id sid tid token lemma pos iajs-1914 1 1 microsoft microsoft PROPN iajs-1914 1 2 word word NOUN iajs-1914 1 3 187 187 NUM iajs-1914 1 4 - - SYM iajs-1914 1 5 196 196 NUM iajs-1914 1 6     SPACE iajs-1914 1 7 187 187 NUM iajs-1914 1 8         SPACE iajs-1914 1 9 ibn ibn PROPN iajs-1914 1 10 al al PROPN iajs-1914 1 11 - - PUNCT iajs-1914 1 12 haitham haitham PROPN iajs-1914 1 13 jour.for jour.for PROPN iajs-1914 1 14 pure&appl.sci pure&appl.sci PROPN iajs-1914 1 15 . . PUNCT iajs-1914 2 1 ihjpas ihjpas PROPN iajs-1914 2 2 https://doi.org/10.30526/32.1.1914 https://doi.org/10.30526/32.1.1914 X iajs-1914 2 3 vol vol NOUN iajs-1914 2 4 . . PUNCT iajs-1914 3 1 32 32 NUM iajs-1914 3 2 ( ( PUNCT iajs-1914 3 3 1 1 NUM iajs-1914 3 4 ) ) PUNCT iajs-1914 3 5 2019 2019 NUM iajs-1914 3 6 bayesian bayesian NOUN iajs-1914 3 7 estimation estimation NOUN iajs-1914 3 8 for for ADP iajs-1914 3 9 two two NUM iajs-1914 3 10 parameters parameter NOUN iajs-1914 3 11 of of ADP iajs-1914 3 12 gamma gamma NOUN iajs-1914 3 13 distribution distribution NOUN iajs-1914 3 14 under under ADP iajs-1914 3 15 precautionary precautionary ADJ iajs-1914 3 16 loss loss NOUN iajs-1914 3 17 function function NOUN iajs-1914 3 18 loaiy loaiy PROPN iajs-1914 3 19 f. f. PROPN iajs-1914 3 20 naji naji PROPN iajs-1914 3 21 department department PROPN iajs-1914 3 22 of of ADP iajs-1914 3 23 mathematics mathematics PROPN iajs-1914 3 24 , , PUNCT iajs-1914 3 25 collage collage NOUN iajs-1914 3 26 of of ADP iajs-1914 3 27 science science NOUN iajs-1914 3 28 , , PUNCT iajs-1914 3 29 mustansiriyah mustansiriyah NOUN iajs-1914 3 30 university university NOUN iajs-1914 3 31 , , PUNCT iajs-1914 3 32 baghdad baghdad PROPN iajs-1914 3 33 , , PUNCT iajs-1914 3 34 iraq iraq PROPN iajs-1914 3 35 . . PUNCT iajs-1914 4 1 louy.faeq@gmail.com louy.faeq@gmail.com PROPN iajs-1914 4 2 huda huda PROPN iajs-1914 4 3 a. a. PROPN iajs-1914 4 4 rasheed rasheed PROPN iajs-1914 4 5 department department PROPN iajs-1914 4 6 of of ADP iajs-1914 4 7 mathematics mathematic NOUN iajs-1914 4 8 , , PUNCT iajs-1914 4 9 collage collage NOUN iajs-1914 4 10 of of ADP iajs-1914 4 11 science science NOUN iajs-1914 4 12 , , PUNCT iajs-1914 4 13 mustansiriyah mustansiriyah NOUN iajs-1914 4 14 university university NOUN iajs-1914 4 15 , , PUNCT iajs-1914 4 16 baghdad baghdad PROPN iajs-1914 4 17 , , PUNCT iajs-1914 4 18 iraq iraq PROPN iajs-1914 4 19 . . PUNCT iajs-1914 5 1 hudamath@uomustansiriyah.edu.iq hudamath@uomustansiriyah.edu.iq NOUN iajs-1914 5 2     SPACE iajs-1914 5 3 article article NOUN iajs-1914 5 4 history history NOUN iajs-1914 5 5 : : PUNCT iajs-1914 5 6 