 1.2964 1.2964 NUM iajs-3349 234 28 0.08246 0.08246 NUM iajs-3349 234 29 0.1682 0.1682 NUM iajs-3349 234 30 beex beex NOUN iajs-3349 234 31 0.5323 0.5323 NUM iajs-3349 234 32 3.0980 3.0980 NUM iajs-3349 234 33 0.1149 0.1149 NUM iajs-3349 234 34 0.0162 0.0162 NUM iajs-3349 234 35 kuex kuex NOUN iajs-3349 234 36 0.5134 0.5134 NUM iajs-3349 234 37 2.9906 2.9906 NUM iajs-3349 234 38 0.1068 0.1068 NUM iajs-3349 234 39 0.0313 0.0313 NUM iajs-3349 234 40 egex egex NOUN iajs-3349 234 41 0.53635 0.53635 NUM iajs-3349 234 42 3.1207 3.1207 NUM iajs-3349 234 43 0.1169 0.1169 NUM iajs-3349 234 44 0.0138 0.0138 NUM iajs-3349 234 45 weex weex NOUN iajs-3349 234 46 0.2172 0.2172 NUM iajs-3349 234 47 1.3341 1.3341 NUM iajs-3349 234 48 0.0908 0.0908 NUM iajs-3349 234 49 0.0994 0.0994 NUM iajs-3349 234 50 goex goex NOUN iajs-3349 234 51 0.2172 0.2172 NUM iajs-3349 234 52 1.3341 1.3341 NUM iajs-3349 234 53 0.0773 0.0773 NUM iajs-3349 234 54 0.2259 0.2259 NUM iajs-3349 234 55 moex moex NOUN iajs-3349 234 56 0.5079 0.5079 NOUN iajs-3349 234 57 2.9607 2.9607 NUM iajs-3349 234 58 0.1218 0.1218 NUM iajs-3349 234 59 0.0090 0.0090 NUM iajs-3349 234 60 ex ex NOUN iajs-3349 234 61 0.5289 0.5289 NUM iajs-3349 234 62 3.0797 3.0797 NUM iajs-3349 234 63 0.1267 0.1267 NUM iajs-3349 234 64 0.0057 0.0057 NUM iajs-3349 234 65 model model NOUN iajs-3349 234 66 mles mle NOUN iajs-3349 234 67 𝒍 𝒍 PROPN iajs-3349 234 68 aic aic PROPN iajs-3349 234 69 caic caic PROPN iajs-3349 234 70 bic bic PROPN iajs-3349 234 71 hqic hqic PROPN iajs-3349 235 1 [ [ X iajs-3349 235 2 0,1 0,1 NUM iajs-3349 235 3 ] ] PUNCT iajs-3349 235 4 tnhe tnhe VERB iajs-3349 235 5 οΏ½ οΏ½ PROPN iajs-3349 235 6 Μ‚ Μ‚ NUM iajs-3349 235 7 οΏ½ οΏ½ NOUN iajs-3349 235 8 = = SYM iajs-3349 235 9 1.154 1.154 NUM iajs-3349 235 10 οΏ½ οΏ½ PROPN iajs-3349 235 11 Μ‚ Μ‚ SYM iajs-3349 235 12 οΏ½ οΏ½ NOUN iajs-3349 235 13 = = PROPN iajs-3349 235 14 2.388 2.388 NUM iajs-3349 235 15 οΏ½ οΏ½ PROPN iajs-3349 235 16 Μ‚ Μ‚ NUM iajs-3349 235 17 οΏ½ οΏ½ NOUN iajs-3349 235 18 = = NOUN iajs-3349 235 19 0.010 0.010 NUM iajs-3349 235 20 868.143 868.143 NUM iajs-3349 235 21 1742.286 1742.286 NUM iajs-3349 235 22 1742.421 1742.421 NUM iajs-3349 235 23 1751.898 1751.898 NUM iajs-3349 235 24 1746.183 1746.183 NUM iajs-3349 235 25 bee bee PROPN iajs-3349 235 26 οΏ½ οΏ½ PROPN iajs-3349 235 27 Μ‚ Μ‚ VERB iajs-3349 235 28 οΏ½ οΏ½ NOUN iajs-3349 235 29 = = SYM iajs-3349 235 30 0.903 0.903 NUM iajs-3349 235 31 οΏ½ οΏ½ PROPN iajs-3349 235 32 Μ‚ Μ‚ VERB iajs-3349 235 33 οΏ½ οΏ½ NOUN iajs-3349 235 34 = = SYM