id sid tid token lemma pos iajs-1885 1 1 microsoft microsoft PROPN iajs-1885 1 2 word word NOUN iajs-1885 1 3 101 101 NUM iajs-1885 1 4 - - SYM iajs-1885 1 5 109 109 NUM iajs-1885 1 6     SPACE iajs-1885 1 7 101 101 NUM iajs-1885 1 8     SPACE iajs-1885 1 9 ibn ibn PROPN iajs-1885 1 10 al al PROPN iajs-1885 1 11 - - PUNCT iajs-1885 1 12 haitham haitham PROPN iajs-1885 1 13 jour jour X iajs-1885 1 14 . . PROPN iajs-1885 2 1 for for ADP iajs-1885 2 2 pure pure ADJ iajs-1885 2 3 & & CCONJ iajs-1885 2 4 appl appl PROPN iajs-1885 2 5 . . PUNCT iajs-1885 3 1 sci sci PROPN iajs-1885 3 2 . . PROPN iajs-1885 4 1 ihjpas ihjpas PROPN iajs-1885 4 2 https://doi.org/10.30526/32.1.1885 https://doi.org/10.30526/32.1.1885 X iajs-1885 4 3 vol vol NOUN iajs-1885 4 4 . . PUNCT iajs-1885 5 1 32 32 NUM iajs-1885 5 2 ( ( PUNCT iajs-1885 5 3 1 1 NUM iajs-1885 5 4 ) ) PUNCT iajs-1885 5 5 2019 2019 NUM iajs-1885 5 6 the the DET iajs-1885 5 7 comparison comparison NOUN iajs-1885 5 8 between between ADP iajs-1885 5 9 the the DET iajs-1885 5 10 mle mle PROPN iajs-1885 5 11 and and CCONJ iajs-1885 5 12 standard standard ADJ iajs-1885 5 13 bayes bayes NOUN iajs-1885 5 14 estimators estimator NOUN iajs-1885 5 15 of of ADP iajs-1885 5 16 the the DET iajs-1885 5 17 reliability reliability NOUN iajs-1885 5 18 function function NOUN iajs-1885 5 19 of of ADP iajs-1885 5 20 exponential exponential ADJ iajs-1885 5 21 distribution distribution NOUN iajs-1885 5 22 mohammed mohammed PROPN iajs-1885 5 23 jamel jamel PROPN iajs-1885 5 24 ali ali PROPN iajs-1885 5 25 moham198999@gmail.com moham198999@gmail.com PROPN iajs-1885 5 26 hazim hazim PROPN iajs-1885 5 27 mansoor mansoor PROPN iajs-1885 5 28 gorgees gorgees PROPN iajs-1885 5 29 hazim5656@yahoo.com hazim5656@yahoo.com PROPN iajs-1885 5 30     SPACE iajs-1885 5 31 department department NOUN iajs-1885 5 32 of of ADP iajs-1885 5 33 mathematics mathematic NOUN iajs-1885 5 34 , , PUNCT iajs-1885 5 35 collage collage NOUN iajs-1885 5 36 of of ADP iajs-1885 5 37 education education NOUN iajs-1885 5 38 for for ADP iajs-1885 5 39 pure pure ADJ iajs-1885 5 40 sciences science NOUN iajs-1885 5 41 ibn ibn PROPN iajs-1885 5 42 al al PROPN iajs-1885 5 43 - - PUNCT iajs-1885 5 44 haitham haitham PROPN iajs-1885 5 45 , , PUNCT iajs-1885 5 46 university university PROPN iajs-1885 5 47 of of ADP iajs-1885 5 48 baghdad baghdad PROPN iajs-1885 5 49 , , PUNCT iajs-1885 5 50 baghdad baghdad PROPN iajs-1885 5 51 , , PUNCT iajs-1885 5 52 iraq iraq PROPN iajs-1885 5 53 . . PUNCT iajs-1885 6 1 article article NOUN iajs-1885 6 2 history history NOUN iajs-1885 6 3 : : PUNCT iajs-1885 6 4 received receive VERB iajs-1885 6 5 10 10 NUM iajs-1885 6 6 june june PROPN iajs-1885 6 7 2018 2018 NUM iajs-1885 6 8 , , PUNCT iajs-1885 6 9 accepted accept VERB iajs-1885 6 10 1 1 NUM iajs-1885 6 11 october october NOUN iajs-1885 6 12 2018 2018 NUM iajs-1885 6 13 , , PUNCT iajs-1885 6 14 publish publish VERB iajs-1885 6 15 january january PROPN iajs-1885 6 16 2019 2019 NUM iajs-1885 6 17 abstract abstract ADV iajs-1885 6 18 in in ADP iajs-1885 6 19 this this DET iajs-1885 6 20 paper paper NOUN iajs-1885 6 21 , , PUNCT iajs-1885 6 22 a a DET iajs-1885 6 23 monte monte PROPN iajs-1885 6 24 carlo carlo PROPN iajs-1885 6 25 simulation simulation NOUN iajs-1885 6 26 technique technique NOUN iajs-1885 6 27 is be AUX iajs-1885 6 28 used use VERB iajs-1885 6 29 to to PART iajs-1885 6 30 compare compare VERB iajs-1885 6 31 the the DET iajs-1885 6 32 performance performance NOUN iajs-1885 6 33 of of ADP iajs-1885 6 34 mle mle PROPN iajs-1885 6 35 and and CCONJ iajs-1885 6 36 the the DET iajs-1885 6 37 standard standard ADJ iajs-1885 6 38 bayes bayes NOUN iajs-1885 6 39 estimators estimator NOUN iajs-1885 6 40 of of ADP iajs-1885 6 41 the the DET iajs-1885 6 42 reliability reliability NOUN iajs-1885 6 43 function function NOUN iajs-1885 6 44 of of ADP iajs-1885 6 45 the the DET iajs-1885 6 46 one one NUM iajs-1885 6 47 parameter parameter NOUN iajs-1885 6 48 exponential exponential ADJ iajs-1885 6 49 distribution distribution NOUN iajs-1885 6 50 . . PUNCT iajs-1885 7 1 two two NUM iajs-1885 7 2 types type NOUN iajs-1885 7 3 of of ADP iajs-1885 7 4 loss loss NOUN iajs-1885 7 5 functions function NOUN iajs-1885 7 6 are be AUX iajs-1885 7 7 adopted adopt VERB iajs-1885 7 8 , , PUNCT iajs-1885 7 9 namely namely ADV iajs-1885 7 10 , , PUNCT iajs-1885 7 11 squared square VERB iajs-1885 7 12 error error NOUN iajs-1885 7 13 loss loss NOUN iajs-1885 7 14 function function NOUN iajs-1885 7 15 ( ( PUNCT iajs-1885 7 16 self self NOUN iajs-1885 7 17 ) ) PUNCT iajs-1885 7 18 and and CCONJ iajs-1885 7 19 modified modify VERB iajs-1885 7 20 square square ADJ iajs-1885 7 21 error error NOUN iajs-1885 7 22 loss loss NOUN iajs-1885 7 23 function function NOUN iajs-1885 7 24 ( ( PUNCT iajs-1885 7 25 mself mself NUM iajs-1885 7 26 ) ) PUNCT iajs-1885 7 27 with with ADP iajs-1885 7 28 informative informative ADJ iajs-1885 7 29 and and CCONJ iajs-1885 7 30 non non ADJ iajs-1885 7 31 informative informative ADJ iajs-1885 7 32 prior prior ADV iajs-1885 7 33 . . PUNCT iajs-1885 8 1 the the DET iajs-1885 8 2 criterion criterion NOUN iajs-1885 8 3 integrated integrate VERB iajs-1885 8 4 mean mean ADJ iajs-1885 8 5 square square NOUN iajs-1885 8 6 error error NOUN iajs-1885 8 7 ( ( PUNCT iajs-1885 8 8 imse imse NOUN iajs-1885 8 9 ) ) PUNCT iajs-1885 8 10 is be AUX iajs-1885 8 11 employed employ VERB iajs-1885 8 12 to to PART iajs-1885 8 13 assess assess VERB iajs-1885 8 14 the the DET iajs-1885 8 15 performance performance NOUN iajs-1885 8 16 of of ADP iajs-1885 8 17 such such ADJ iajs-1885 8 18 estimators estimator NOUN iajs-1885 8 19 . . PUNCT iajs-1885 9 1 keywords keyword NOUN iajs-1885 9 2 : : PUNCT iajs-1885 9 3 mle mle PROPN iajs-1885 9 4 , , PUNCT iajs-1885 9 5 standard standard ADJ iajs-1885 9 6 bayes bayes PROPN iajs-1885 9 7 estimator estimator NOUN iajs-1885 9 8 , , PUNCT iajs-1885 9 9 loss loss NOUN iajs-1885 9 10 function function NOUN iajs-1885 9 11 , , PUNCT iajs-1885 9 12 imse imse NOUN iajs-1885 9 13 . . PUNCT iajs-1885 10 1 1 1 X iajs-1885 10 2 . . X iajs-1885 10 3 introduction introduction NOUN iajs-1885 10 4 the the DET iajs-1885 10 5 reliability reliability NOUN iajs-1885 10 6 theory theory NOUN iajs-1885 10 7 is be AUX iajs-1885 10 8 related relate VERB iajs-1885 10 9 with with ADP iajs-1885 10 10 random random ADJ iajs-1885 10 11 happening happening NOUN iajs-1885 10 12 of of ADP iajs-1885 10 13 unwanted unwanted ADJ iajs-1885 10 14 events event NOUN iajs-1885 10 15 or or CCONJ iajs-1885 10 16 failure failure NOUN iajs-1885 10 17 during during ADP iajs-1885 10 18 the the DET iajs-1885 10 19 life life NOUN iajs-1885 10 20 of of ADP iajs-1885 10 21 a a DET iajs-1885 10 22 physical physical ADJ iajs-1885 10 23 or or CCONJ iajs-1885 10 24 biological biological ADJ iajs-1885 10 25 system system NOUN iajs-1885 11 1 [ [ X iajs-1885 11 2 1 1 NUM iajs-1885 11 3 ] ] PUNCT iajs-1885 11 4 . . PUNCT iajs-1885 12 1 reliability reliability NOUN iajs-1885 12 2 is be AUX iajs-1885 12 3 an an DET iajs-1885 12 4 essential essential ADJ iajs-1885 12 5 feature feature NOUN iajs-1885 12 6 of of ADP iajs-1885 12 7 a a DET iajs-1885 12 8 system system NOUN iajs-1885 12 9 . . PUNCT iajs-1885 13 1 basic basic ADJ iajs-1885 13 2 concepts concept NOUN iajs-1885 13 3 associated associate VERB iajs-1885 13 4 with with ADP iajs-1885 13 5 reliability reliability NOUN iajs-1885 13 6 has have AUX iajs-1885 13 7 been be AUX iajs-1885 13 8 known know VERB iajs-1885 13 9 for for ADP iajs-1885 13 10 a a DET iajs-1885 13 11 number number NOUN iajs-1885 13 12 of of ADP iajs-1885 13 13 years year NOUN iajs-1885 13 14 , , PUNCT iajs-1885 13 15 however however ADV iajs-1885 13 16 , , PUNCT iajs-1885 13 17 it it PRON iajs-1885 13 18 has have AUX iajs-1885 13 19 got get VERB iajs-1885 13 20 greatest great ADJ iajs-1885 13 21 significance significance NOUN iajs-1885 13 22 during during ADP iajs-1885 13 23 the the DET iajs-1885 13 24 past past ADJ iajs-1885 13 25 decade decade NOUN iajs-1885 13 26 as as ADP iajs-1885 13 27 a a DET iajs-1885 13 28 consequence consequence NOUN iajs-1885 13 29 of of ADP iajs-1885 13 30 the the DET iajs-1885 13 31 use use NOUN iajs-1885 13 32 of of ADP iajs-1885 13 33 highly highly ADV iajs-1885 13 34 complex complex ADJ iajs-1885 13 35 systems system NOUN iajs-1885 13 36 . . PUNCT iajs-1885 14 1 in in ADP iajs-1885 14 2 reliability reliability NOUN iajs-1885 14 3 theory theory NOUN iajs-1885 14 4 , , PUNCT iajs-1885 14 5 the the DET iajs-1885 14 6 exponential exponential ADJ iajs-1885 14 7 distribution distribution NOUN iajs-1885 14 8 plays play VERB iajs-1885 14 9 an an DET iajs-1885 14 10 important important ADJ iajs-1885 14 11 role role NOUN iajs-1885 14 12 in in ADP iajs-1885 14 13 life life NOUN iajs-1885 14 14 testing testing NOUN iajs-1885 14 15 experiments experiment NOUN iajs-1885 14 16 . . PUNCT iajs-1885 15 1 historically historically ADV iajs-1885 15 2 , , PUNCT iajs-1885 15 3 it it PRON iajs-1885 15 4 was be AUX iajs-1885 15 5 the the DET iajs-1885 15 6 first first ADJ iajs-1885 15 7 life life NOUN iajs-1885 15 8 time time NOUN iajs-1885 15 9 model model NOUN iajs-1885 15 10 for for ADP iajs-1885 15 11 which which PRON iajs-1885 15 12 statistical statistical ADJ iajs-1885 15 13 procedures procedure NOUN iajs-1885 15 14 were be AUX iajs-1885 15 15 widely widely ADV iajs-1885 15 16 developed develop VERB iajs-1885 15 17 . . PUNCT iajs-1885 16 1 many many ADJ iajs-1885 16 2 researchers researcher NOUN iajs-1885 16 3 gave give VERB iajs-1885 16 4 numerous numerous ADJ iajs-1885 16 5 results result NOUN iajs-1885 16 6 and and CCONJ iajs-1885 16 7 