received receive VERB iajs-1914 5 7 29 29 NUM iajs-1914 5 8 july july PROPN iajs-1914 5 9 2018 2018 NUM iajs-1914 5 10 , , PUNCT iajs-1914 5 11 accepted accept VERB iajs-1914 5 12 26 26 NUM iajs-1914 5 13 november november PROPN iajs-1914 5 14 2018 2018 NUM iajs-1914 5 15 , , PUNCT iajs-1914 5 16 publish publish VERB iajs-1914 5 17 january january PROPN iajs-1914 5 18 2019 2019 NUM iajs-1914 5 19 abstract abstract ADV iajs-1914 5 20 in in ADP iajs-1914 5 21 the the DET iajs-1914 5 22 current current ADJ iajs-1914 5 23 study study NOUN iajs-1914 5 24 , , PUNCT iajs-1914 5 25 the the DET iajs-1914 5 26 researchers researcher NOUN iajs-1914 5 27 have have AUX iajs-1914 5 28 been be AUX iajs-1914 5 29 obtained obtain VERB iajs-1914 5 30 bayes bayes NOUN iajs-1914 5 31 estimators estimator NOUN iajs-1914 5 32 for for ADP iajs-1914 5 33 the the DET iajs-1914 5 34 shape shape NOUN iajs-1914 5 35 and and CCONJ iajs-1914 5 36 scale scale NOUN iajs-1914 5 37 parameters parameter NOUN iajs-1914 5 38 of of ADP iajs-1914 5 39 gamma gamma NOUN iajs-1914 5 40 distribution distribution NOUN iajs-1914 5 41 under under ADP iajs-1914 5 42 the the DET iajs-1914 5 43 precautionary precautionary ADJ iajs-1914 5 44 loss loss NOUN iajs-1914 5 45 function function NOUN iajs-1914 5 46 , , PUNCT iajs-1914 5 47 assuming assume VERB iajs-1914 5 48 the the DET iajs-1914 5 49 priors prior NOUN iajs-1914 5 50 , , PUNCT iajs-1914 5 51 represented represent VERB iajs-1914 5 52 by by ADP iajs-1914 5 53 gamma gamma NOUN iajs-1914 5 54 and and CCONJ iajs-1914 5 55 exponential exponential ADJ iajs-1914 5 56 priors prior NOUN iajs-1914 5 57 for for ADP iajs-1914 5 58 the the DET iajs-1914 5 59 shape shape NOUN iajs-1914 5 60 and and CCONJ iajs-1914 5 61 scale scale NOUN iajs-1914 5 62 parameters parameter NOUN iajs-1914 5 63 respectively respectively ADV iajs-1914 5 64 . . PUNCT iajs-1914 6 1 moment moment NOUN iajs-1914 6 2 , , PUNCT iajs-1914 6 3 maximum maximum ADJ iajs-1914 6 4 likelihood likelihood NOUN iajs-1914 6 5 estimators estimator NOUN iajs-1914 6 6 and and CCONJ iajs-1914 6 7 lindley lindley NOUN iajs-1914 6 8 ’s ’s PART iajs-1914 6 9 approximation approximation NOUN iajs-1914 6 10 have have AUX iajs-1914 6 11 been be AUX iajs-1914 6 12 used use VERB iajs-1914 6 13 effectively effectively ADV iajs-1914 6 14 in in ADP iajs-1914 6 15 bayesian bayesian NOUN iajs-1914 6 16 estimation estimation NOUN iajs-1914 6 17 . . PUNCT iajs-1914 7 1 based base VERB iajs-1914 7 2 on on ADP iajs-1914 7 3 monte monte PROPN iajs-1914 7 4 carlo carlo PROPN iajs-1914 7 5 simulation simulation PROPN iajs-1914 7 6 method method NOUN