iajs-3349 235 35 1.482 1.482 NUM iajs-3349 235 36 οΏ½ οΏ½ PROPN iajs-3349 235 37 Μ‚ Μ‚ SYM iajs-3349 235 38 οΏ½ οΏ½ NOUN iajs-3349 235 39 = = PUNCT iajs-3349 235 40 0.013 0.013 NUM iajs-3349 235 41 876.633 876.633 NUM iajs-3349 235 42 1759.266 1759.266 NUM iajs-3349 235 43 1759.401 1759.401 NUM iajs-3349 235 44 1768.878 1768.878 NUM iajs-3349 235 45 1763.162 1763.162 NUM iajs-3349 235 46 kue kue PROPN iajs-3349 235 47 οΏ½ οΏ½ PROPN iajs-3349 235 48 Μ‚ Μ‚ VERB iajs-3349 235 49 οΏ½ οΏ½ PROPN iajs-3349 235 50 = = SYM iajs-3349 235 51 0.892 0.892 NUM iajs-3349 235 52 οΏ½ οΏ½ PROPN iajs-3349 235 53 Μ‚ Μ‚ VERB iajs-3349 235 54 οΏ½ οΏ½ NOUN iajs-3349 235 55 = = SYM iajs-3349 235 56 2.327 2.327 NUM iajs-3349 235 57 οΏ½ οΏ½ PROPN iajs-3349 235 58 Μ‚ Μ‚ NUM iajs-3349 235 59 οΏ½ οΏ½ NOUN iajs-3349 235 60 = = SYM iajs-3349 235 61 0.008 0.008 NUM iajs-3349 235 62 876.051 876.051 NUM iajs-3349 235 63 1758.103 1758.103 NUM iajs-3349 235 64 1758.238 1758.238 NUM iajs-3349 235 65 1767.715 1767.715 NUM iajs-3349 235 66 1762 1762 NUM iajs-3349 235 67 ege ege PROPN iajs-3349 235 68 οΏ½ οΏ½ PROPN iajs-3349 235 69 Μ‚ Μ‚ VERB iajs-3349 235 70 οΏ½ οΏ½ NOUN iajs-3349 235 71 = = SYM iajs-3349 235 72 1.667 1.667 NUM iajs-3349 235 73 οΏ½ οΏ½ PROPN iajs-3349 235 74 Μ‚ Μ‚ NUM iajs-3349 235 75 οΏ½ οΏ½ NOUN iajs-3349 235 76 = = PUNCT iajs-3349 235 77 0.911 0.911 NUM iajs-3349 235 78 οΏ½ οΏ½ PROPN iajs-3349 235 79 Μ‚ Μ‚ VERB iajs-3349 235 80 οΏ½ οΏ½ NOUN iajs-3349 235 81 = = SYM iajs-3349 236 1 0.012 0.012 NUM iajs-3349 236 2 876.782 876.782 NUM iajs-3349 236 3 1759.565 1759.565 NUM iajs-3349 236 4 1759.7 1759.7 NUM iajs-3349 236 5 1769.177 1769.177 NUM iajs-3349 236 6 1763.462 1763.462 NUM iajs-3349 236 7 wee wee ADJ iajs-3349 236 8 οΏ½ οΏ½ PROPN iajs-3349 236 9 Μ‚ Μ‚ NOUN iajs-3349 236 10 οΏ½ οΏ½ NOUN iajs-3349 236 11 = = SYM iajs-3349 236 12 0.883 0.883 NUM iajs-3349 236 13 οΏ½ οΏ½ PROPN iajs-3349 236 14 Μ‚ Μ‚ VERB iajs-3349 236 15 οΏ½ οΏ½ NOUN iajs-3349 236 16 = = SYM iajs-3349 236 17 0.259 0.259 NUM iajs-3349 236 18 οΏ½ οΏ½ PROPN iajs-3349 236 19 Μ‚ Μ‚ SYM iajs-3349 236 20 οΏ½ οΏ½ NOUN iajs-3349 236 21 = = NOUN iajs-3349 236 22 0.006 0.006 NUM iajs-3349 236 23 874.402 874.402 NUM iajs-3349 237 1 1745.208 1745.208 NUM iajs-3349 237 2 1745.343 1745.343 PROPN iajs-3349 237 3 1754.82 1754.82 NUM iajs-3349 237 4 1749.105 1749.105 PROPN iajs-3349 237 5 goe goe NOUN iajs-3349 237 6 οΏ½ οΏ½ NOUN iajs-3349 237 7 Μ‚ Μ‚ NOUN iajs-3349 237 8 οΏ½ οΏ½ NOUN iajs-3349 237 9 = = SYM iajs-3349 237 10 1.006 1.006 NUM iajs-3349 237 11 οΏ½ οΏ½ PROPN