generalized generalize VERB iajs-1885 16 8 the the DET iajs-1885 16 9 exponential exponential ADJ iajs-1885 16 10 distribution distribution NOUN iajs-1885 16 11 as as ADP iajs-1885 16 12 a a DET iajs-1885 16 13 life life NOUN iajs-1885 16 14 time time NOUN iajs-1885 16 15 distribution distribution NOUN iajs-1885 16 16 , , PUNCT iajs-1885 16 17 particularly particularly ADV iajs-1885 16 18 , , PUNCT iajs-1885 16 19 in in ADP iajs-1885 16 20 the the DET iajs-1885 16 21 field field NOUN iajs-1885 16 22 of of ADP iajs-1885 16 23 industrial industrial ADJ iajs-1885 16 24 life life NOUN iajs-1885 16 25 testing testing NOUN iajs-1885 16 26 . . PUNCT iajs-1885 17 1 the the DET iajs-1885 17 2 exponential exponential ADJ iajs-1885 17 3 distribution distribution NOUN iajs-1885 17 4 is be AUX iajs-1885 17 5 desirable desirable ADJ iajs-1885 17 6 because because SCONJ iajs-1885 17 7 of of ADP iajs-1885 17 8 its its PRON iajs-1885 17 9 simplicity simplicity NOUN iajs-1885 17 10 and and CCONJ iajs-1885 17 11 its its PRON iajs-1885 17 12 own own ADJ iajs-1885 17 13 features feature NOUN iajs-1885 17 14 such such ADJ iajs-1885 17 15 as as ADP iajs-1885 17 16 lacks lack NOUN iajs-1885 17 17 memory memory NOUN iajs-1885 17 18 and and CCONJ iajs-1885 17 19 self self NOUN iajs-1885 17 20 - - PUNCT iajs-1885 17 21 producing produce VERB iajs-1885 17 22 property property NOUN iajs-1885 17 23 . . PUNCT iajs-1885 18 1 the the DET iajs-1885 18 2 probability probability NOUN iajs-1885 18 3 density density NOUN iajs-1885 18 4 , , PUNCT iajs-1885 18 5 cumulative cumulative ADJ iajs-1885 18 6 distribution distribution NOUN iajs-1885 18 7 and and CCONJ iajs-1885 18 8 reliability reliability NOUN iajs-1885 18 9 functions function NOUN iajs-1885 18 10 of of ADP iajs-1885 18 11 one one NUM iajs-1885 18 12 parameter parameter NOUN iajs-1885 18 13 exponential exponential ADJ iajs-1885 18 14 distribution distribution NOUN iajs-1885 18 15 are be AUX iajs-1885 18 16 respectively respectively ADV iajs-1885 18 17 defined define VERB iajs-1885 18 18 as as ADP iajs-1885 18 19 [ [ X iajs-1885 18 20 2 2 NUM iajs-1885 18 21 ] ] PUNCT iajs-1885 18 22 : : PUNCT iajs-1885 18 23 𝑓 𝑓 DET iajs-1885 18 24 𝑡 𝑡 NOUN iajs-1885 18 25 , , PUNCT iajs-1885 18 26 ɵ ɵ NOUN iajs-1885 18 27 ɵ𝑒 ɵ𝑒 PROPN iajs-1885 18 28 ɵ ɵ SYM iajs-1885 18 29 , , PUNCT iajs-1885 18 30 𝑡 𝑡 PROPN iajs-1885 18 31 , , PUNCT iajs-1885 18 32 ɵ ɵ NOUN iajs-1885 18 33 0 0 NUM iajs-1885 18 34 1 1 NUM iajs-1885 18 35 the the DET iajs-1885 18 36 cumulative cumulative ADJ iajs-1885 18 37 distribution distribution NOUN iajs-1885 18 38 function function NOUN iajs-1885 18 39 is be AUX iajs-1885 18 40 given give VERB iajs-1885 18 41 by by ADP iajs-1885 18 42 : : PUNCT iajs-1885 18 43 f f PROPN iajs-1885 18 44 t t PROPN iajs-1885 18 45 pr pr VERB iajs-1885 18 46 t t PROPN iajs-1885 18 47 t t PROPN iajs-1885 18 48 1 1 NUM iajs-1885 18 49 e e NOUN iajs-1885 18 50 ɵ ɵ SYM iajs-1885 18 51 2 2 NUM iajs-1885 18 52 r r NOUN iajs-1885 18 53 t t NOUN iajs-1885 18 54 1 1 NUM iajs-1885 18 55 f f NOUN iajs-1885 18 56 t t X iajs-1885 18 57 e e X iajs-1885 18 58 ɵ ɵ ADP iajs-1885 18 59 3 3 NUM iajs-1885 18 60 2.1 2.1 NUM iajs-1885 18 61 . . PUNCT iajs-1885 19 1 maximum maximum ADJ iajs-1885 19 2 likelihood likelihood NOUN iajs-1885 19 3 estimator estimator NOUN iajs-1885 19 4 let let VERB iajs-1885 19 5 𝑡 𝑡 PROPN iajs-1885 19 6 , , PUNCT iajs-1885 19 7 𝑡 𝑡 PROPN iajs-1885 19 8 , , PUNCT iajs-1885 19 9 … … PUNCT iajs-1885 19 10 , , PUNCT iajs-1885 19 11 𝑡 𝑡 PROPN iajs-1885 19 12 be be VERB iajs-1885 19 13 the the DET iajs-1885 19 14 set set NOUN iajs-1885 19 15 of of ADP iajs-1885 19 16 n n CCONJ iajs-1885 19 17 random random ADJ iajs-1885 19 18 lifetimes lifetime NOUN iajs-1885 19 19 from from ADP iajs-1885 19 20 the the DET iajs-1885 19 21 one one NUM iajs-1885 19 22 parameter parameter NOUN iajs-1885 19 23 exponential exponential ADJ iajs-1885 19 24 distribution distribution NOUN iajs-1885 19 25 then then ADV iajs-1885 19 26     SPACE iajs-1885 19 27 102 102 NUM iajs-1885 19 28     SPACE iajs-1885 19 29 ibn ibn PROPN iajs-1885 19 30 al al PROPN iajs-1885 19 31 - - PUNCT iajs-1885 19 32 haitham haitham PROPN iajs-1885 19 33 jour jour X iajs-1885 19 34 . . PROPN iajs-1885 20 1 for for ADP iajs-1885 20 2 pure pure ADJ iajs-1885 20 3 & & CCONJ iajs-1885 20 4 appl appl PROPN iajs-1885 20 5 . . PUNCT iajs-1885 21 1 sci sci PROPN iajs-1885 21 2 . . PROPN iajs-1885 22 1 ihjpas ihjpas PROPN iajs-1885 22 2 https://doi.org/10.30526/32.1.1885 https://doi.org/10.30526/32.1.1885 X iajs-1885 22 3 vol vol NOUN iajs-1885 22 4 . . PUNCT iajs-1885 23 1 32 32 NUM iajs-1885 23 2 ( ( PUNCT iajs-1885 23 3 1 1 NUM iajs-1885 23 4 ) ) PUNCT iajs-1885 23 5 2019 2019 NUM iajs-1885 23 6 l l NOUN iajs-1885 23 7 ɵ ɵ NUM iajs-1885 23 8 ; ; PUNCT iajs-1885 23 9 t t PROPN iajs-1885 23 10 , , PUNCT iajs-1885 23 11 t t PROPN iajs-1885 23 12 , , PUNCT iajs-1885 23 13 … … PUNCT iajs-1885 23 14 , , PUNCT iajs-1885 23 15 t t PROPN iajs-1885 23 16 ɵ ɵ NOUN iajs-1885 23 17 e e PROPN iajs-1885 23 18 ɵ ɵ ADP iajs-1885 23 19 ∑ ∑ PROPN iajs-1885 23 20 4 4 NUM iajs-1885 23 21 the the DET iajs-1885 23 22 value value NOUN iajs-1885 23 23 of of ADP iajs-1885 23 24 ɵ ɵ SYM iajs-1885 23 25 maximize maximize VERB iajs-1885 23 26 l l NOUN iajs-1885 23 27 , , PUNCT iajs-1885 23 28 also also ADV iajs-1885 23 29 maximize maximize VERB iajs-1885 23 30 ln ln ADJ iajs-1885 23 31 l. l. PROPN iajs-1885 23 32 hence hence ADV iajs-1885 23 33 , , PUNCT iajs-1885 23 34 by by ADP iajs-1885 23 35 taking take VERB iajs-1885 23 36 the the DET iajs-1885 23 37 natural natural ADJ iajs-1885 23 38 logarithm logarithm NOUN iajs-1885 23 39 for for ADP iajs-1885 23 40 the the DET iajs-1885 23 41 likelihood likelihood NOUN iajs-1885 23 42 function function NOUN iajs-1885 23 43 we we PRON iajs-1885 23 44 get get VERB iajs-1885 23 45 [ [ X iajs-1885 23 46 3 3 X iajs-1885 23 47 ] ] PUNCT iajs-1885 23 48 lnl lnl PROPN iajs-1885 23 49 ɵ ɵ PRON iajs-1885 23 50 ; ; PUNCT iajs-1885 23 51 t t PROPN iajs-1885 23 52 , , PUNCT iajs-1885 23 53 t t PROPN iajs-1885 23 54 , , PUNCT iajs-1885 23 55 t t PROPN iajs-1885 23 56 nlnɵ nlnɵ PROPN iajs-1885 23 57 ɵ ɵ ADP iajs-1885 23 58 𝑡 𝑡 NOUN iajs-1885 23 59 differentiating differentiate VERB iajs-1885 23 60 the the DET iajs-1885 23 61 log log NOUN iajs-1885 23 62 likelihood likelihood NOUN iajs-1885 23 63 function function NOUN iajs-1885 23 64 with with ADP iajs-1885 23 65 respect respect NOUN iajs-1885 23 66 to to ADP iajs-1885 23 67 ɵ ɵ NOUN iajs-1885 23 68 and and CCONJ iajs-1885 23 69 setting set VERB iajs-1885 23 70 the the DET iajs-1885 23 71 resultant resultant NOUN iajs-1885 23 72 derivative derivative NOUN iajs-1885 23 73 equal equal ADJ iajs-1885 23 74 to to ADP iajs-1885 23 75 zero zero NUM iajs-1885 23 76 , , PUNCT iajs-1885 23 77 yield yield NOUN iajs-1885 23 78 : : PUNCT iajs-1885 23 79 𝜕 𝜕 PROPN iajs-1885 23 80 ln ln PROPN iajs-1885 23 81 𝐿 𝐿 PROPN iajs-1885 23 82 ; ; PUNCT iajs-1885 23 83 𝑡 𝑡 PROPN iajs-1885 23 84 , , PUNCT iajs-1885 23 85 𝑡 𝑡 PROPN iajs-1885 23 86 , , PUNCT iajs-1885 23 87 … … PUNCT iajs-1885 23 88 , , PUNCT iajs-1885 23 89 𝑡 𝑡 PROPN iajs-1885 23 90 𝜕ɵ 𝜕ɵ NOUN iajs-1885 23 91 𝑛 𝑛 PRON iajs-1885 23 92 ɵ ɵ SYM iajs-1885 23 93 𝑡 𝑡 PROPN iajs-1885 23 94 0 0 VERB iajs-1885 23 95 therefore therefore ADV iajs-1885 23 96 the the DET iajs-1885 23 97 mle mle NOUN iajs-1885 23 98 for for ADP iajs-1885 23 99 ɵ ɵ NUM iajs-1885 23 100 denoted denote VERB iajs-1885 23 101 by by ADP iajs-1885 23 102 ɵ ɵ NOUN iajs-1885 23 103 is be AUX iajs-1885 23 104 : : PUNCT iajs-1885 23 105 ɵ ɵ ADP iajs-1885 23 106 𝑛 𝑛 PROPN iajs-1885 23 107 ∑ ∑ PART iajs-1885 23 108 𝑡 𝑡 X iajs-1885 23 109 𝑛 𝑛 PRON iajs-1885 23 110 𝑇 𝑇 PROPN iajs-1885 23 111 5 5 NUM iajs-1885 23 112 where where SCONJ iajs-1885 23 113 we we PRON iajs-1885 23 114 suppose suppose VERB iajs-1885 23 115 that that SCONJ iajs-1885 23 116 ∑ ∑ PROPN iajs-1885 23 117 𝑡 𝑡 PROPN iajs-1885 23 118 . . PUNCT iajs-1885 24 1 hence hence ADV iajs-1885 24 2 , , PUNCT iajs-1885 24 3 the the DET iajs-1885 24 4 mle mle NOUN iajs-1885 24 5 of of ADP iajs-1885 24 6 the the DET iajs-1885 24 7 reliability reliability NOUN iajs-1885 24 8 function function NOUN iajs-1885 24 9 will will AUX iajs-1885 24 10 be be AUX iajs-1885 24 11 𝑅 𝑅 PROPN iajs-1885 24 12 𝑡 𝑡 NOUN iajs-1885 24 13 𝑒 𝑒 PROPN iajs-1885 24 14 ɵ ɵ NUM iajs-1885 24 15 6 6 NUM iajs-1885 24 16 2.2 2.2 NUM iajs-1885 24 17 . . PUNCT iajs-1885 25 1 standard standard ADJ iajs-1885 25 2 bayes bayes PROPN iajs-1885 25 3 estimators estimator VERB iajs-1885 25 4 the the DET iajs-1885 25 5 researchers researcher NOUN iajs-1885 25 6 employed employ VERB iajs-1885 25 7 two two NUM iajs-1885 25 8 types type NOUN iajs-1885 25 9 of of ADP iajs-1885 25 10 loss loss NOUN iajs-1885 25 11 functions function NOUN iajs-1885 25 12 , , PUNCT iajs-1885 25 13 namely namely ADV iajs-1885 25 14 , , PUNCT iajs-1885 25 15 the the DET iajs-1885 25 16 squared square VERB iajs-1885 25 17 error error NOUN iajs-1885 25 18 loss loss NOUN iajs-1885 25 19 function function NOUN iajs-1885 25 20 ( ( PUNCT iajs-1885 25 21 self self NOUN iajs-1885 25 22 ) ) PUNCT iajs-1885 25 23 and and CCONJ iajs-1885 25 24 modified modify VERB iajs-1885 25 25 squared square VERB iajs-1885 25 26 error error NOUN iajs-1885 25 27 loss loss NOUN iajs-1885 25 28 function function NOUN iajs-1885 25 29 ( ( PUNCT iajs-1885 25 30 mself mself NUM iajs-1885 25 31 ) ) PUNCT iajs-1885 25 32 . . PUNCT iajs-1885 26 1 the the DET iajs-1885 26 2 bayes bayes PROPN iajs-1885 26 3 estimator estimator NOUN iajs-1885 26 4 of of ADP iajs-1885 26 5 the the DET iajs-1885 26 6 parameter parameter NOUN iajs-1885 26 7 ɵ ɵ NOUN