iajs-1914 7 7 , , PUNCT iajs-1914 7 8 those those DET iajs-1914 7 9 estimators estimator NOUN iajs-1914 7 10 are be AUX iajs-1914 7 11 compared compare VERB iajs-1914 7 12 depending depend VERB iajs-1914 7 13 on on ADP iajs-1914 7 14 the the DET iajs-1914 7 15 mean mean ADJ iajs-1914 7 16 squared square VERB iajs-1914 7 17 errors error NOUN iajs-1914 7 18 ( ( PUNCT iajs-1914 7 19 mse mse PROPN iajs-1914 7 20 ’s ’s PART iajs-1914 7 21 ) ) PUNCT iajs-1914 7 22 . . PUNCT iajs-1914 8 1 the the DET iajs-1914 8 2 results result NOUN iajs-1914 8 3 show show VERB iajs-1914 8 4 that that SCONJ iajs-1914 8 5 , , PUNCT iajs-1914 8 6 the the DET iajs-1914 8 7 performance performance NOUN iajs-1914 8 8 of of ADP iajs-1914 8 9 bayes bayes PROPN iajs-1914 8 10 estimator estimator NOUN iajs-1914 8 11 under under ADP iajs-1914 8 12 precautionary precautionary ADJ iajs-1914 8 13 loss loss NOUN iajs-1914 8 14 function function NOUN iajs-1914 8 15 with with ADP iajs-1914 8 16 gamma gamma NOUN iajs-1914 8 17 and and CCONJ iajs-1914 8 18 exponential exponential ADJ iajs-1914 8 19 priors prior NOUN iajs-1914 8 20 is be AUX iajs-1914 8 21 better well ADJ iajs-1914 8 22 than than ADP iajs-1914 8 23 other other ADJ iajs-1914 8 24 estimates estimate NOUN iajs-1914 8 25 in in ADP iajs-1914 8 26 all all DET iajs-1914 8 27 cases case NOUN iajs-1914 8 28 . . PUNCT iajs-1914 9 1 keywords keyword NOUN iajs-1914 9 2 : : PUNCT iajs-1914 9 3 gamma gamma NOUN iajs-1914 9 4 distribution distribution NOUN iajs-1914 9 5 ; ; PUNCT iajs-1914 9 6 maximum maximum ADJ iajs-1914 9 7 likelihood likelihood NOUN iajs-1914 9 8 estimator estimator NOUN iajs-1914 9 9 ; ; PUNCT iajs-1914 9 10 precautionary precautionary ADJ iajs-1914 9 11 loss loss NOUN iajs-1914 9 12 function function NOUN iajs-1914 9 13 ; ; PUNCT iajs-1914 9 14 exponential exponential NOUN iajs-1914 9 15 prior prior NOUN iajs-1914 9 16 ; ; PUNCT iajs-1914 9 17 lindley lindley PROPN iajs-1914 9 18 ’s ’s PART iajs-1914 9 19 approximation approximation NOUN iajs-1914 9 20 . . PUNCT iajs-1914 10 1 1 1 X iajs-1914 10 2 . . X iajs-1914 10 3 introduction introduction NOUN iajs-1914 10 4 the the DET iajs-1914 10 5 gamma gamma NOUN iajs-1914 10 6 distribution distribution NOUN iajs-1914 10 7 is be AUX iajs-1914 10 8 extremely extremely ADV iajs-1914 10 9 important important ADJ iajs-1914 10 10 in in ADP iajs-1914 10 11 reliability reliability NOUN iajs-1914 10 12 analysis analysis NOUN iajs-1914 10 13 and and CCONJ iajs-1914 10 14 life life NOUN iajs-1914 10 15 testing testing NOUN iajs-1914 10 16 . . PUNCT iajs-1914 11 1 hogg hogg PROPN iajs-1914 11 2 and and CCONJ iajs-1914 