iajs-3349 237 12 Μ‚ Μ‚ SYM iajs-3349 237 13 οΏ½ οΏ½ NOUN iajs-3349 237 14 = = SYM iajs-3349 237 15 0.171 0.171 NUM iajs-3349 237 16 οΏ½ οΏ½ PROPN iajs-3349 237 17 Μ‚ Μ‚ NUM iajs-3349 237 18 οΏ½ οΏ½ NOUN iajs-3349 237 19 = = SYM iajs-3349 237 20 0.028 0.028 NUM iajs-3349 237 21 869.604 869.604 NUM iajs-3349 237 22 1745.208 1745.208 NUM iajs-3349 237 23 1745.343 1745.343 PROPN iajs-3349 237 24 1754.82 1754.82 NUM iajs-3349 237 25 1749.105 1749.105 NUM iajs-3349 237 26 moe moe PROPN iajs-3349 237 27 οΏ½ οΏ½ PROPN iajs-3349 237 28 Μ‚ Μ‚ NOUN iajs-3349 237 29 οΏ½ οΏ½ NOUN iajs-3349 237 30 = = PUNCT iajs-3349 237 31 0.944 0.944 NUM iajs-3349 237 32 οΏ½ οΏ½ PROPN iajs-3349 237 33 Μ‚ Μ‚ SYM iajs-3349 237 34 οΏ½ οΏ½ NOUN iajs-3349 237 35 = = SYM iajs-3349 237 36 0.021 0.021 NUM iajs-3349 237 37 876.340 876.340 NUM iajs-3349 237 38 1756.683 1756.683 NUM iajs-3349 237 39 1756.75 1756.75 NUM iajs-3349 237 40 1763.091 1763.091 NUM iajs-3349 237 41 1759.281 1759.281 NUM iajs-3349 237 42 e e X iajs-3349 237 43 οΏ½ οΏ½ PROPN iajs-3349 237 44 Μ‚ Μ‚ VERB iajs-3349 237 45 οΏ½ οΏ½ NOUN iajs-3349 237 46 = = SYM iajs-3349 237 47 0.021 0.021 NUM iajs-3349 237 48 877.236 877.236 NUM iajs-3349 237 49 1756.473 1756.473 NUM iajs-3349 237 50 1756.495 1756.495 NUM iajs-3349 237 51 1759.677 1759.677 NUM iajs-3349 237 52 1757.772 1757.772 NUM iajs-3349 237 53 ihjpas ihjpa NOUN iajs-3349 237 54 . . PUNCT iajs-3349 238 1 37 37 NUM iajs-3349 238 2 ( ( PUNCT iajs-3349 238 3 2 2 NUM iajs-3349 238 4 ) ) PUNCT iajs-3349 238 5 2024 2024 NUM iajs-3349 238 6 374 374 NUM iajs-3349 238 7 figure figure NOUN iajs-3349 238 8 1 1 NUM iajs-3349 238 9 . . PUNCT iajs-3349 238 10 estimated estimate VERB iajs-3349 238 11 fitted fit VERB iajs-3349 238 12 densities density NOUN iajs-3349 238 13 of of ADP iajs-3349 238 14 model model NOUN iajs-3349 238 15 for for ADP iajs-3349 238 16 dataset dataset NOUN iajs-3349 238 17 . . PUNCT iajs-3349 239 1 figure figure NOUN iajs-3349 239 2 2 2 NUM iajs-3349 239 3 . . PUNCT iajs-3349 239 4 estimated estimate VERB iajs-3349 239 5 fitted fit VERB iajs-3349 239 6 cdf cdf PROPN iajs-3349 239 7 for for ADP iajs-3349 239 8 data datum NOUN iajs-3349 239 9 set set VERB iajs-3349 239 10 . . PUNCT iajs-3349 240 1 ihjpas ihjpas PROPN iajs-3349 240 2 . . PUNCT iajs-3349 241 1 37 37 NUM iajs-3349 241 2 ( ( PUNCT iajs-3349 241 3 2 2 NUM iajs-3349 241 4 ) ) PUNCT iajs-3349 241 5 2024 2024 NUM iajs-3349 241 6 375 375 NUM iajs-3349 241 7 figure figure NOUN iajs-3349 241 8 3 3 NUM iajs-3349 241 9 . . X iajs-3349 241 10 ttt ttt PROPN