iajs-1885 26 8 is be AUX iajs-1885 26 9 the the DET iajs-1885 26 10 value value NOUN iajs-1885 26 11 of of ADP iajs-1885 26 12 ɵ ɵ PRON iajs-1885 26 13 that that PRON iajs-1885 26 14 minimize minimize VERB iajs-1885 26 15 the the DET iajs-1885 26 16 risk risk NOUN iajs-1885 26 17 function function NOUN iajs-1885 26 18 r(ɵ r(ɵ NOUN iajs-1885 26 19 , , PUNCT iajs-1885 26 20 ɵ ɵ NUM iajs-1885 26 21 where where SCONJ iajs-1885 26 22 [ [ X iajs-1885 26 23 4 4 NUM iajs-1885 26 24 ] ] X iajs-1885 26 25 𝑅 𝑅 PROPN iajs-1885 26 26 ɵ ɵ NOUN iajs-1885 26 27 , , PUNCT iajs-1885 26 28 ɵ ɵ NOUN iajs-1885 26 29 𝐸 𝐸 PROPN iajs-1885 26 30 𝐿 𝐿 PROPN iajs-1885 26 31 ɵ ɵ NOUN iajs-1885 26 32 , , PUNCT iajs-1885 26 33 ɵ ɵ PROPN iajs-1885 26 34 𝐿 𝐿 PROPN iajs-1885 26 35 ɵ ɵ NOUN iajs-1885 26 36 , , PUNCT iajs-1885 26 37 ɵ ɵ ADJ iajs-1885 26 38 ℎ ℎ PROPN iajs-1885 26 39 ɵ׀𝑡 ɵ׀𝑡 NOUN iajs-1885 26 40 𝑑𝜃 𝑑𝜃 PROPN iajs-1885 26 41 ɵ ɵ SYM iajs-1885 26 42 7 7 NUM iajs-1885 26 43 in in ADP iajs-1885 26 44 the the DET iajs-1885 26 45 case case NOUN iajs-1885 26 46 of of ADP iajs-1885 26 47 squared square VERB iajs-1885 26 48 error error NOUN iajs-1885 26 49 loss loss NOUN iajs-1885 26 50 function function NOUN iajs-1885 26 51 , , PUNCT iajs-1885 26 52 we we PRON iajs-1885 26 53 have have VERB iajs-1885 26 54 : : PUNCT iajs-1885 26 55 𝐿 𝐿 PROPN iajs-1885 26 56 𝜃 𝜃 PROPN iajs-1885 26 57 , , PUNCT iajs-1885 26 58 𝜃 𝜃 NOUN iajs-1885 26 59 𝜃 𝜃 NOUN iajs-1885 26 60 𝜃 𝜃 NOUN iajs-1885 26 61 8 8 NUM iajs-1885 26 62 then then ADV iajs-1885 26 63 , , PUNCT iajs-1885 26 64 the the DET iajs-1885 26 65 risk risk NOUN iajs-1885 26 66 function function NOUN iajs-1885 26 67 will will AUX iajs-1885 26 68 be be AUX iajs-1885 26 69 r r NOUN iajs-1885 26 70 ɵ ɵ NUM iajs-1885 26 71 , , PUNCT iajs-1885 26 72 ɵ ɵ PRON iajs-1885 26 73 ɵ ɵ ADP iajs-1885 26 74 ɵ ɵ ADP iajs-1885 26 75 ɵ ɵ ADP iajs-1885 26 76 h h NOUN iajs-1885 26 77 ɵ׀t ɵ׀t INTJ iajs-1885 26 78 dɵ dɵ VERB iajs-1885 26 79 ɵ ɵ ADP iajs-1885 26 80 ℎ ℎ PROPN iajs-1885 26 81 ɵ׀𝑡 ɵ׀𝑡 NOUN iajs-1885 26 82 𝑑ɵ 𝑑ɵ NOUN iajs-1885 26 83 2ɵ 2ɵ NUM iajs-1885 26 84 ɵℎ ɵℎ PRON iajs-1885 26 85 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 26 86 𝑑ɵ 𝑑ɵ NOUN iajs-1885 26 87 ɵ ɵ ADP iajs-1885 26 88 ɵ ɵ PROPN iajs-1885 26 89 ɵ ɵ ADP iajs-1885 26 90 ℎ ℎ PROPN iajs-1885 26 91 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 26 92 𝑑ɵ 𝑑ɵ NOUN iajs-1885 26 93 ɵ ɵ PROPN iajs-1885 26 94 𝑅 𝑅 PROPN iajs-1885 26 95 ɵ ɵ NOUN iajs-1885 26 96 , , PUNCT iajs-1885 26 97 ɵ ɵ PRON iajs-1885 26 98 ɵ ɵ SYM iajs-1885 26 99 2ɵ𝐸 2ɵ𝐸 NUM iajs-1885 26 100 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 26 101 𝐸 𝐸 PROPN iajs-1885 26 102 ɵ ɵ ADP iajs-1885 26 103 𝑡׀ 𝑡׀ ADP iajs-1885 26 104 differentiating differentiate VERB iajs-1885 26 105 𝑅 𝑅 PROPN iajs-1885 26 106 ɵ ɵ NOUN iajs-1885 26 107 , , PUNCT iajs-1885 26 108 ɵ ɵ ADJ iajs-1885 26 109 with with ADP iajs-1885 26 110 respect respect NOUN iajs-1885 26 111 to to ADP iajs-1885 26 112 ɵ ɵ NOUN iajs-1885 26 113 and and CCONJ iajs-1885 26 114 setting set VERB iajs-1885 26 115 the the DET iajs-1885 26 116 resultant resultant NOUN iajs-1885 26 117 derivative derivative NOUN iajs-1885 26 118 equal equal ADJ iajs-1885 26 119 to to ADP iajs-1885 26 120 zero zero NUM iajs-1885 26 121 , , PUNCT iajs-1885 26 122 we we PRON iajs-1885 26 123 get get VERB iajs-1885 26 124 : : PUNCT iajs-1885 26 125 2ɵ 2ɵ NUM iajs-1885 26 126 2e 2e PROPN iajs-1885 26 127 ɵ׀𝑡 ɵ׀𝑡 NOUN iajs-1885 26 128 0 0 NUM iajs-1885 26 129 solving solve VERB iajs-1885 26 130 for for ADP iajs-1885 26 131 ɵ ɵ PRON iajs-1885 26 132 implies imply VERB iajs-1885 26 133 that that SCONJ iajs-1885 26 134 ɵ ɵ NUM iajs-1885 26 135 e e X iajs-1885 26 136 ɵ׀t ɵ׀t PROPN iajs-1885 26 137 9 9 NUM iajs-1885 26 138 the the DET iajs-1885 26 139 modified modified ADJ iajs-1885 26 140 square square ADJ iajs-1885 26 141 error error NOUN iajs-1885 26 142 loss loss NOUN iajs-1885 26 143 function function NOUN iajs-1885 26 144 is be AUX iajs-1885 26 145 defined define VERB iajs-1885 26 146 as as ADP iajs-1885 26 147 [ [ X iajs-1885 26 148 5 5 NUM iajs-1885 26 149 ] ] PUNCT iajs-1885 26 150 : : PUNCT iajs-1885 26 151 l l NOUN iajs-1885 26 152 ɵ ɵ X iajs-1885 26 153 , , PUNCT iajs-1885 26 154 ɵ ɵ PRON iajs-1885 26 155 ɵ ɵ SYM iajs-1885 26 156 ɵ ɵ ADP iajs-1885 26 157 ɵ ɵ NUM iajs-1885 26 158 10 10 NUM iajs-1885 26 159 where where SCONJ iajs-1885 26 160 r r NOUN iajs-1885 26 161 is be AUX iajs-1885 26 162 a a DET iajs-1885 26 163 positive positive ADJ iajs-1885 26 164 integer integer NOUN iajs-1885 26 165 . . PUNCT iajs-1885 27 1 if if SCONJ iajs-1885 27 2 ( ( PUNCT iajs-1885 27 3 mself mself PRON iajs-1885 27 4 ) ) PUNCT iajs-1885 27 5 is be AUX iajs-1885 27 6 adopted adopt VERB iajs-1885 27 7 , , PUNCT iajs-1885 27 8 it it PRON iajs-1885 27 9 can can AUX iajs-1885 27 10 be be AUX iajs-1885 27 11 in in ADP iajs-1885 27 12 the the DET iajs-1885 27 13 same same ADJ iajs-1885 27 14 manner manner NOUN iajs-1885 27 15 show show VERB iajs-1885 27 16 that that SCONJ iajs-1885 27 17 the the DET iajs-1885 27 18 bayes bayes PROPN iajs-1885 27 19 estimator estimator NOUN iajs-1885 27 20 of of ADP iajs-1885 27 21 ɵ ɵ NOUN iajs-1885 27 22 is be AUX iajs-1885 27 23 ɵ ɵ ADP iajs-1885 27 24 𝐸 𝐸 PROPN iajs-1885 27 25 ɵ ɵ SYM iajs-1885 27 26 𝑡׀ 𝑡׀ NOUN iajs-1885 27 27 𝐸 𝐸 PROPN iajs-1885 27 28 ɵ ɵ ADP iajs-1885 27 29 𝑡׀ 𝑡׀ ADP iajs-1885 27 30 11 11 NUM iajs-1885 27 31     SPACE iajs-1885 27 32 103 103 NUM iajs-1885 27 33     SPACE iajs-1885 27 34 ibn ibn PROPN iajs-1885 27 35 al al PROPN iajs-1885 27 36 - - PUNCT iajs-1885 27 37 haitham haitham PROPN iajs-1885 27 38 jour jour X iajs-1885 27 39 . . PROPN iajs-1885 28 1 for for ADP iajs-1885 28 2 pure pure ADJ iajs-1885 28 3 & & CCONJ iajs-1885 28 4 appl appl PROPN iajs-1885 28 5 . . PUNCT iajs-1885 29 1 sci sci PROPN iajs-1885 29 2 . . PROPN iajs-1885 30 1 ihjpas ihjpas PROPN iajs-1885 30 2 https://doi.org/10.30526/32.1.1885 https://doi.org/10.30526/32.1.1885 X iajs-1885 30 3 vol vol NOUN iajs-1885 30 4 . . PUNCT iajs-1885 31 1 32 32 NUM iajs-1885 31 2 ( ( PUNCT iajs-1885 31 3 1 1 NUM iajs-1885 31 4 ) ) PUNCT iajs-1885 31 5 2019 2019 NUM iajs-1885 31 6 2.3 2.3 NUM iajs-1885 31 7 . . PUNCT iajs-1885 32 1 posterior posterior ADJ iajs-1885 32 2 density density NOUN iajs-1885 32 3 based base VERB iajs-1885 32 4 on on ADP iajs-1885 32 5 jeffrey jeffrey PROPN iajs-1885 32 6 's 's PART iajs-1885 32 7 prior prior ADJ iajs-1885 32 8 information information NOUN iajs-1885 32 9 let let VERB iajs-1885 32 10 us we PRON iajs-1885 32 11 assume assume VERB iajs-1885 32 12 that that SCONJ iajs-1885 32 13 ɵ ɵ PRON iajs-1885 32 14 has have VERB iajs-1885 32 15 non non ADJ iajs-1885 32 16 informative informative ADJ iajs-1885 32 17 prior prior ADJ iajs-1885 32 18 density density NOUN iajs-1885 32 19 . . PUNCT iajs-1885 33 1 jeffrey jeffrey PROPN iajs-1885 33 2 's 's PART iajs-1885 33 3 ( ( PUNCT iajs-1885 33 4 1961 1961 NUM iajs-1885 33 5 ) ) PUNCT iajs-1885 33 6 developed develop VERB iajs-1885 33 7 a a DET iajs-1885 33 8 general general ADJ iajs-1885 33 9 rule rule NOUN iajs-1885 33 10 for for ADP iajs-1885 33 11 obtaining obtain VERB iajs-1885 33 12 the the DET iajs-1885 33 13 prior prior ADJ iajs-1885 33 14 distribution distribution NOUN iajs-1885 33 15 of of ADP iajs-1885 33 16 ɵ ɵ PRON iajs-1885 33 17 [ [ X iajs-1885 33 18 6 6 NUM iajs-1885 33 19 ] ] PUNCT iajs-1885 33 20 . . PUNCT iajs-1885 34 1 he he PRON iajs-1885 34 2 established establish VERB iajs-1885 34 3 that that SCONJ iajs-1885 34 4 the the DET iajs-1885 34 5 single single ADJ iajs-1885 34 6 unknown unknown ADJ iajs-1885 34 7 parameter parameter NOUN iajs-1885 34 8 ɵ ɵ ADP iajs-1885 34 9 which which PRON iajs-1885 34 10 is be AUX iajs-1885 34 11 regarded regard VERB iajs-1885 34 12 as as ADP iajs-1885 34 13 a a DET iajs-1885 34 14 random random ADJ iajs-1885 34 15 variable variable NOUN iajs-1885 34 16 follows follow VERB iajs-1885 34 17 such such DET iajs-1885 34 18 a a DET iajs-1885 34 19 distribution distribution NOUN iajs-1885 34 20 that that PRON iajs-1885 34 21 is be AUX iajs-1885 34 22 proportional proportional ADJ iajs-1885 34 23 to to ADP iajs-1885 34 24 the the DET iajs-1885 34 25 square square ADJ iajs-1885 34 26 root root NOUN iajs-1885 34 27 of of ADP iajs-1885 34 28 the the DET iajs-1885 34 29 fisher fisher PROPN iajs-1885 34 30 information information NOUN iajs-1885 34 31 on on ADP iajs-1885 34 32 ɵ ɵ SYM iajs-1885 34 33 , , PUNCT iajs-1885 34 34 that that PRON iajs-1885 34 35 is be AUX iajs-1885 34 36 [ [ X iajs-1885 34 37 5 5 NUM iajs-1885 34 38 ] ] PUNCT iajs-1885 34 39 : : PUNCT iajs-1885 34 40 𝑔 𝑔 PROPN iajs-1885 34 41 ɵ ɵ SYM iajs-1885 34 42 𝛼 𝛼 VERB iajs-1885 34 43 𝐼 𝐼 ADP iajs-1885 34 44 ɵ ɵ SYM iajs-1885 34 45 12 12 NUM iajs-1885 34 46 that that PRON iajs-1885 34 47 is be AUX iajs-1885 34 48 𝑔 𝑔 PROPN iajs-1885 34 49 ɵ ɵ ADP iajs-1885 34 50 𝑐 𝑐 NOUN iajs-1885 34 51 𝐼 𝐼 NOUN iajs-1885 34 52 ɵ ɵ ADP iajs-1885 34 53 where where SCONJ iajs-1885 34 54 c c NOUN iajs-1885 34 55 is be AUX iajs-1885 34 56 a a DET iajs-1885 34 57 constant constant ADJ iajs-1885 34 58 of of ADP iajs-1885 34 59 proportionality proportionality NOUN iajs-1885 34 60 and and CCONJ iajs-1885 34 61 i(ɵ i(ɵ PROPN iajs-1885 34 62 ) ) PUNCT iajs-1885 34 63 represent represent VERB iajs-1885 34 64 fisher fisher PROPN iajs-1885 34 65 information information NOUN iajs-1885 34 66 defined define VERB iajs-1885 34 67 as as SCONJ iajs-1885 34 68 follows follow VERB iajs-1885 34 69 : : PUNCT iajs-1885 34 70 𝐼 𝐼 PROPN iajs-1885 34 71 ɵ ɵ PROPN iajs-1885 34 72 𝑛𝐸 𝑛𝐸 PROPN iajs-1885 34 73 𝜕 𝜕 PROPN iajs-1885 34 74 𝑙𝑛𝑓 𝑙𝑛𝑓 PROPN iajs-1885 34 75 𝑡 𝑡 PROPN iajs-1885 34 76 , , PUNCT iajs-1885 