11 3 et et PROPN iajs-1914 11 4 al al PROPN iajs-1914 11 5 . . PROPN iajs-1914 12 1 ( ( PUNCT iajs-1914 12 2 2013 2013 NUM iajs-1914 12 3 ) ) PUNCT iajs-1914 12 4 , , PUNCT iajs-1914 12 5 showed show VERB iajs-1914 12 6 that that SCONJ iajs-1914 12 7 , , PUNCT iajs-1914 12 8 the the DET iajs-1914 12 9 gamma gamma NOUN iajs-1914 12 10 distribution distribution NOUN iajs-1914 12 11 is be AUX iajs-1914 12 12 not not PART iajs-1914 12 13 only only ADV iajs-1914 12 14 a a DET iajs-1914 12 15 good good ADJ iajs-1914 12 16 model model NOUN iajs-1914 12 17 for for ADP iajs-1914 12 18 waiting waiting NOUN iajs-1914 12 19 times time NOUN iajs-1914 12 20 , , PUNCT iajs-1914 12 21 but but CCONJ iajs-1914 12 22 one one NUM iajs-1914 12 23 for for ADP iajs-1914 12 24 many many ADJ iajs-1914 12 25 nonnegative nonnegative ADJ iajs-1914 12 26 random random ADJ iajs-1914 12 27 variables variable NOUN iajs-1914 12 28 of of ADP iajs-1914 12 29 the the DET iajs-1914 12 30 continuous continuous ADJ iajs-1914 12 31 type type NOUN iajs-1914 12 32 [ [ X iajs-1914 12 33 1 1 NUM iajs-1914 12 34 ] ] PUNCT iajs-1914 12 35 . . PUNCT iajs-1914 13 1 also also ADV iajs-1914 13 2 , , PUNCT iajs-1914 13 3 it it PRON iajs-1914 13 4 is be AUX iajs-1914 13 5 a a DET iajs-1914 13 6 flexible flexible ADJ iajs-1914 13 7 distribution distribution NOUN iajs-1914 13 8 that that PRON iajs-1914 13 9 commonly commonly ADV iajs-1914 13 10 offers offer VERB iajs-1914 13 11 a a DET iajs-1914 13 12 good good ADJ iajs-1914 13 13 fit fit NOUN iajs-1914 13 14 to to ADP iajs-1914 13 15 any any DET iajs-1914 13 16 variable variable NOUN iajs-1914 13 17 such such ADJ iajs-1914 13 18 as as ADP iajs-1914 13 19 in in ADP iajs-1914 13 20 environmental environmental ADJ iajs-1914 13 21 , , PUNCT iajs-1914 13 22 meteorology meteorology NOUN iajs-1914 13 23 , , PUNCT iajs-1914 13 24 climatology climatology NOUN iajs-1914 13 25 and and CCONJ iajs-1914 13 26 other other ADJ iajs-1914 13 27 physical physical ADJ iajs-1914 13 28 situations situation NOUN iajs-1914 13 29 [ [ X iajs-1914 13 30 2 2 NUM iajs-1914 13 31 ] ] PUNCT iajs-1914 13 32 . . PUNCT iajs-1914 14 1 the the DET iajs-1914 14 2 probability probability NOUN iajs-1914 14 3 density density NOUN iajs-1914 14 4 function function NOUN iajs-1914 14 5 of of ADP iajs-1914 14 6 the the DET iajs-1914 14 7 gamma gamma NOUN iajs-1914 14 8 distribution distribution NOUN iajs-1914 14 9 is be AUX iajs-1914 14 10 defined define VERB iajs-1914 14 11 as as SCONJ iajs-1914 14 12 follows follow VERB iajs-1914 14 13 [ [ X iajs-1914 14 14 3 3 X iajs-1914 14 15 ] ] PUNCT iajs-1914 14 16 𝑓 𝑓 PRON iajs-1914 