iajs-3349 241 11 plot plot NOUN iajs-3349 241 12 of of ADP iajs-3349 241 13 [ [ X iajs-3349 241 14 0,1 0,1 NUM iajs-3349 241 15 ] ] PUNCT iajs-3349 241 16 tnhe tnhe VERB iajs-3349 241 17 distribution distribution NOUN iajs-3349 241 18 for for ADP iajs-3349 241 19 data datum NOUN iajs-3349 241 20 set set VERB iajs-3349 241 21 . . PUNCT iajs-3349 242 1 5 5 X iajs-3349 242 2 . . X iajs-3349 242 3 conclusion conclusion NOUN iajs-3349 242 4 this this DET iajs-3349 242 5 paper paper NOUN iajs-3349 242 6 proposes propose VERB iajs-3349 242 7 a a DET iajs-3349 242 8 new new ADJ iajs-3349 242 9 extension extension NOUN iajs-3349 242 10 of of ADP iajs-3349 242 11 the the DET iajs-3349 242 12 exponential exponential ADJ iajs-3349 242 13 distribution distribution NOUN iajs-3349 242 14 based base VERB iajs-3349 242 15 on on ADP iajs-3349 242 16 the the DET iajs-3349 242 17 [ [ NOUN iajs-3349 242 18 0,1 0,1 NUM iajs-3349 242 19 ] ] PUNCT iajs-3349 242 20 truncated truncated ADJ iajs-3349 242 21 nadarajah nadarajah PROPN iajs-3349 242 22 - - PUNCT iajs-3349 242 23 haghighi haghighi PROPN iajs-3349 242 24 - - PUNCT iajs-3349 242 25 g g PROPN iajs-3349 242 26 family family NOUN iajs-3349 242 27 of of ADP iajs-3349 242 28 distributions distribution NOUN iajs-3349 242 29 called call VERB iajs-3349 243 1 [ [ X iajs-3349 243 2 0,1 0,1 NUM iajs-3349 243 3 ] ] PUNCT iajs-3349 243 4 truncated truncate VERB iajs-3349 243 5 nadarajah nadarajah PROPN iajs-3349 243 6 - - PUNCT iajs-3349 243 7 haghighi haghighi PROPN iajs-3349 243 8 . . PUNCT iajs-3349 244 1 the the DET iajs-3349 244 2 exponential exponential ADJ iajs-3349 244 3 distribution distribution NOUN iajs-3349 244 4 , , PUNCT iajs-3349 244 5 which which PRON iajs-3349 244 6 is be AUX iajs-3349 244 7 a a DET iajs-3349 244 8 new new ADJ iajs-3349 244 9 distribution distribution NOUN iajs-3349 244 10 with with ADP iajs-3349 244 11 three three NUM iajs-3349 244 12 parameters parameter NOUN iajs-3349 244 13 , , PUNCT iajs-3349 244 14 is be AUX iajs-3349 244 15 more more ADV iajs-3349 244 16 flexible flexible ADJ iajs-3349 244 17 than than ADP iajs-3349 244 18 some some DET iajs-3349 244 19 other other ADJ iajs-3349 244 20 distributions distribution NOUN iajs-3349 244 21 , , PUNCT iajs-3349 244 22 such such ADJ iajs-3349 244 23 as as ADP iajs-3349 244 24 the the DET iajs-3349 244 25 exponential exponential ADJ iajs-3349 244 26 distribution distribution NOUN iajs-3349 244 27 and and CCONJ iajs-3349 244 28 the the DET iajs-3349 244 29 weibull weibull NOUN iajs-3349 244 30 exponential