34 77 ɵ ɵ X iajs-1885 34 78 𝜕ɵ 𝜕ɵ NOUN iajs-1885 34 79 if if SCONJ iajs-1885 34 80 𝑔 𝑔 PROPN iajs-1885 34 81 ɵ ɵ SYM iajs-1885 34 82 denote denote VERB iajs-1885 34 83 jeffrey jeffrey PROPN iajs-1885 34 84 's 's PART iajs-1885 34 85 prior prior ADJ iajs-1885 34 86 information information NOUN iajs-1885 34 87 then then ADV iajs-1885 34 88 𝑔 𝑔 PROPN iajs-1885 34 89 ɵ ɵ PROPN iajs-1885 34 90 𝑐 𝑐 PROPN iajs-1885 34 91 𝑛𝐸 𝑛𝐸 PROPN iajs-1885 34 92 𝜕 𝜕 PROPN iajs-1885 34 93 𝑙𝑛𝑓 𝑙𝑛𝑓 PROPN iajs-1885 34 94 𝑡 𝑡 PROPN iajs-1885 34 95 ; ; PUNCT iajs-1885 34 96 ɵ ɵ X iajs-1885 34 97 𝜕ɵ 𝜕ɵ NOUN iajs-1885 34 98 13 13 NUM iajs-1885 34 99 for for ADP iajs-1885 34 100 the the DET iajs-1885 34 101 exponential exponential ADJ iajs-1885 34 102 distribution distribution NOUN iajs-1885 34 103 we we PRON iajs-1885 34 104 have have VERB iajs-1885 34 105 𝑙𝑛𝑓 𝑙𝑛𝑓 NOUN iajs-1885 34 106 𝑡 𝑡 PROPN iajs-1885 34 107 , , PUNCT iajs-1885 34 108 ɵ ɵ PRON iajs-1885 34 109 𝑙𝑛ɵ 𝑙𝑛ɵ NOUN iajs-1885 34 110 ɵ𝑡 ɵ𝑡 X iajs-1885 34 111 ; ; PUNCT iajs-1885 34 112 ɵ ɵ X iajs-1885 34 113 ɵ ɵ ADP iajs-1885 34 114 ɵ ɵ ADP iajs-1885 34 115 t t NOUN iajs-1885 34 116 the the DET iajs-1885 34 117 second second ADJ iajs-1885 34 118 derivative derivative NOUN iajs-1885 34 119 is be AUX iajs-1885 34 120 ∂ ∂ NUM iajs-1885 34 121 lnf lnf PROPN iajs-1885 34 122 t t PROPN iajs-1885 34 123 ; ; PUNCT iajs-1885 34 124 ɵ ɵ NUM iajs-1885 34 125 ∂ɵ ∂ɵ PROPN iajs-1885 34 126 1 1 NUM iajs-1885 34 127 ɵ ɵ NOUN iajs-1885 34 128 hence hence ADV iajs-1885 34 129 , , PUNCT iajs-1885 34 130 𝐸 𝐸 PROPN iajs-1885 34 131 𝜕 𝜕 PROPN iajs-1885 34 132 𝑙𝑛𝑓 𝑙𝑛𝑓 PROPN iajs-1885 34 133 𝑡 𝑡 PROPN iajs-1885 34 134 ; ; PUNCT iajs-1885 34 135 ɵ ɵ X iajs-1885 34 136 𝜕ɵ 𝜕ɵ NOUN iajs-1885 34 137 1 1 NUM iajs-1885 34 138 ɵ ɵ NOUN iajs-1885 34 139 substituting substitute VERB iajs-1885 34 140 in in ADP iajs-1885 34 141 equation equation NOUN iajs-1885 34 142 ( ( PUNCT iajs-1885 34 143 13 13 NUM iajs-1885 34 144 ) ) PUNCT iajs-1885 34 145 it it PRON iajs-1885 34 146 follows follow VERB iajs-1885 34 147 that that SCONJ iajs-1885 34 148 𝑔 𝑔 PROPN iajs-1885 34 149 ɵ ɵ SYM iajs-1885 34 150 𝑐 𝑐 PROPN iajs-1885 34 151 ɵ ɵ NUM iajs-1885 34 152 √𝑛 √𝑛 ADP iajs-1885 34 153 from from ADP iajs-1885 34 154 bayes bayes PROPN iajs-1885 34 155 theorem theorem VERB iajs-1885 34 156 the the DET iajs-1885 34 157 posterior posterior ADJ iajs-1885 34 158 density density NOUN iajs-1885 34 159 function function NOUN iajs-1885 34 160 of of ADP iajs-1885 34 161 ɵ ɵ PRON iajs-1885 34 162 denoted denote VERB iajs-1885 34 163 by by ADP iajs-1885 34 164 ℎ ℎ PROPN iajs-1885 34 165 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 34 166 can can AUX iajs-1885 34 167 be be AUX iajs-1885 34 168 derived derive VERB iajs-1885 34 169 as as ADP iajs-1885 34 170 [ [ X iajs-1885 34 171 4 4 NUM iajs-1885 34 172 ] ] PUNCT iajs-1885 34 173 : : PUNCT iajs-1885 34 174 ℎ ℎ PROPN iajs-1885 34 175 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 34 176 , , PUNCT iajs-1885 34 177 … … PUNCT iajs-1885 34 178 , , PUNCT iajs-1885 34 179 𝑡 𝑡 PROPN iajs-1885 34 180 ɵ ɵ VERB iajs-1885 34 181 ɵ ɵ NUM iajs-1885 34 182 ; ; PUNCT iajs-1885 34 183 , , PUNCT iajs-1885 34 184 … … PUNCT iajs-1885 34 185 , , PUNCT iajs-1885 34 186 ɵ ɵ NOUN iajs-1885 34 187 ɵ ɵ NUM iajs-1885 34 188 ; ; PUNCT iajs-1885 34 189 , , PUNCT iajs-1885 34 190 … … PUNCT iajs-1885 34 191 , , PUNCT iajs-1885 34 192 ɵ ɵ SYM iajs-1885 34 193 ℎ ℎ PROPN iajs-1885 34 194 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 34 195 , , PUNCT iajs-1885 34 196 … … PUNCT iajs-1885 34 197 , , PUNCT iajs-1885 34 198 𝑡 𝑡 PROPN iajs-1885 34 199 ɵ ɵ VERB iajs-1885 34 200 ɵ ɵ SYM iajs-1885 34 201 ɵ ɵ ADP iajs-1885 34 202 ɵ ɵ NUM iajs-1885 34 203 ɵ ɵ NOUN iajs-1885 34 204 , , PUNCT iajs-1885 34 205 t=∑ t=∑ NOUN iajs-1885 34 206 𝑡 𝑡 PROPN iajs-1885 34 207 hence hence ADV iajs-1885 34 208 , , PUNCT iajs-1885 34 209 the the DET iajs-1885 34 210 posterior posterior ADJ iajs-1885 34 211 density density NOUN iajs-1885 34 212 function function NOUN iajs-1885 34 213 for for ADP iajs-1885 34 214 ɵ ɵ NUM iajs-1885 34 215 based base VERB iajs-1885 34 216 on on ADP iajs-1885 34 217 jeffery jeffery PROPN iajs-1885 34 218 's 's PART iajs-1885 34 219 prior prior ADJ iajs-1885 34 220 information information NOUN iajs-1885 34 221 will will AUX iajs-1885 34 222 be be AUX iajs-1885 34 223 : : PUNCT iajs-1885 34 224 ℎ ℎ PROPN iajs-1885 34 225 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 34 226 , , PUNCT iajs-1885 34 227 … … PUNCT iajs-1885 34 228 , , PUNCT iajs-1885 34 229 𝑡 𝑡 PROPN iajs-1885 34 230 ɵ ɵ VERB iajs-1885 34 231 ɵ ɵ NOUN iajs-1885 34 232 г г PROPN iajs-1885 34 233 14 14 NUM iajs-1885 34 234 the the DET iajs-1885 34 235 posterior posterior ADJ iajs-1885 34 236 density density NOUN iajs-1885 34 237 in in ADP iajs-1885 34 238 equation equation NOUN iajs-1885 34 239 ( ( PUNCT iajs-1885 34 240 14 14 NUM iajs-1885 34 241 ) ) PUNCT iajs-1885 34 242 is be AUX iajs-1885 34 243 defined define VERB iajs-1885 34 244 identified identify VERB iajs-1885 34 245 as as ADP iajs-1885 34 246 a a DET iajs-1885 34 247 density density NOUN iajs-1885 34 248 of of ADP iajs-1885 34 249 the the DET iajs-1885 34 250 gamma gamma NOUN iajs-1885 34 251 distribution distribution NOUN iajs-1885 34 252 , , PUNCT iajs-1885 34 253 that that PRON iajs-1885 34 254 is be AUX iajs-1885 34 255 : : PUNCT iajs-1885 34 256 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 34 257 , , PUNCT iajs-1885 34 258 𝑡 𝑡 PROPN iajs-1885 34 259 , , PUNCT iajs-1885 34 260 … … PUNCT iajs-1885 34 261 , , PUNCT iajs-1885 34 262 𝑡 𝑡 X iajs-1885 34 263 ~ ~ PUNCT iajs-1885 34 264 gamma gamma NOUN iajs-1885 34 265 ( ( PUNCT iajs-1885 34 266 n n CCONJ iajs-1885 34 267 , , PUNCT iajs-1885 34 268 ) ) PUNCT iajs-1885 34 269 with with ADP iajs-1885 34 270 e(ɵ e(ɵ PROPN iajs-1885 34 271 ) ) PUNCT iajs-1885 34 272 = = PUNCT iajs-1885 34 273 and and CCONJ iajs-1885 34 274 var(ɵ var(ɵ NOUN iajs-1885 34 275 ) ) PUNCT iajs-1885 34 276 = = SYM iajs-1885 35 1 ɵ ɵ PROPN iajs-1885 35 2 ~ ~ NOUN iajs-1885 35 3 gamma gamma NOUN iajs-1885 35 4 ( ( PUNCT iajs-1885 35 5 n n NOUN iajs-1885 35 6 , , PUNCT iajs-1885 35 7 ) ) PUNCT iajs-1885 35 8     SPACE iajs-1885 35 9 104 104 NUM iajs-1885 35 10     SPACE iajs-1885 35 11 ibn ibn PROPN iajs-1885 35 12 al al PROPN iajs-1885 35 13 - - PUNCT iajs-1885 35 14 haitham haitham PROPN iajs-1885 35 15 jour jour X iajs-1885 35 16 . . PROPN iajs-1885 36 1 for for ADP iajs-1885 36 2 pure pure ADJ iajs-1885 36 3 & & CCONJ iajs-1885 36 4 appl appl PROPN iajs-1885 36 5 . . PUNCT iajs-1885 37 1 sci sci PROPN iajs-1885 37 2 . . PROPN iajs-1885 38 1 ihjpas ihjpas PROPN iajs-1885 38 2 https://doi.org/10.30526/32.1.1885 https://doi.org/10.30526/32.1.1885 X iajs-1885 38 3 vol vol NOUN iajs-1885 38 4 . . PUNCT iajs-1885 39 1 32 32 NUM iajs-1885 39 2 ( ( PUNCT iajs-1885 39 3 1 1 NUM iajs-1885 39 4 ) ) PUNCT iajs-1885 39 5 2019 2019 NUM iajs-1885 39 6 2.4 2.4 NUM iajs-1885 39 7 . . PUNCT iajs-1885 40 1 posterior posterior ADJ iajs-1885 40 2 density density NOUN iajs-1885 40 3 based base VERB iajs-1885 40 4 on on ADP iajs-1885 40 5 gamma gamma NOUN iajs-1885 40 6 prior prior ADJ iajs-1885 40 7 distribution distribution NOUN iajs-1885 40 8 assuming assume VERB iajs-1885 40 9 that that SCONJ iajs-1885 40 10 ɵ ɵ PRON iajs-1885 40 11 has have VERB iajs-1885 40 12 informative informative ADJ iajs-1885 40 13 prior prior ADV iajs-1885 40 14 as as ADP iajs-1885 40 15 gamma gamma NOUN iajs-1885 40 16 distribution distribution NOUN iajs-1885 40 17 which which PRON iajs-1885 40 18 takes take VERB iajs-1885 40 19 the the DET iajs-1885 40 20 following follow VERB iajs-1885 40 21 form form NOUN iajs-1885 40 22 : : PUNCT iajs-1885 40 23 𝑔 𝑔 PROPN iajs-1885 40 24 ɵ ɵ SYM iajs-1885 40 25 𝛽 𝛽 PROPN iajs-1885 40 26 ɵ ɵ PRON iajs-1885 40 27 𝑒 𝑒 PROPN iajs-1885 40 28 ɵ ɵ ADJ iajs-1885 40 29 г г PROPN iajs-1885 40 30 𝛼 𝛼 NOUN iajs-1885 40 31 ; ; PUNCT iajs-1885 40 32 ɵ ɵ X iajs-1885 40 33 0 0 NUM iajs-1885 40 34 , , PUNCT iajs-1885 40 35 𝛽 𝛽 PROPN iajs-1885 40 36 0 0 NUM iajs-1885 40 37 , , PUNCT iajs-1885 40 38 𝛼 𝛼 PROPN iajs-1885 40 39 0 0 NUM iajs-1885 40 40 15 15 NUM iajs-1885 40 41 where where SCONJ iajs-1885 40 42 α α X iajs-1885 40 43 , , PUNCT iajs-1885 40 44 β β X iajs-1885 40 45 are be AUX iajs-1885 40 46 the the DET iajs-1885 40 47 shape shape NOUN iajs-1885 40 48 parameter parameter NOUN iajs-1885 40 49 and and CCONJ iajs-1885 40 50 scale scale NOUN iajs-1885 40 51 parameter parameter NOUN iajs-1885 40 52 respectively respectively ADV iajs-1885 40 53 . . PUNCT iajs-1885 41 1 the the DET iajs-1885 41 2 posterior posterior ADJ iajs-1885 41 3 density density NOUN iajs-1885 41 4 function function NOUN iajs-1885 41 5 is be AUX iajs-1885 41 6 ℎ ℎ PROPN iajs-1885 41 7 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 41 8 𝑔 𝑔 PROPN iajs-1885 41 9 ɵ ɵ SYM iajs-1885 41 10 𝐿 𝐿 PROPN iajs-1885 41 11 ɵ ɵ NUM iajs-1885 41 12 ; ; PUNCT iajs-1885 41 13 𝑡 𝑡 X iajs-1885 41 14 , , PUNCT iajs-1885 41 15 … … PUNCT iajs-1885 41 16 , , PUNCT iajs-1885 41 17 𝑡 𝑡 PROPN iajs-1885 41 18 𝑔 𝑔 PROPN iajs-1885 41 19 ɵ ɵ SYM iajs-1885 41 20 𝐿 𝐿 PROPN iajs-1885 41 21 ɵ ɵ NUM iajs-1885 41 22 ; ; PUNCT iajs-1885 41 23 𝑡 𝑡 X iajs-1885 41 24 , , PUNCT iajs-1885 41 25 … … PUNCT iajs-1885 41 26 , , PUNCT iajs-1885 41 27 𝑡 𝑡 X iajs-1885 41 28 𝑑ɵ 𝑑ɵ NOUN iajs-1885 41 29 thus thus ADV iajs-1885 41 30 ℎ ℎ ADP iajs-1885 41 31 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 41 32 𝑃 𝑃 PROPN iajs-1885 41 33 ɵ ɵ NOUN iajs-1885 41 34 𝑒 𝑒 X iajs-1885 41 35 ɵ ɵ ADJ iajs-1885 41 36 г г PROPN iajs-1885 41 37 𝛼 𝛼 NOUN iajs-1885 41 38 𝑛 𝑛 PROPN iajs-1885 41 39 16 16 NUM iajs-1885 41 40 where where