14 17 𝑥 𝑥 NOUN iajs-1914 14 18 ; ; PUNCT iajs-1914 14 19 α α X iajs-1914 14 20 , , PUNCT iajs-1914 14 21 β β X iajs-1914 14 22 γ γ X iajs-1914 14 23 ; ; PUNCT iajs-1914 14 24 x x SYM iajs-1914 14 25 > > X iajs-1914 14 26 0 0 NUM iajs-1914 14 27 , , PUNCT iajs-1914 14 28 α α X iajs-1914 14 29 > > X iajs-1914 14 30 0 0 PUNCT iajs-1914 14 31 , , PUNCT iajs-1914 14 32 β β X iajs-1914 14 33 > > X iajs-1914 14 34 0 0 PUNCT iajs-1914 15 1 ( ( PUNCT iajs-1914 15 2 1 1 NUM iajs-1914 15 3 ) ) PUNCT iajs-1914 15 4 where where SCONJ iajs-1914 15 5 , , PUNCT iajs-1914 15 6 α α PROPN iajs-1914 15 7 and and CCONJ iajs-1914 15 8 β β X iajs-1914 15 9 are be AUX iajs-1914 15 10 often often ADV iajs-1914 15 11 called call VERB iajs-1914 15 12 the the DET iajs-1914 15 13 shape shape NOUN iajs-1914 15 14 and and CCONJ iajs-1914 15 15 scale scale NOUN iajs-1914 15 16 parameters parameter NOUN iajs-1914 15 17 , , PUNCT iajs-1914 15 18 respectively respectively ADV iajs-1914 15 19 . . PUNCT iajs-1914 16 1 the the DET iajs-1914 16 2 gamma gamma PROPN iajs-1914 16 3 function function NOUN iajs-1914 16 4 is be AUX iajs-1914 16 5 γ γ X iajs-1914 16 6 𝛼 𝛼 PART iajs-1914 16 7 𝑥 𝑥 NOUN iajs-1914 16 8 𝑒 𝑒 PROPN iajs-1914 16 9 ∞ ∞ NOUN iajs-1914 16 10 𝑑𝑥 𝑑𝑥 VERB iajs-1914 16 11 , , PUNCT iajs-1914 16 12 for for ADP iajs-1914 16 13 α α NOUN iajs-1914 16 14 0 0 PUNCT iajs-1914 17 1 the the DET iajs-1914 17 2 cumulative cumulative ADJ iajs-1914 17 3 distribution distribution NOUN iajs-1914 17 4 function function NOUN iajs-1914 17 5 ( ( PUNCT iajs-1914 17 6 cdf cdf PROPN iajs-1914 17 7 ) ) PUNCT iajs-1914 17 8 is be AUX iajs-1914 17 9 f f PROPN iajs-1914 17 10 ( ( PUNCT iajs-1914 17 11 𝑥 𝑥 NOUN iajs-1914 17 12 ; ; PUNCT iajs-1914 17 13 α α NOUN iajs-1914 17 14 , , PUNCT iajs-1914 17 15 β β NOUN iajs-1914 17 16 ) ) PUNCT iajs-1914 17 17 = = PUNCT iajs-1914 17 18 γ γ X iajs-1914 17 19 𝑢 𝑢 PROPN iajs-1914 17 20 𝑒 𝑒 ADP iajs-1914 17 21 𝑑𝑢 𝑑𝑢 PRON iajs-1914 17 22     SPACE iajs-1914 17 23 188 188 NUM iajs-1914 17 24         SPACE iajs-1914 17 25 ibn ibn PROPN iajs-1914 17 26 al al PROPN iajs-1914 17 27 - - PUNCT iajs-1914 17 28 haitham haitham PROPN iajs-1914 17 29 jour.for jour.for PROPN iajs-1914 17 30 pure&appl.sci pure&appl.sci PROPN iajs-1914 17 31 . . PUNCT iajs-1914 18 1 ihjpas ihjpas PROPN iajs-1914 18 2 https://doi.org/10.30526/32.1.1914 https://doi.org/10.30526/32.1.1914 X iajs-1914 18 3 vol vol NOUN iajs-1914 18 4 . . PUNCT iajs-1914 19 1 32 32 NUM iajs-1914 19 2 ( ( PUNCT iajs-1914 19 3 1 1 NUM iajs-1914 19 4 ) ) PUNCT iajs-1914 19 5 2019 2019 NUM iajs-1914 20 1 this this DET iajs-1914 20 2 function function NOUN