exponential ADJ iajs-3349 244 31 distribution distribution NOUN iajs-3349 244 32 . . PUNCT iajs-3349 245 1 also also ADV iajs-3349 245 2 , , PUNCT iajs-3349 245 3 we we PRON iajs-3349 245 4 derive derive VERB iajs-3349 245 5 some some DET iajs-3349 245 6 statistical statistical ADJ iajs-3349 245 7 properties property NOUN iajs-3349 245 8 for for ADP iajs-3349 245 9 the the DET iajs-3349 245 10 new new ADJ iajs-3349 245 11 distribution distribution NOUN iajs-3349 245 12 , , PUNCT iajs-3349 245 13 such such ADJ iajs-3349 245 14 as as ADP iajs-3349 245 15 the the DET iajs-3349 245 16 quantile quantile ADJ iajs-3349 245 17 function function NOUN iajs-3349 245 18 , , PUNCT iajs-3349 245 19 moments moment NOUN iajs-3349 245 20 , , PUNCT iajs-3349 245 21 incomplete incomplete ADJ iajs-3349 245 22 moments moment NOUN iajs-3349 245 23 and and CCONJ iajs-3349 245 24 entropy entropy NOUN iajs-3349 245 25 . . PUNCT iajs-3349 246 1 finally finally ADV iajs-3349 246 2 , , PUNCT iajs-3349 246 3 we we PRON iajs-3349 246 4 use use VERB iajs-3349 246 5 the the DET iajs-3349 246 6 maximum maximum ADJ iajs-3349 246 7 likelihood likelihood NOUN iajs-3349 246 8 method method NOUN iajs-3349 246 9 to to PART iajs-3349 246 10 estimate estimate VERB iajs-3349 246 11 the the DET iajs-3349 246 12 parameters parameter NOUN iajs-3349 246 13 for for ADP iajs-3349 246 14 the the DET iajs-3349 246 15 new new ADJ iajs-3349 246 16 distribution distribution NOUN iajs-3349 246 17 . . 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PUNCT iajs-3349 262 1 journal journal NOUN iajs-3349 262 2 of of ADP iajs-3349 262 3 statistical statistical ADJ iajs-3349 262 4 theory theory NOUN iajs-3349 262 5 and and CCONJ iajs-3349 262 6 practice practice NOUN iajs-3349 262 7 2017 2017 NUM iajs-3349 262 8 , , PUNCT iajs-3349 262 9 11 11 NUM iajs-3349 262 10 , , PUNCT iajs-3349 262 11 179 179 NUM iajs-3349 262 12 - - SYM iajs-3349 262 13 207 207 NUM iajs-3349 262 14 . . PUNCT iajs-3349 263 1 https://doi.org/10.1080/15598608.2016.1267668 https://doi.org/10.1080/15598608.2016.1267668 NOUN iajs-3349 264 1 6 6 X iajs-3349 264 2 . . X iajs-3349 264 3 afify afify NOUN iajs-3349 264 4 , , PUNCT iajs-3349 264 5 a. a. NOUN iajs-3349 264 6 z z PROPN iajs-3349 264 7 ; ; PUNCT iajs-3349 264 8 cordeiro cordeiro PROPN iajs-3349 264 9 , , PUNCT iajs-3349 264 10 g. g. PROPN iajs-3349 264 11 m m PROPN iajs-3349 264 12 ; ; PUNCT iajs-3349 264 13 maed maed NOUN iajs-3349 264 14 , , PUNCT iajs-3349 264 15 m. m. 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