SCONJ iajs-1885 41 41 p p NOUN iajs-1885 41 42 β β X iajs-1885 41 43 t t NOUN iajs-1885 41 44 it it PRON iajs-1885 41 45 can can AUX iajs-1885 41 46 easily easily ADV iajs-1885 41 47 be be AUX iajs-1885 41 48 noted note VERB iajs-1885 41 49 that that SCONJ iajs-1885 41 50 ( ( PUNCT iajs-1885 41 51 ɵ׀𝑡 ɵ׀𝑡 NOUN iajs-1885 41 52 ~𝐺𝑎𝑚𝑚𝑎 ~𝐺𝑎𝑚𝑚𝑎 PUNCT iajs-1885 41 53 𝛼 𝛼 PRON iajs-1885 41 54 𝑛 𝑛 NOUN iajs-1885 41 55 , , PUNCT iajs-1885 41 56 with with ADP iajs-1885 41 57 e(ɵ e(ɵ PROPN iajs-1885 41 58 ) ) PUNCT iajs-1885 41 59 = = PRON iajs-1885 41 60 , , PUNCT iajs-1885 41 61 var(ɵ var(ɵ NOUN iajs-1885 41 62 ) ) PUNCT iajs-1885 41 63 = = SYM iajs-1885 41 64 ( ( PUNCT iajs-1885 41 65 ) ) PUNCT iajs-1885 41 66 2.5 2.5 NUM iajs-1885 41 67 . . PUNCT iajs-1885 42 1 bayes bayes PROPN iajs-1885 42 2 estimator estimator NOUN iajs-1885 42 3 when when SCONJ iajs-1885 42 4 ( ( PUNCT iajs-1885 42 5 self self NOUN iajs-1885 42 6 ) ) PUNCT iajs-1885 42 7 is be AUX iajs-1885 42 8 adopted adopt VERB iajs-1885 42 9 a a PRON iajs-1885 42 10 : : PUNCT iajs-1885 42 11 the the DET iajs-1885 42 12 case case NOUN iajs-1885 42 13 of of ADP iajs-1885 42 14 jeffrey jeffrey PROPN iajs-1885 42 15 's 's PART iajs-1885 42 16 prior prior ADJ iajs-1885 42 17 information information NOUN iajs-1885 42 18 . . PUNCT iajs-1885 43 1 from from ADP iajs-1885 43 2 equation equation NOUN iajs-1885 43 3 ( ( PUNCT iajs-1885 43 4 9 9 X iajs-1885 43 5 ) ) PUNCT iajs-1885 43 6 we we PRON iajs-1885 43 7 found find VERB iajs-1885 43 8 that that SCONJ iajs-1885 43 9 : : PUNCT iajs-1885 43 10 ɵ ɵ NUM iajs-1885 43 11 e e X iajs-1885 43 12 ɵ׀t ɵ׀t X iajs-1885 43 13 ɵℎ ɵℎ PROPN iajs-1885 43 14 ɵ׀𝑡 ɵ׀𝑡 NOUN iajs-1885 43 15 𝑑𝜃 𝑑𝜃 PROPN iajs-1885 43 16 ɵ ɵ ADP iajs-1885 43 17 n n NOUN iajs-1885 43 18 t t PROPN iajs-1885 43 19 17 17 NUM iajs-1885 43 20 similarly similarly ADV iajs-1885 43 21 , , PUNCT iajs-1885 43 22 the the DET iajs-1885 43 23 bayes bayes PROPN iajs-1885 43 24 estimator estimator NOUN iajs-1885 43 25 of of ADP iajs-1885 43 26 the the DET iajs-1885 43 27 reliability reliability NOUN iajs-1885 43 28 function function NOUN iajs-1885 43 29 can can AUX iajs-1885 43 30 be be AUX iajs-1885 43 31 obtained obtain VERB iajs-1885 43 32 : : PUNCT iajs-1885 43 33 as as SCONJ iajs-1885 43 34 follows follow VERB iajs-1885 43 35 : : PUNCT iajs-1885 44 1 r r NOUN iajs-1885 44 2 t t NOUN iajs-1885 44 3 e e NOUN iajs-1885 44 4 r r NOUN iajs-1885 44 5 t t NOUN iajs-1885 44 6 t׀ t׀ NOUN iajs-1885 44 7 r r NOUN iajs-1885 44 8 t t NOUN iajs-1885 44 9 h h NOUN iajs-1885 44 10 ɵ׀t ɵ׀t PROPN iajs-1885 44 11 dθ dθ PROPN iajs-1885 44 12 r r NOUN iajs-1885 44 13 t t PROPN iajs-1885 44 14 t t PROPN iajs-1885 44 15 t t PROPN iajs-1885 44 16 t t PROPN iajs-1885 44 17 18 18 NUM iajs-1885 44 18 b b NOUN iajs-1885 44 19 : : PUNCT iajs-1885 44 20 the the DET iajs-1885 44 21 case case NOUN iajs-1885 44 22 of of ADP iajs-1885 44 23 gamma gamma NOUN iajs-1885 44 24 prior prior ADJ iajs-1885 44 25 distribution distribution NOUN iajs-1885 44 26 . . PUNCT iajs-1885 45 1 in in ADP iajs-1885 45 2 this this DET iajs-1885 45 3 case case NOUN iajs-1885 45 4 we we PRON iajs-1885 45 5 have have VERB iajs-1885 45 6 ɵ ɵ NUM iajs-1885 45 7 e e NOUN iajs-1885 45 8 ɵ׀t ɵ׀t INTJ iajs-1885 46 1 ɵh ɵh INTJ iajs-1885 46 2 ɵ׀t ɵ׀t PROPN iajs-1885 46 3 dθ dθ PROPN iajs-1885 46 4 19 19 NUM iajs-1885 46 5 the the DET iajs-1885 46 6 estimator estimator NOUN iajs-1885 46 7 of of ADP iajs-1885 46 8 the the DET iajs-1885 46 9 reliability reliability NOUN iajs-1885 46 10 function function NOUN iajs-1885 46 11 can can AUX iajs-1885 46 12 be be AUX iajs-1885 46 13 obtained obtain VERB iajs-1885 46 14 as as ADP iajs-1885 46 15 : : PUNCT iajs-1885 46 16 𝑅 𝑅 PROPN iajs-1885 46 17 𝑡 𝑡 PROPN iajs-1885 46 18 e e NOUN iajs-1885 46 19 r r NOUN iajs-1885 46 20 t t NOUN iajs-1885 46 21 t׀ t׀ NOUN iajs-1885 46 22 𝑅 𝑅 PROPN iajs-1885 46 23 𝑡 𝑡 PROPN iajs-1885 46 24 ℎ ℎ ADP iajs-1885 46 25 ɵ׀𝑡 ɵ׀𝑡 NOUN iajs-1885 46 26 𝑑𝜃 𝑑𝜃 ADP iajs-1885 46 27 𝑃 𝑃 NOUN iajs-1885 46 28 𝑃 𝑃 NOUN iajs-1885 46 29 𝑡 𝑡 NOUN iajs-1885 46 30 20 20 NUM iajs-1885 46 31 2.6 2.6 NUM iajs-1885 46 32 . . PUNCT iajs-1885 47 1 bayes bayes PROPN iajs-1885 47 2 estimator estimator NOUN iajs-1885 47 3 when when SCONJ iajs-1885 47 4 ( ( PUNCT iajs-1885 47 5 mself mself PRON iajs-1885 47 6 ) ) PUNCT iajs-1885 47 7 is be AUX iajs-1885 47 8 adopted adopt VERB iajs-1885 47 9 a a PRON iajs-1885 47 10 : : PUNCT iajs-1885 47 11 the the DET iajs-1885 47 12 case case NOUN iajs-1885 47 13 of of ADP iajs-1885 47 14 jeffrey jeffrey PROPN iajs-1885 47 15 's 's PART iajs-1885 47 16 prior prior ADJ iajs-1885 47 17 information information NOUN iajs-1885 47 18 from from ADP iajs-1885 47 19 equation equation NOUN iajs-1885 47 20 ( ( PUNCT iajs-1885 47 21 11 11 NUM iajs-1885 47 22 ) ) PUNCT iajs-1885 47 23 we we PRON iajs-1885 47 24 have have VERB iajs-1885 47 25 : : PUNCT iajs-1885 47 26 ɵ ɵ X iajs-1885 47 27 𝐸 𝐸 PROPN iajs-1885 47 28 ɵ ɵ SYM iajs-1885 47 29 𝑡׀ 𝑡׀ ADP iajs-1885 47 30 𝐸 𝐸 PROPN iajs-1885 47 31 ɵ ɵ NUM iajs-1885 47 32 𝑡׀ 𝑡׀ ADP iajs-1885 47 33 the the DET iajs-1885 47 34 𝑟 𝑟 NOUN iajs-1885 47 35 moment moment NOUN iajs-1885 47 36 of of ADP iajs-1885 47 37 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 47 38 can can AUX iajs-1885 47 39 be be AUX iajs-1885 47 40 evaluated evaluate VERB iajs-1885 47 41 as as SCONJ iajs-1885 47 42 follows follow VERB iajs-1885 47 43 : : PUNCT iajs-1885 47 44     SPACE iajs-1885 47 45 105 105 NUM iajs-1885 47 46     SPACE iajs-1885 47 47 ibn ibn PROPN iajs-1885 47 48 al al PROPN iajs-1885 47 49 - - PUNCT iajs-1885 47 50 haitham haitham PROPN iajs-1885 47 51 jour jour X iajs-1885 47 52 . . PROPN iajs-1885 48 1 for for ADP iajs-1885 48 2 pure pure ADJ iajs-1885 48 3 & & CCONJ iajs-1885 48 4 appl appl PROPN iajs-1885 48 5 . . PUNCT iajs-1885 49 1 sci sci PROPN iajs-1885 49 2 . . PROPN iajs-1885 50 1 ihjpas ihjpas PROPN iajs-1885 50 2 https://doi.org/10.30526/32.1.1885 https://doi.org/10.30526/32.1.1885 X iajs-1885 50 3 vol vol NOUN iajs-1885 50 4 . . PUNCT iajs-1885 51 1 32 32 NUM iajs-1885 51 2 ( ( PUNCT iajs-1885 51 3 1 1 NUM iajs-1885 51 4 ) ) PUNCT iajs-1885 51 5 2019 2019 NUM iajs-1885 51 6 e(ɵ e(ɵ PROPN iajs-1885 51 7 𝑡׀ 𝑡׀ NOUN iajs-1885 51 8 ɵ ɵ SYM iajs-1885 51 9 ℎ ℎ PROPN iajs-1885 51 10 ɵ׀𝑡 ɵ׀𝑡 NOUN iajs-1885 51 11 𝑑ɵ 𝑑ɵ NOUN iajs-1885 51 12 hence hence ADV iajs-1885 51 13 , , PUNCT iajs-1885 51 14 e(ɵ e(ɵ PROPN iajs-1885 51 15 𝑡׀ 𝑡׀ ADP iajs-1885 51 16 г г PROPN iajs-1885 51 17 г г PROPN iajs-1885 51 18 21 21 NUM iajs-1885 51 19 let let VERB iajs-1885 51 20 us we PRON iajs-1885 51 21 assume assume VERB iajs-1885 51 22 that that SCONJ iajs-1885 51 23 r=1,3 r=1,3 VERB iajs-1885 51 24 and and CCONJ iajs-1885 51 25 ɵ ɵ CCONJ iajs-1885 51 26 , , PUNCT iajs-1885 51 27 ɵ ɵ PRON iajs-1885 51 28 represent represent VERB iajs-1885 51 29 the the DET iajs-1885 51 30 estimator estimator NOUN iajs-1885 51 31 of of ADP iajs-1885 51 32 the the DET iajs-1885 51 33 parameter parameter NOUN iajs-1885 51 34 ɵ ɵ AUX iajs-1885 51 35 corresponding correspond VERB iajs-1885 51 36 to to PART iajs-1885 51 37 r=1,3 r=1,3 VERB iajs-1885 51 38 respectively respectively ADV iajs-1885 51 39 then then ADV iajs-1885 51 40 by by ADP iajs-1885 51 41 applying apply VERB iajs-1885 51 42 the the DET iajs-1885 51 43 formulas formula NOUN iajs-1885 51 44 in in ADP iajs-1885 51 45 ( ( PUNCT iajs-1885 51 46 11 11 NUM iajs-1885 51 47 ) ) PUNCT iajs-1885 51 48 and and CCONJ iajs-1885 51 49 ( ( PUNCT iajs-1885 51 50 21 21 NUM iajs-1885 51 51 ) ) PUNCT iajs-1885 51 52 we we PRON iajs-1885 51 53 get get VERB iajs-1885 51 54 ɵ ɵ ADP iajs-1885 51 55 ɵ ɵ X iajs-1885 51 56 ׀ ׀ PROPN iajs-1885 51 57 ɵ ɵ NUM iajs-1885 51 58 ׀ ׀ PROPN iajs-1885 51 59 г г PROPN iajs-1885 51 60 г г PROPN iajs-1885 51 61 г г PROPN iajs-1885 51 62 г г PROPN iajs-1885 51 63 22 22 NUM iajs-1885 52 1 ɵ ɵ SYM iajs-1885 52 2 ɵ ɵ X iajs-1885 52 3 ׀ ׀ PROPN iajs-1885 52 4 ɵ ɵ NUM iajs-1885 52 5 ׀ ׀ PROPN iajs-1885 52 6 г г PROPN iajs-1885 52 7 г г PROPN iajs-1885 52 8 г г PROPN iajs-1885 52 9 г г PROPN iajs-1885 52 10 23 23 NUM iajs-1885 52 11 similarly similarly ADV iajs-1885 52 12 , , PUNCT iajs-1885 52 13 the the DET iajs-1885 52 14 bayes bayes NOUN iajs-1885 52 15 estimators estimator NOUN iajs-1885 52 16 of of ADP iajs-1885 52 17 the the DET iajs-1885 52 18 reliability reliability NOUN iajs-1885 52 19 function function NOUN iajs-1885 52 20 can can AUX iajs-1885 52 21 be be AUX iajs-1885 52 22 obtained obtain VERB iajs-1885 52 23 as as SCONJ iajs-1885 52 24 follows follow VERB iajs-1885 52 25 𝑅 𝑅 PROPN iajs-1885 52 26 𝑡 𝑡 PROPN iajs-1885 52 27 ׀ ׀ VERB iajs-1885 52 28 ׀ ׀ VERB iajs-1885 52 29 𝑅 𝑅 PROPN iajs-1885 52 30 𝑡 𝑡 PROPN iajs-1885 52 31 𝐸 𝐸 PROPN iajs-1885 52 32 𝑅 𝑅 PROPN iajs-1885 52 33 𝑡 𝑡 PROPN iajs-1885 52 34 𝑡׀ 𝑡׀ ADP iajs-1885 52 35 𝐸 𝐸 PROPN iajs-1885 52 36 𝑅 𝑅 PROPN iajs-1885 52 37 𝑡 𝑡 PROPN iajs-1885 52 38 𝑡׀ 𝑡׀ ADP iajs-1885 52 39 now now ADV iajs-1885 52 40 , , PUNCT iajs-1885 52 41 we we PRON iajs-1885 52 42 have have VERB iajs-1885 52 43 to to PART iajs-1885 52 44 determine determine VERB iajs-1885 52 45 each each PRON iajs-1885 52 46 of of ADP iajs-1885 52 47 e e NOUN iajs-1885 52 48 [ [ X iajs-1885 52 49 ( ( PUNCT iajs-1885 52 50 𝑅 𝑅 PROPN iajs-1885 52 51 𝑡 𝑡 PROPN iajs-1885 52 52 𝑡׀ 𝑡׀ NOUN iajs-1885 52 53 , , PUNCT iajs-1885 52 54 e e PROPN iajs-1885 52 55 [ [ PUNCT iajs-1885 52 56 𝑅 𝑅 PROPN iajs-1885 52 57 𝑡 𝑡 NOUN iajs-1885 52 58 𝑡׀ 𝑡׀ NOUN iajs-1885 52 59 and and CCONJ iajs-1885 52 60 e e PROPN iajs-1885 52 61 𝑅 𝑅 PROPN iajs-1885 52 62 𝑡 𝑡 PROPN iajs-1885 52 63 𝑡׀ 𝑡׀ ADP iajs-1885 52 64 𝐸 𝐸 PROPN iajs-1885 52 65 𝑅 𝑅 PROPN iajs-1885 52 66 𝑡 𝑡 PROPN iajs-1885 52 67 