iajs-1914 20 3 is be AUX iajs-1914 20 4 called call VERB iajs-1914 20 5 incomplete incomplete ADJ iajs-1914 20 6 gamma gamma NOUN iajs-1914 20 7 function function NOUN iajs-1914 20 8 . . PUNCT iajs-1914 21 1 the the DET iajs-1914 21 2 formula formula NOUN iajs-1914 21 3 for for ADP iajs-1914 21 4 the the DET iajs-1914 21 5 cumulative cumulative ADJ iajs-1914 21 6 distribution distribution NOUN iajs-1914 21 7 can can AUX iajs-1914 21 8 be be AUX iajs-1914 21 9 written write VERB iajs-1914 21 10 as as ADP iajs-1914 21 11 f f PROPN iajs-1914 21 12 ( ( PUNCT iajs-1914 21 13 𝑥 𝑥 NOUN iajs-1914 21 14 ; ; PUNCT iajs-1914 21 15 α α NOUN iajs-1914 21 16 , , PUNCT iajs-1914 21 17 β β X iajs-1914 21 18 ) ) PUNCT iajs-1914 21 19 = = SYM iajs-1914 22 1 1 1 NUM iajs-1914 22 2 ∑ ∑ PUNCT iajs-1914 22 3 ! ! PUNCT iajs-1914 23 1 𝑒 𝑒 PROPN iajs-1914 23 2 ∑ ∑ PROPN iajs-1914 23 3 ! ! PUNCT iajs-1914 24 1 𝑒∞ 𝑒∞ PROPN iajs-1914 24 2 therefore therefore ADV iajs-1914 24 3 , , PUNCT iajs-1914 24 4 the the DET iajs-1914 24 5 reliability reliability NOUN iajs-1914 24 6 functions function NOUN iajs-1914 24 7 for for ADP iajs-1914 24 8 γ γ PRON iajs-1914 24 9 𝛼 𝛼 PROPN iajs-1914 24 10 , , PUNCT iajs-1914 24 11 𝛽 𝛽 NOUN iajs-1914 24 12 is be AUX iajs-1914 24 13 [ [ X iajs-1914 24 14 3 3 NUM iajs-1914 24 15 ] ] X iajs-1914 24 16 : : PUNCT iajs-1914 24 17 𝑅 𝑅 PROPN iajs-1914 24 18 𝑥 𝑥 PROPN iajs-1914 24 19 ; ; PUNCT iajs-1914 24 20 α α X iajs-1914 24 21 , , PUNCT iajs-1914 24 22 β β X iajs-1914 24 23 ∑ ∑ PUNCT iajs-1914 24 24 ! ! PUNCT iajs-1914 25 1 𝑒 𝑒 PROPN iajs-1914 25 2 2 2 NUM iajs-1914 25 3 . . PUNCT iajs-1914 25 4 estimation estimation NOUN iajs-1914 25 5 methods method NOUN iajs-1914 25 6 in in ADP iajs-1914 25 7 this this DET iajs-1914 25 8 paper paper NOUN iajs-1914 25 9 , , PUNCT iajs-1914 25 10 the the DET iajs-1914 25 11 moment moment NOUN iajs-1914 25 12 estimators estimator NOUN iajs-1914 25 13 are be AUX iajs-1914 25 14 used use VERB iajs-1914 25 15 as as ADP iajs-1914 25 16 primary primary ADJ iajs-1914 25 17 estimators estimator NOUN iajs-1914 25 18 for for ADP iajs-1914 25 19 maximum maximum ADJ iajs-1914 25 20 likelihood likelihood NOUN iajs-1914 25 21 estimators estimator NOUN iajs-1914 25 22 of of ADP iajs-1914 25 23 each each PRON iajs-1914 25 24 of of ADP iajs-1914 25 25 α α NOUN iajs-1914 25 26 and and CCONJ iajs-1914 25 27 β.on β.on ADP iajs-1914 25 28 the the DET iajs-1914 25 29 other other ADJ iajs-1914 25 30 hand hand NOUN iajs-1914 25 31 , , PUNCT iajs-1914 25 32 the the DET iajs-1914 25 33 maximum maximum ADJ iajs-1914 25 34 likelihood likelihood NOUN iajs-1914 25 35 estimators