𝑡׀ 𝑡׀ ADP iajs-1885 52 68 𝑇 𝑇 PROPN iajs-1885 52 69 𝑇 𝑇 PROPN iajs-1885 52 70 𝑡 𝑡 PART iajs-1885 52 71 e e PROPN iajs-1885 52 72 𝑅 𝑅 PROPN iajs-1885 52 73 𝑡 𝑡 PROPN iajs-1885 52 74 𝑡׀ 𝑡׀ ADP iajs-1885 52 75 𝑅 𝑅 PROPN iajs-1885 52 76 𝑡 𝑡 NOUN iajs-1885 52 77 ℎ ℎ NOUN iajs-1885 52 78 ɵ׀𝑡 ɵ׀𝑡 NOUN iajs-1885 52 79 𝑑ɵ 𝑑ɵ NOUN iajs-1885 52 80 = = PUNCT iajs-1885 52 81 by by ADP iajs-1885 52 82 the the DET iajs-1885 52 83 same same ADJ iajs-1885 52 84 way way NOUN iajs-1885 52 85 we we PRON iajs-1885 52 86 find find VERB iajs-1885 52 87 that that SCONJ iajs-1885 52 88 , , PUNCT iajs-1885 52 89 e(𝑅 e(𝑅 PROPN iajs-1885 53 1 𝑡 𝑡 VERB iajs-1885 53 2 𝑡׀ 𝑡׀ ADP iajs-1885 53 3 e e PROPN iajs-1885 53 4 𝑅 𝑅 PROPN iajs-1885 53 5 𝑡 𝑡 PROPN iajs-1885 53 6 hence hence ADV iajs-1885 53 7 , , PUNCT iajs-1885 53 8 𝑅 𝑅 PROPN iajs-1885 53 9 𝑡 𝑡 PROPN iajs-1885 53 10 𝑇 𝑇 PROPN iajs-1885 53 11 𝑇 𝑇 PROPN iajs-1885 53 12 2𝑡 2𝑡 NOUN iajs-1885 54 1 𝑇 𝑇 PROPN iajs-1885 54 2 𝑇 𝑇 PROPN iajs-1885 54 3 𝑡 𝑡 PRON iajs-1885 54 4 𝑇 𝑇 PROPN iajs-1885 54 5 𝑡 𝑡 PRON iajs-1885 54 6 𝑇 𝑇 PROPN iajs-1885 54 7 2𝑡 2𝑡 NOUN iajs-1885 54 8 24 24 NUM iajs-1885 54 9 𝑅 𝑅 PROPN iajs-1885 54 10 𝑡 𝑡 PROPN iajs-1885 54 11 25 25 NUM iajs-1885 54 12 b b NOUN iajs-1885 54 13 : : PUNCT iajs-1885 54 14 the the DET iajs-1885 54 15 case case NOUN iajs-1885 54 16 of of ADP iajs-1885 54 17 gamma gamma NOUN iajs-1885 54 18 prior prior ADJ iajs-1885 54 19 distribution distribution NOUN iajs-1885 54 20 from from ADP iajs-1885 54 21 equation equation NOUN iajs-1885 54 22 ( ( PUNCT iajs-1885 54 23 11 11 NUM iajs-1885 54 24 ) ) PUNCT iajs-1885 54 25 we we PRON iajs-1885 54 26 have have VERB iajs-1885 54 27 : : PUNCT iajs-1885 54 28 ɵ ɵ X iajs-1885 54 29 𝐸 𝐸 PROPN iajs-1885 54 30 ɵ ɵ SYM iajs-1885 54 31 𝑡׀ 𝑡׀ ADP iajs-1885 54 32 𝐸 𝐸 PROPN iajs-1885 54 33 ɵ ɵ NUM iajs-1885 54 34 𝑡׀ 𝑡׀ ADP iajs-1885 54 35 the the DET iajs-1885 54 36 𝑟 𝑟 NOUN iajs-1885 54 37 moment moment NOUN iajs-1885 54 38 of of ADP iajs-1885 54 39 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 54 40 can can AUX iajs-1885 54 41 be be AUX iajs-1885 54 42 evaluated evaluate VERB iajs-1885 54 43 as as SCONJ iajs-1885 54 44 follows follow VERB iajs-1885 54 45 : : PUNCT iajs-1885 54 46 𝐸 𝐸 PROPN iajs-1885 54 47 ɵ ɵ SYM iajs-1885 54 48 𝑡׀ 𝑡׀ NOUN iajs-1885 54 49 ɵ ɵ SYM iajs-1885 54 50 ℎ ℎ PROPN iajs-1885 54 51 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 54 52 𝑑ɵ 𝑑ɵ NOUN iajs-1885 54 53 𝐸 𝐸 PROPN iajs-1885 54 54 ɵ ɵ NUM iajs-1885 54 55 𝑡׀ 𝑡׀ ADP iajs-1885 54 56 г г PROPN iajs-1885 54 57 𝛼 𝛼 VERB iajs-1885 54 58 𝑛 𝑛 ADP iajs-1885 54 59 𝑟 𝑟 PRON iajs-1885 54 60 г г X iajs-1885 54 61 𝛼 𝛼 PRON iajs-1885 54 62 𝑛 𝑛 PRON iajs-1885 54 63 𝑃 𝑃 NOUN iajs-1885 54 64 26 26 NUM iajs-1885 54 65 let let VERB iajs-1885 54 66 us we PRON iajs-1885 54 67 assume assume VERB iajs-1885 54 68 that that SCONJ iajs-1885 54 69 r=1,3 r=1,3 VERB iajs-1885 54 70 and and CCONJ iajs-1885 54 71 ɵ ɵ CCONJ iajs-1885 54 72 , , PUNCT iajs-1885 54 73 ɵ ɵ PRON iajs-1885 54 74 represent represent VERB iajs-1885 54 75 the the DET iajs-1885 54 76 estimator estimator NOUN iajs-1885 54 77 of of ADP iajs-1885 54 78 the the DET iajs-1885 54 79 parameter parameter NOUN iajs-1885 54 80 ɵ ɵ AUX iajs-1885 54 81 corresponding correspond VERB iajs-1885 54 82 to to PART iajs-1885 54 83 r=1,3 r=1,3 VERB iajs-1885 54 84 respectively respectively ADV iajs-1885 54 85 then then ADV iajs-1885 54 86 by by ADP iajs-1885 54 87 applying apply VERB iajs-1885 54 88 the the DET iajs-1885 54 89 formulas formula NOUN iajs-1885 54 90 in in ADP iajs-1885 54 91 ( ( PUNCT iajs-1885 54 92 11 11 NUM iajs-1885 54 93 ) ) PUNCT iajs-1885 54 94 and and CCONJ iajs-1885 54 95 ( ( PUNCT iajs-1885 54 96 26 26 NUM iajs-1885 54 97 ) ) PUNCT iajs-1885 54 98 we we PRON iajs-1885 54 99 get get VERB iajs-1885 54 100 : : PUNCT iajs-1885 54 101     SPACE iajs-1885 54 102 106 106 NUM iajs-1885 54 103     SPACE iajs-1885 54 104 ibn ibn PROPN iajs-1885 54 105 al al PROPN iajs-1885 54 106 - - PUNCT iajs-1885 54 107 haitham haitham PROPN iajs-1885 54 108 jour jour X iajs-1885 54 109 . . PROPN iajs-1885 55 1 for for ADP iajs-1885 55 2 pure pure ADJ iajs-1885 55 3 & & CCONJ iajs-1885 55 4 appl appl PROPN iajs-1885 55 5 . . PUNCT iajs-1885 56 1 sci sci PROPN iajs-1885 56 2 . . PROPN iajs-1885 57 1 ihjpas ihjpas PROPN iajs-1885 57 2 https://doi.org/10.30526/32.1.1885 https://doi.org/10.30526/32.1.1885 X iajs-1885 57 3 vol vol NOUN iajs-1885 57 4 . . PUNCT iajs-1885 58 1 32 32 NUM iajs-1885 58 2 ( ( PUNCT iajs-1885 58 3 1 1 NUM iajs-1885 58 4 ) ) PUNCT iajs-1885 58 5 2019 2019 NUM iajs-1885 59 1 ɵ ɵ SYM iajs-1885 59 2 𝐸 𝐸 PRON iajs-1885 59 3 ɵ ɵ NUM iajs-1885 59 4 𝑡׀ 𝑡׀ NOUN iajs-1885 59 5 e e X iajs-1885 59 6 ɵ׀t ɵ׀t X iajs-1885 59 7 𝛼 𝛼 PROPN iajs-1885 59 8 𝑛 𝑛 ADP iajs-1885 59 9 𝛼 𝛼 NOUN iajs-1885 59 10 𝑛 𝑛 DET iajs-1885 59 11 1 1 NUM iajs-1885 59 12 𝑃 𝑃 NOUN iajs-1885 59 13 𝛼 𝛼 NOUN iajs-1885 59 14 𝑛 𝑛 DET iajs-1885 59 15 𝑃 𝑃 NOUN iajs-1885 59 16 𝛼 𝛼 NOUN iajs-1885 59 17 𝑛 𝑛 PRON iajs-1885 59 18 1 1 NUM iajs-1885 59 19 𝑃 𝑃 NOUN iajs-1885 59 20 27 27 NUM iajs-1885 59 21 ɵ ɵ SYM iajs-1885 59 22 𝐸 𝐸 PROPN iajs-1885 59 23 ɵ ɵ NUM iajs-1885 59 24 𝑡׀ 𝑡׀ NOUN iajs-1885 59 25 𝐸 𝐸 PROPN iajs-1885 59 26 ɵ ɵ NUM iajs-1885 59 27 𝑡׀ 𝑡׀ NOUN iajs-1885 59 28 𝛼 𝛼 PART iajs-1885 59 29 𝑛 𝑛 DET iajs-1885 59 30 𝛼 𝛼 NOUN iajs-1885 59 31 𝑛 𝑛 DET iajs-1885 59 32 1 1 NUM iajs-1885 59 33 𝛼 𝛼 NOUN iajs-1885 59 34 𝑛 𝑛 DET iajs-1885 59 35 2 2 NUM iajs-1885 59 36 𝛼 𝛼 NOUN iajs-1885 59 37 𝑛 𝑛 DET iajs-1885 59 38 3 3 NUM iajs-1885 59 39 𝑃 𝑃 NOUN iajs-1885 59 40 𝛼 𝛼 NOUN iajs-1885 59 41 𝑛 𝑛 DET iajs-1885 59 42 𝛼 𝛼 NOUN iajs-1885 59 43 𝑛 𝑛 DET iajs-1885 59 44 1 1 NUM iajs-1885 59 45 𝛼 𝛼 NOUN iajs-1885 59 46 𝑛 𝑛 DET iajs-1885 59 47 2 2 NUM iajs-1885 59 48 𝑃 𝑃 NOUN iajs-1885 59 49 = = SYM iajs-1885 59 50 28 28 NUM iajs-1885 59 51 similarly similarly ADV iajs-1885 59 52 , , PUNCT iajs-1885 59 53 the the DET iajs-1885 59 54 bayes bayes NOUN iajs-1885 59 55 estimators estimator NOUN iajs-1885 59 56 of of ADP iajs-1885 59 57 the the DET iajs-1885 59 58 reliability reliability NOUN iajs-1885 59 59 function function NOUN iajs-1885 59 60 can can AUX iajs-1885 59 61 be be AUX iajs-1885 59 62 obtained obtain VERB iajs-1885 59 63 as as SCONJ iajs-1885 59 64 follows follow VERB iajs-1885 59 65 𝑅 𝑅 PROPN iajs-1885 59 66 𝑡 𝑡 PROPN iajs-1885 59 67 ׀ ׀ VERB iajs-1885 59 68 ׀ ׀ VERB iajs-1885 59 69 𝑅 𝑅 PROPN iajs-1885 59 70 𝑡 𝑡 PROPN iajs-1885 59 71 𝐸 𝐸 PROPN iajs-1885 59 72 𝑅 𝑅 PROPN iajs-1885 59 73 𝑡 𝑡 PROPN iajs-1885 59 74 𝑡׀ 𝑡׀ ADP iajs-1885 59 75 𝐸 𝐸 PROPN iajs-1885 59 76 𝑅 𝑅 PROPN iajs-1885 59 77 𝑡 𝑡 PROPN iajs-1885 59 78 𝑡׀ 𝑡׀ ADP iajs-1885 59 79 now now ADV iajs-1885 59 80 , , PUNCT iajs-1885 59 81 we we PRON iajs-1885 59 82 have have VERB iajs-1885 59 83 to to PART iajs-1885 59 84 determine determine VERB iajs-1885 59 85 each each PRON iajs-1885 59 86 of of ADP iajs-1885 59 87 e e NOUN iajs-1885 59 88 [ [ X iajs-1885 59 89 ( ( PUNCT iajs-1885 59 90 𝑅 𝑅 PROPN iajs-1885 59 91 𝑡 𝑡 PROPN iajs-1885 59 92 𝑡׀ 𝑡׀ NOUN iajs-1885 59 93 , , PUNCT iajs-1885 59 94 e e X iajs-1885 59 95 [ [ PUNCT iajs-1885 59 96 𝑅 𝑅 PROPN iajs-1885 59 97 𝑡 𝑡 NOUN iajs-1885 59 98 𝑡׀ 𝑡׀ NOUN iajs-1885 59 99 and and CCONJ iajs-1885 59 100 e e PROPN iajs-1885 59 101 𝑅 𝑅 PROPN iajs-1885 59 102 𝑡 𝑡 PROPN iajs-1885 59 103 𝑡׀ 𝑡׀ ADP iajs-1885 59 104 e(r(t)׀𝑡 e(r(t)׀𝑡 PROPN iajs-1885 59 105 𝑅 𝑅 PROPN iajs-1885 59 106 𝑡 𝑡 NOUN iajs-1885 59 107 ℎ ℎ NOUN iajs-1885 59 108 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 59 109 𝑑ɵ 𝑑ɵ NOUN iajs-1885 59 110 = = SYM iajs-1885 59 111 𝐸 𝐸 PROPN iajs-1885 59 112 𝑅 𝑅 PROPN iajs-1885 59 113 𝑡 𝑡 PROPN iajs-1885 59 114 𝑡׀ 𝑡׀ NOUN iajs-1885 59 115 is be AUX iajs-1885 59 116 obtained obtain VERB iajs-1885 59 117 as as SCONJ iajs-1885 59 118 follows follow VERB iajs-1885 59 119 e e PROPN iajs-1885 59 120 𝑅 𝑅 PROPN iajs-1885 59 121 𝑡 𝑡 PROPN iajs-1885 59 122 𝑡׀ 𝑡׀ ADP iajs-1885 59 123 𝑅 𝑅 PROPN iajs-1885 59 124 𝑡 𝑡 NOUN iajs-1885 59 125 ℎ ℎ PROPN iajs-1885 59 126 ɵ׀𝑡 ɵ׀𝑡 PROPN iajs-1885 59 127 𝑑ɵ 𝑑ɵ NOUN iajs-1885 59 128 = = PUNCT iajs-1885 59 129 similarly similarly ADV iajs-1885 59 130 , , PUNCT iajs-1885 59 131 we we PRON iajs-1885 59 132 find find VERB iajs-1885 59 133 that that SCONJ iajs-1885 59 134 e e PROPN iajs-1885 59 135 𝑅 𝑅 PROPN iajs-1885 59 136 𝑡 𝑡 PROPN iajs-1885 59 137 𝑡׀ 𝑡׀ ADP iajs-1885 59 138 𝑃 𝑃 NOUN iajs-1885 59 139 𝑃 𝑃 NOUN iajs-1885 59 140 3𝑡 3𝑡 NUM iajs-1885 59 141 e e PROPN iajs-1885 59 142 𝑅 𝑅 PROPN iajs-1885 59 143 𝑡 𝑡 PROPN iajs-1885 59 144 𝑡׀ 𝑡׀ ADP iajs-1885 59 145 𝑃 𝑃 NOUN iajs-1885 59 146 𝑃 𝑃 NOUN iajs-1885 59 147 4𝑡 4𝑡 NOUN iajs-1885 59 148 hence hence ADV iajs-1885 59 149 , , PUNCT iajs-1885 59 150 𝑅 𝑅 PROPN iajs-1885 59 151 𝑡 𝑡 PROPN iajs-1885 59 152 ׀ ׀ VERB iajs-1885 59 153 ׀ ׀ VERB iajs-1885 59 154 29 29 NUM iajs-1885 59 155 𝑅 𝑅 PROPN iajs-1885 59 156 𝑡 𝑡 PROPN iajs-1885 59 157 ׀ ׀ VERB iajs-1885 59 158 ׀ ׀ VERB iajs-1885 59 159 30 30 NUM iajs-1885 59 160 3 3 NUM iajs-1885 59 161 . . PUNCT iajs-1885 60 1 simulation simulation NOUN iajs-1885 60 2 study study VERB iajs-1885 60 3 the the DET iajs-1885 60 4 simulation simulation NOUN iajs-1885 60 5 study study NOUN iajs-1885 60 6 was be AUX iajs-1885 60 7 conducted conduct VERB iajs-1885 60 8 in in ADP iajs-1885 60 9 order order NOUN iajs-1885 60 10 to to PART iajs-1885 60 11 compare compare VERB iajs-1885 60 12 the the DET iajs-1885 60 13 performance performance NOUN iajs-1885 60 14 of of ADP iajs-1885 60 15 the the DET iajs-1885 60 16 maximum maximum ADJ iajs-1885 60 17 likelihood likelihood NOUN iajs-1885 60 18 estimator estimator NOUN iajs-1885 60 19 ( ( PUNCT iajs-1885 60 20 mle mle PROPN iajs-1885 60 21 ) ) PUNCT iajs-1885 60 22 and and CCONJ iajs-1885 61 1 bayesian bayesian NOUN iajs-1885 61 2 estimators estimator NOUN iajs-1885 61 3 of of ADP iajs-1885 61 4 the the DET iajs-1885 61 5 reliability reliability NOUN iajs-1885 61 6 function function NOUN iajs-1885 61 7 r(t)of r(t)of NUM iajs-1885 61 8 one one NUM iajs-1885 61 9 parameter parameter NOUN iajs-1885 61 10 exponential exponential ADJ iajs-1885 61 11 distribution distribution NOUN iajs-1885 61 12 . . PUNCT iajs-1885 62 1 we we PRON iajs-1885 62 2 adopted adopt VERB iajs-1885 62 3 the the DET iajs-1885 62 4 integrated integrate VERB iajs-1885 62 5 mean mean NOUN iajs-1885 62 6 squared square VERB iajs-1885 62 7 error error NOUN iajs-1885 62 8 ( ( PUNCT iajs-1885 62 9 imse imse NOUN iajs-1885 62 10 ) ) PUNCT iajs-1885 62 11 as as ADP iajs-1885 62 12 a a DET iajs-1885 62 13 criterion criterion NOUN iajs-1885 62 14 of of ADP iajs-1885 62 15 comparison comparison NOUN iajs-1885 62 16 where where SCONJ iajs-1885 62 17 𝐼𝑀𝑆𝐸 𝐼𝑀𝑆𝐸 PROPN iajs-1885 62 18 𝑅 𝑅 PROPN iajs-1885 62 19 𝑡𝑖 𝑡𝑖 NOUN iajs-1885 62 20 1 1 NUM iajs-1885 62 21 𝐿 𝐿 PROPN iajs-1885 62 22 1 1 NUM iajs-1885 62 23 𝑛 𝑛 PRON iajs-1885 62 24 𝑅 𝑅 PROPN iajs-1885 62 25 𝑡 𝑡 PROPN iajs-1885 62 26 𝑅 𝑅 PROPN iajs-1885 62 27 𝑡 𝑡 PROPN iajs-1885 62 28 1 1 NUM iajs-1885 62 29 𝑛 𝑛 DET iajs-1885 62 30 𝑀𝑆𝐸 𝑀𝑆𝐸 PROPN iajs-1885 62 31 𝑅 𝑅 PROPN iajs-1885 62 32 𝑡 𝑡 PROPN iajs-1885 62 33     SPACE iajs-1885 62 34 107 107 NUM iajs-1885 62 35     SPACE iajs-1885 62 36 ibn ibn PROPN iajs-1885 62 37 al al PROPN iajs-1885 62 38 - - PUNCT iajs-1885 62 39 haitham haitham PROPN iajs-1885 62 40 jour jour X iajs-1885 62 41 . . PROPN iajs-1885 63 1 for for ADP iajs-1885 63 2 pure pure ADJ iajs-1885 63 3 & & CCONJ iajs-1885 63 4 appl appl PROPN iajs-1885 63 5 . . PUNCT iajs-1885 64 1 sci sci PROPN iajs-1885 64 2 . . PROPN iajs-1885 65 1 ihjpas ihjpas PROPN iajs-1885 65 2 https://doi.org/10.30526/32.1.1885 https://doi.org/10.30526/32.1.1885 X iajs-1885 65 3 vol vol NOUN iajs-1885 65 4 . . PUNCT iajs-1885 66 1 32 32 NUM iajs-1885 66 2 ( ( PUNCT iajs-1885 66 3 1 1 NUM iajs-1885 66 4 ) ) PUNCT iajs-1885 66 5 2019 2019 NUM iajs-1885 67 1 where where SCONJ iajs-1885 67 2 𝑛 𝑛 PROPN iajs-1885 67 3 is be AUX iajs-1885 67 4 the the DET iajs-1885 67 5 random random ADJ iajs-1885 67 6 limits limit NOUN iajs-1885 67 7 of of ADP iajs-1885 67 8 𝑡 𝑡 PROPN iajs-1885 67 9 , , PUNCT iajs-1885 67 10 using use VERB iajs-1885 67 11 t=(0.1,0.2,0.3,0.4,0.5,0.6,0.7 t=(0.1,0.2,0.3,0.4,0.5,0.6,0.7 NOUN iajs-1885 67 12 , , PUNCT iajs-1885 67 13 0.8,0.9,1 0.8,0.9,1 NUM iajs-1885 67 14 ) ) PUNCT iajs-1885 68 1 . . PUNCT iajs-1885 69 1 l l NOUN iajs-1885 69 2 is be AUX iajs-1885 69 3 the the DET iajs-1885 69 4 number number NOUN iajs-1885 69 5 of of ADP iajs-1885 69 6 replications replication NOUN iajs-1885 69 7 which which PRON iajs-1885 69 8 we we PRON iajs-1885 69 9 assumed assume VERB iajs-1885 69 10 that that SCONJ iajs-1885 69 11 l=1000 l=1000 PROPN iajs-1885 69 12 in in ADP iajs-1885 69 13 our our PRON iajs-1885 69 14 study study NOUN iajs-1885 69 15 , , PUNCT iajs-1885 69 16 𝑅(𝑡 𝑅(𝑡 X iajs-1885 69 17 ) ) PUNCT iajs-1885 69 18 is be AUX iajs-1885 69 19 the the DET iajs-1885 69 20 estimator estimator NOUN iajs-1885 69 21 of of ADP iajs-1885 69 22 r(t r(t NOUN iajs-1885 69 23 ) ) PUNCT iajs-1885 69 24 at at ADP iajs-1885 69 25 the the DET iajs-1885 69 26 𝐿 𝐿 PROPN iajs-1885 69 27 replication replication NOUN iajs-1885 69 28 . . PUNCT iajs-1885 70 1 the the DET iajs-1885 70 2 bayesian bayesian NOUN iajs-1885 70 3 estimators estimator NOUN iajs-1885 70 4 of of ADP iajs-1885 70 5 r(t r(t NOUN iajs-1885 70 6 ) ) PUNCT iajs-1885 70 7 are be AUX iajs-1885 70 8 derived derive VERB iajs-1885 70 9 with with ADP iajs-1885 70 10 respect respect NOUN iajs-1885 70 11 to to ADP iajs-1885 70 12 two two NUM iajs-1885 70 13 loss loss NOUN iajs-1885 70 14 function function NOUN iajs-1885 70 15 which which PRON iajs-1885 70 16 are be AUX iajs-1885 70 17 the the DET iajs-1885 70 18 square square ADJ iajs-1885 70 19 error error NOUN iajs-1885 70 20 loss loss NOUN iajs-1885 70 21 function function NOUN iajs-1885 70 22 ( ( PUNCT iajs-1885 70 23 self self NOUN iajs-1885 70 24 ) ) PUNCT iajs-1885 70 25 and and CCONJ iajs-1885 70 26 modified modify VERB iajs-1885 70 27 squared square VERB iajs-1885 70 28 error error NOUN iajs-1885 70 29 loss loss NOUN iajs-1885 70 30 function function NOUN iajs-1885 70 31 ( ( PUNCT iajs-1885 70 32 mself mself NUM iajs-1885 70 33 ) ) PUNCT iajs-1885 70 34 , , PUNCT iajs-1885 70 35 moreover moreover ADV iajs-1885 70 36 , , PUNCT iajs-1885 70 37 the the DET iajs-1885 70 38 informative informative ADJ iajs-1885 70 39 and and CCONJ iajs-1885 70 40 non non ADJ iajs-1885 70 41 - - ADJ iajs-1885 70 42 informative informative ADJ iajs-1885 70 43 prior prior ADV iajs-1885 70 44 were be AUX iajs-1885 70 45 postulated postulate VERB iajs-1885 70 46 . . PUNCT iajs-1885 71 1 the the DET iajs-1885 71 2 sample sample NOUN iajs-1885 71 3 sizes size VERB iajs-1885 71 4 n=10,50 n=10,50 PROPN iajs-1885 71 5 , , PUNCT iajs-1885 71 6 100 100 NUM iajs-1885 71 7 and and CCONJ iajs-1885 71 8 200 200 NUM iajs-1885 71 9 were be AUX iajs-1885 71 10 chosen choose VERB iajs-1885 71 11 to to PART iajs-1885 71 12 represent represent VERB iajs-1885 71 13 small small ADJ iajs-1885 71 14 , , PUNCT iajs-1885 71 15 moderate moderate ADJ iajs-1885 71 16 , , PUNCT iajs-1885 71 17 large large ADJ iajs-1885 71 18 and and CCONJ iajs-1885 71 19 very very ADV iajs-1885 71 20 large large ADJ iajs-1885 71 21 sample sample NOUN iajs-1885 71 22 sizes size NOUN iajs-1885 71 23 from from ADP iajs-1885 71 24 the the DET iajs-1885 71 25 one one NUM iajs-1885 71 26 parameter parameter NOUN iajs-1885 71 27 exponential exponential ADJ iajs-1885 71 28 distribution distribution NOUN iajs-1885 71 29 . . PUNCT iajs-1885 72 1 the the DET iajs-1885 72 2 postulated postulate VERB iajs-1885 72 3 values value NOUN iajs-1885 72 4 of of ADP iajs-1885 72 5 the the DET iajs-1885 72 6 unique unique ADJ iajs-1885 72 7 parameter parameter NOUN iajs-1885 72 8 ɵ ɵ PART iajs-1885 72 9 were be AUX iajs-1885 72 10 ɵ=0.5,1.5 ɵ=0.5,1.5 PROPN iajs-1885 72 11 and and CCONJ iajs-1885 72 12 the the DET iajs-1885 72 13 values value NOUN iajs-1885 72 14 of of ADP iajs-1885 72 15 the the DET iajs-1885 72 16 parameters parameter NOUN iajs-1885 72 17 for for ADP iajs-1885 72 18 gamma gamma PROPN iajs-1885 72 19 prior prior ADV iajs-1885 72 20 were be AUX iajs-1885 72 21 α=0.3,1 α=0.3,1 NOUN iajs-1885 72 22 and and CCONJ iajs-1885 72 23 β=1.2,3 β=1.2,3 PROPN iajs-1885 72 24 . . PUNCT iajs-1885 73 1 the the DET iajs-1885 73 2 values value NOUN iajs-1885 73 3 assumed assume VERB iajs-1885 73 4 for for ADP iajs-1885 73 5 the the DET iajs-1885 73 6 loss loss NOUN iajs-1885 73 7 parameter parameter NOUN iajs-1885 73 8 of of ADP iajs-1885 73 9 modified modify VERB iajs-1885 73 10 loss loss NOUN iajs-1885 73 11 function function NOUN iajs-1885 73 12 were be AUX iajs-1885 73 13 r=1,3 r=1,3 ADJ iajs-1885 73 14 . . PUNCT iajs-1885 74 1 the the DET iajs-1885 74 2 results result NOUN iajs-1885 74 3 are be AUX iajs-1885 74 4 presented present VERB iajs-1885 74 5 in in ADP iajs-1885 74 6 tables table NOUN iajs-1885 74 7 ( ( PUNCT iajs-1885 74 8 1 1 NUM iajs-1885 74 9 - - SYM iajs-1885 74 10 4 4 NUM iajs-1885 74 11 ) ) PUNCT iajs-1885 74 12 . . PUNCT iajs-1885 75 1 table table NOUN iajs-1885 75 2 1 1 NUM iajs-1885 75 3 . . PUNCT iajs-1885 76 1 ( ( PUNCT iajs-1885 76 2 imse imse ADJ iajs-1885 76 3 ) ) PUNCT iajs-1885 76 4 values value NOUN iajs-1885 76 5 of of ADP iajs-1885 76 6 the the DET iajs-1885 76 7 reliability reliability NOUN iajs-1885 76 8 function function NOUN iajs-1885 76 9 estimators estimator NOUN iajs-1885 76 10 by by ADP iajs-1885 76 11 using use VERB iajs-1885 76 12 jeffrey jeffrey PROPN iajs-1885 76 13 's 's PART iajs-1885 76 14 prior prior ADJ iajs-1885 76 15 information information NOUN iajs-1885 76 16 at at ADP iajs-1885 76 17 ɵ=0.5 ɵ=0.5 VERB iajs-1885 76 18 200 200 NUM iajs-1885 76 19 100 100 NUM iajs-1885 76 20 50 50 NUM iajs-1885 76 21 10 10 NUM iajs-1885 76 22 n n NOUN iajs-1885 76 23 estimator estimator NOUN iajs-1885 76 24 0.002636 0.002636 NUM iajs-1885 76 25 0.0040830.0047130.000574 0.0040830.0047130.000574 PROPN iajs-1885 76 26 mle mle NOUN iajs-1885 76 27 0.002632 0.002632 NUM iajs-1885 76 28 0.0040600.0046350.000526 0.0040600.0046350.000526 NOUN iajs-1885 76 29 jsqu jsqu NOUN iajs-1885 76 30 0.002624 0.002624 NUM iajs-1885 76 31 0.0040140.0044780.000450 0.0040140.0044780.000450 NUM iajs-1885 76 32 jmsq1 jmsq1 VERB iajs-1885 76 33 r=1 r=1 NOUN iajs-1885 76 34 0.002608 0.002608 NUM iajs-1885 76 35 0.0039180.0041710.000370 0.0039180.0041710.000370 NOUN iajs-1885 76 36 jmsq2 jmsq2 NOUN iajs-1885 77 1 r=3 r=3 PROPN iajs-1885 77 2 table table NOUN iajs-1885 77 3 2 2 NUM iajs-1885 77 4 . . PUNCT iajs-1885 78 1 ( ( PUNCT iajs-1885 78 2 imse imse ADJ iajs-1885 78 3 ) ) PUNCT iajs-1885 78 4 values value NOUN iajs-1885 78 5 of of ADP iajs-1885 78 6 the the DET iajs-1885 78 7 reliability reliability NOUN iajs-1885 78 8 function function NOUN iajs-1885 78 9 estimators estimator NOUN iajs-1885 78 10 by by ADP iajs-1885 78 11 using use VERB iajs-1885 78 12 jeffrey jeffrey PROPN iajs-1885 78 13 's 's PART iajs-1885 78 14 prior prior ADJ iajs-1885 78 15 information information NOUN iajs-1885 78 16 at at ADP iajs-1885 78 17 ɵ=1.5 ɵ=1.5 NOUN iajs-1885 78 18 200 200 NUM iajs-1885 78 19 100 100 NUM iajs-1885 78 20 50 50 NUM iajs-1885 78 21 10 10 NUM iajs-1885 78 22 n n NOUN iajs-1885 78 23 estimator estimator NOUN iajs-1885 78 24 0.001262 0.001262 NUM iajs-1885 78 25 0.0023160.0034440.001008 0.0023160.0034440.001008 NOUN iajs-1885 78 26 mle mle PROPN iajs-1885 78 27 0.001261 0.001261 PROPN iajs-1885 78 28 0.0023090.0033960.000911 0.0023090.0033960.000911 PROPN iajs-1885 78 29 jsqu jsqu PROPN iajs-1885 78 30 0.001260 0.001260 NUM iajs-1885 78 31 0.0022930.0032930.000865 0.0022930.0032930.000865 NUM iajs-1885 78 32 jmsq1 jmsq1 NOUN iajs-1885 78 33 r=1 r=1 NOUN iajs-1885 78 34 0.001257 0.001257 NUM iajs-1885 78 35 0.0022580.0030720.001262 0.0022580.0030720.001262 NUM iajs-1885 78 36 jmsq2 jmsq2 NOUN iajs-1885 78 37 r=3 r=3 PROPN iajs-1885 78 38 table table NOUN iajs-1885 78 39 3 3 NUM iajs-1885 78 40 . . 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PUNCT iajs-1885 94 1   X iajs-1885 95 1 from from ADP iajs-1885 95 2 table table NOUN iajs-1885 95 3 4 4 NUM iajs-1885 95 4 : : PUNCT iajs-1885 95 5 when when SCONJ iajs-1885 95 6 ɵ=1.5 ɵ=1.5 ADJ iajs-1885 95 7 for for ADP iajs-1885 95 8 ( ( PUNCT iajs-1885 95 9 n=10 n=10 NOUN iajs-1885 95 10 ) ) PUNCT iajs-1885 95 11 the the DET iajs-1885 95 12 bayes bayes PROPN iajs-1885 95 13 estimator estimator NOUN iajs-1885 95 14 under under ADP iajs-1885 95 15 square square ADJ iajs-1885 95 16 error error NOUN iajs-1885 95 17 loss loss NOUN iajs-1885 95 18 function function NOUN iajs-1885 95 19 with with ADP iajs-1885 95 20 gamma gamma PROPN iajs-1885 95 21 prior prior ADV iajs-1885 95 22 ( ( PUNCT iajs-1885 95 23 α=1 α=1 NOUN iajs-1885 95 24 , , PUNCT iajs-1885 95 25 β=1.2 β=1.2 NOUN iajs-1885 95 26 ) ) PUNCT iajs-1885 95 27 is be AUX iajs-1885 95 28 the the DET iajs-1885 95 29 best good ADJ iajs-1885 95 30 , , PUNCT iajs-1885 95 31 and and CCONJ iajs-1885 95 32 for for ADP iajs-1885 95 33 ( ( PUNCT iajs-1885 95 34 n=50,200 n=50,200 NOUN iajs-1885 95 35 ) ) PUNCT iajs-1885 95 36 the the DET iajs-1885 95 37 bayes bayes PROPN iajs-1885 95 38 estimator estimator NOUN iajs-1885 95 39 under under ADP iajs-1885 95 40 modified modify VERB iajs-1885 95 41 squared square VERB iajs-1885 95 42 error error NOUN iajs-1885 95 43 loss loss NOUN iajs-1885 95 44 function function NOUN iajs-1885 95 45 when when SCONJ iajs-1885 95 46 r=3 r=3 VERB iajs-1885 95 47 with with ADP iajs-1885 95 48 gamma gamma PROPN iajs-1885 95 49 prior prior ADV iajs-1885 95 50 ( ( PUNCT iajs-1885 95 51 α=0.3 α=0.3 ADV iajs-1885 95 52 , , PUNCT iajs-1885 95 53 β=3 β=3 PUNCT iajs-1885 95 54 ) ) PUNCT iajs-1885 95 55 is be AUX iajs-1885 95 56 the the DET iajs-1885 95 57 best good ADJ iajs-1885 95 58 and and CCONJ iajs-1885 95 59 for for ADP iajs-1885 95 60 ( ( PUNCT iajs-1885 95 61 n=100 n=100 NUM iajs-1885 95 62 ) ) PUNCT iajs-1885 95 63 the the DET iajs-1885 95 64 bayes bayes PROPN iajs-1885 95 65 estimator estimator NOUN iajs-1885 95 66 under under ADP iajs-1885 95 67 modified modify VERB iajs-1885 95 68 squared square VERB iajs-1885 95 69 error error NOUN iajs-1885 95 70 loss loss NOUN iajs-1885 95 71 function function NOUN iajs-1885 96 1 when when SCONJ iajs-1885 96 2 r=3 r=3 VERB iajs-1885 96 3 with with ADP iajs-1885 96 4 gamma gamma PROPN iajs-1885 96 5 prior prior ADV iajs-1885 96 6 ( ( PUNCT iajs-1885 96 7 α=1 α=1 X iajs-1885 96 8 , , PUNCT iajs-1885 96 9 β=3 β=3 NUM iajs-1885 96 10 ) ) PUNCT iajs-1885 96 11 is be AUX iajs-1885 96 12 the the DET iajs-1885 96 13 best good ADJ iajs-1885 96 14 . . PUNCT iajs-1885 97 1   PUNCT iajs-1885 97 2 according accord VERB iajs-1885 97 3 to to ADP iajs-1885 97 4 the the DET iajs-1885 97 5 simulation simulation NOUN iajs-1885 97 6 results result NOUN iajs-1885 97 7 we we PRON iajs-1885 97 8 conclude conclude VERB iajs-1885 97 9 that that SCONJ iajs-1885 97 10 the the DET iajs-1885 97 11 bayes bayes PROPN iajs-1885 97 12 estimator estimator NOUN iajs-1885 97 13 under under ADP iajs-1885 97 14 modified modify VERB iajs-1885 97 15 squared square VERB iajs-1885 97 16 error error NOUN iajs-1885 97 17 loss loss NOUN iajs-1885 97 18 function function NOUN iajs-1885 97 19 with with ADP iajs-1885 97 20 r=3 r=3 PROPN iajs-1885 97 21 is be AUX iajs-1885 97 22 superior superior ADJ iajs-1885 97 23 to to ADP iajs-1885 97 24 the the DET iajs-1885 97 25 performance performance NOUN iajs-1885 97 26 of of ADP iajs-1885 97 27 other other ADJ iajs-1885 97 28 estimators estimator NOUN iajs-1885 97 29 in in ADP iajs-1885 97 30 almost almost ADV iajs-1885 97 31 cases case NOUN iajs-1885 97 32 that that PRON iajs-1885 97 33 are be AUX iajs-1885 97 34 studied study VERB iajs-1885 97 35 in in ADP iajs-1885 97 36 this this DET iajs-1885 97 37 paper paper NOUN iajs-1885 97 38 , , PUNCT iajs-1885 97 39 where where SCONJ iajs-1885 97 40 the the DET iajs-1885 97 41 integrated integrate VERB iajs-1885 97 42 mean mean NOUN iajs-1885 97 43 squared square VERB iajs-1885 97 44 error error NOUN iajs-1885 97 45 ( ( PUNCT iajs-1885 97 46 imse imse NOUN iajs-1885 97 47 ) ) PUNCT iajs-1885 97 48 is be AUX iajs-1885 97 49 employed employ VERB iajs-1885 97 50 as as ADP iajs-1885 97 51 a a DET iajs-1885 97 52 criterion criterion NOUN iajs-1885 97 53 to to PART iajs-1885 97 54 assess assess VERB iajs-1885 97 55 the the DET iajs-1885 97 56 performance performance NOUN iajs-1885 97 57 of of ADP iajs-1885 97 58 such such ADJ iajs-1885 97 59 estimators estimator NOUN iajs-1885 97 60 . . PUNCT iajs-1885 97 61     SPACE iajs-1885 98 1 109 109 NUM iajs-1885 98 2     SPACE iajs-1885 98 3 ibn ibn PROPN iajs-1885 98 4 al al PROPN iajs-1885 98 5 - - PUNCT iajs-1885 98 6 haitham haitham PROPN iajs-1885 98 7 jour jour X iajs-1885 98 8 . . PROPN iajs-1885 98 9 for for ADP iajs-1885 98 10 pure pure ADJ iajs-1885 98 11 & & CCONJ iajs-1885 98 12 appl appl PROPN iajs-1885 98 13 . . PUNCT iajs-1885 99 1 sci sci PROPN iajs-1885 99 2 . . PROPN iajs-1885 100 1 ihjpas ihjpas PROPN iajs-1885 100 2 https://doi.org/10.30526/32.1.1885 https://doi.org/10.30526/32.1.1885 X iajs-1885 100 3 vol vol NOUN iajs-1885 100 4 . . PUNCT iajs-1885 101 1 32 32 NUM iajs-1885 101 2 ( ( PUNCT iajs-1885 101 3 1 1 NUM iajs-1885 101 4 ) ) PUNCT iajs-1885 101 5 2019 2019 NUM iajs-1885 101 6 references reference NOUN iajs-1885 101 7 1 1 NUM iajs-1885 101 8 . . PUNCT iajs-1885 101 9 meeker meeker PROPN iajs-1885 101 10 , , PUNCT iajs-1885 101 11 w.q w.q PROPN iajs-1885 101 12 . . PROPN iajs-1885 101 13 ; ; PUNCT iajs-1885 102 1 escobar escobar PROPN iajs-1885 102 2 , , PUNCT iajs-1885 102 3 l.u.statistical l.u.statistical ADJ iajs-1885 102 4 methods method NOUN iajs-1885 102 5 for for ADP iajs-1885 102 6 reliability reliability NOUN iajs-1885 102 7 data datum NOUN iajs-1885 102 8 , , PUNCT iajs-1885 102 9 john john PROPN iajs-1885 102 10 wiley wiley PROPN iajs-1885 102 11 & & CCONJ iajs-1885 102 12 sons sons PROPN iajs-1885 102 13 . . PUNCT iajs-1885 103 1 inc inc PROPN iajs-1885 103 2 . . PROPN iajs-1885 103 3 canada canada PROPN iajs-1885 103 4 . . PUNCT iajs-1885 104 1 1998 1998 NUM iajs-1885 104 2 . . PUNCT iajs-1885 105 1 2 2 X iajs-1885 105 2 . . X iajs-1885 105 3 pugh pugh PROPN iajs-1885 105 4 , , PUNCT iajs-1885 105 5 e.l e.l PROPN iajs-1885 105 6 . . PUNCT iajs-1885 106 1 the the DET iajs-1885 106 2 best good ADJ iajs-1885 106 3 estimation estimation NOUN iajs-1885 106 4 of of ADP iajs-1885 106 5 reliability reliability NOUN iajs-1885 106 6 in in ADP iajs-1885 106 7 the the DET iajs-1885 106 8 exponential exponential ADJ iajs-1885 106 9 case case NOUN iajs-1885 106 10 , , PUNCT iajs-1885 106 11 journal journal NOUN iajs-1885 106 12 of of ADP iajs-1885 106 13 operation operation NOUN iajs-1885 106 14 research research NOUN iajs-1885 106 15 . . PUNCT iajs-1885 107 1 1993,11 1993,11 NOUN iajs-1885 107 2 . . PUNCT iajs-1885 108 1 3 3 X iajs-1885 108 2 . . X iajs-1885 108 3 al al PROPN iajs-1885 108 4 - - PUNCT iajs-1885 108 5 timeemy timeemy PROPN iajs-1885 108 6 , , PUNCT iajs-1885 108 7 o.i o.i PROPN iajs-1885 108 8 . . PROPN iajs-1885 108 9 estimation estimation NOUN iajs-1885 108 10 of of ADP iajs-1885 108 11 reliability reliability NOUN iajs-1885 108 12 function function NOUN iajs-1885 108 13 for for ADP iajs-1885 108 14 inverse inverse NOUN iajs-1885 108 15 gaussian gaussian ADJ iajs-1885 108 16 distribution distribution NOUN iajs-1885 108 17 model model NOUN iajs-1885 108 18 with with ADP iajs-1885 108 19 application application NOUN iajs-1885 108 20 by by ADP iajs-1885 108 21 using use VERB iajs-1885 108 22 monte monte PROPN iajs-1885 108 23 carlo carlo PROPN iajs-1885 108 24 simulation simulation PROPN iajs-1885 108 25 , , PUNCT iajs-1885 108 26 a a DET iajs-1885 108 27 thesis thesis NOUN iajs-1885 108 28 submitted submit VERB iajs-1885 108 29 to to ADP iajs-1885 108 30 the the DET iajs-1885 108 31 college college NOUN iajs-1885 108 32 of of ADP iajs-1885 108 33 science science PROPN iajs-1885 108 34 alnahrain alnahrain PROPN iajs-1885 108 35 university university NOUN iajs-1885 108 36 for for ADP iajs-1885 108 37 the the DET iajs-1885 108 38 degree degree NOUN iajs-1885 108 39 of of ADP iajs-1885 108 40 master master NOUN iajs-1885 108 41 of of ADP iajs-1885 108 42 science science NOUN iajs-1885 108 43 in in ADP iajs-1885 108 44 applied applied ADJ iajs-1885 108 45 mathematics mathematic NOUN iajs-1885 108 46 . . 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PUNCT iajs-1885 110 1 2012 2012 NUM iajs-1885 110 2 , , PUNCT iajs-1885 110 3 6 6 NUM iajs-1885 110 4 , , PUNCT iajs-1885 110 5 1 1 NUM iajs-1885 110 6 , , PUNCT iajs-1885 110 7 113124 113124 NUM iajs-1885 110 8 . . PUNCT iajs-1885 111 1 5 5 X iajs-1885 111 2 . . X iajs-1885 111 3 rasheed rasheed NOUN iajs-1885 111 4 , , PUNCT iajs-1885 111 5 h.a h.a PROPN iajs-1885 111 6 . . PROPN iajs-1885 111 7 ; ; PUNCT iajs-1885 111 8 al al PROPN iajs-1885 111 9 - - PUNCT iajs-1885 111 10 gazi gazi PROPN iajs-1885 111 11 , , PUNCT iajs-1885 111 12 n.a.a n.a.a PROPN iajs-1885 111 13 . . 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