estimator NOUN iajs-1914 25 36 are be AUX iajs-1914 25 37 used use VERB iajs-1914 25 38 as as ADP iajs-1914 25 39 initial initial ADJ iajs-1914 25 40 values value NOUN iajs-1914 25 41 for for ADP iajs-1914 25 42 bayesian bayesian NOUN iajs-1914 25 43 estimators estimator NOUN iajs-1914 25 44 . . PUNCT iajs-1914 26 1 2.1 2.1 NUM iajs-1914 26 2 . . PUNCT iajs-1914 26 3 moment moment NOUN iajs-1914 26 4 method method NOUN iajs-1914 26 5 suppose suppose VERB iajs-1914 26 6 that that SCONJ iajs-1914 26 7 , , PUNCT iajs-1914 26 8 x x PRON iajs-1914 26 9 be be AUX iajs-1914 26 10 a a DET iajs-1914 26 11 random random ADJ iajs-1914 26 12 variable variable NOUN iajs-1914 26 13 has have VERB iajs-1914 26 14 a a DET iajs-1914 26 15 gamma gamma NOUN iajs-1914 26 16 distribution distribution NOUN iajs-1914 26 17 defined define VERB iajs-1914 26 18 by by ADP iajs-1914 26 19 ( ( PUNCT iajs-1914 26 20 1 1 NUM iajs-1914 26 21 ) ) PUNCT iajs-1914 26 22 . . PUNCT iajs-1914 27 1 let let VERB iajs-1914 27 2 x1 x1 NUM iajs-1914 27 3 , , PUNCT iajs-1914 27 4 x2 x2 PROPN iajs-1914 27 5 , , PUNCT iajs-1914 27 6 . . PUNCT iajs-1914 27 7 . . PUNCT iajs-1914 28 1 . . PUNCT iajs-1914 29 1 , , PUNCT iajs-1914 29 2 xn xn PROPN iajs-1914 29 3 be be AUX iajs-1914 29 4 a a DET iajs-1914 29 5 random random ADJ iajs-1914 29 6 sample sample NOUN iajs-1914 29 7 of of ADP iajs-1914 29 8 size size NOUN iajs-1914 29 9 n n CCONJ iajs-1914 29 10 from from ADP iajs-1914 29 11 x. x. NOUN iajs-1914 29 12 defining define VERB iajs-1914 29 13 the the DET iajs-1914 29 14 first first ADJ iajs-1914 29 15 k k PROPN iajs-1914 29 16 sample sample NOUN iajs-1914 29 17 moments moment NOUN iajs-1914 29 18 about about ADP iajs-1914 29 19 origin origin NOUN iajs-1914 29 20 as as ADP iajs-1914 29 21 𝑚′ 𝑚′ NUM iajs-1914 29 22 ∑ ∑ SYM iajs-1914 29 23 𝑥 𝑥 X iajs-1914 29 24 ` ` PUNCT iajs-1914 29 25 , , PUNCT iajs-1914 29 26 r r NOUN iajs-1914 29 27 = = SYM iajs-1914 29 28 1 1 NUM iajs-1914 29 29 , , PUNCT iajs-1914 29 30 2 2 NUM iajs-1914 29 31 , , PUNCT iajs-1914 29 32 . . PUNCT iajs-1914 29 33 . . PUNCT iajs-1914 30 1 . . PUNCT iajs-1914 31 1 , , PUNCT iajs-1914 31 2 k. k. PROPN iajs-1914 31 3 the the DET iajs-1914 31 4 first first ADJ iajs-1914 31 5 k k PROPN iajs-1914 31 6 population population NOUN iajs-1914 31 7 moments moment NOUN iajs-1914 31 8 about about ADP iajs-1914 31 9 origin origin NOUN iajs-1914 31 10 are be AUX iajs-1914 31 11 given give VERB iajs-1914 31 12 by by ADP iajs-1914 31 13 𝜇′ 𝜇′ ADJ iajs-1914 31 14 𝐸 𝐸 PROPN iajs-1914 31 15 𝑋 𝑋 NOUN iajs-1914 31 16 . . PUNCT iajs-1914 32 1 now now ADV iajs-1914 32 2 , , PUNCT iajs-1914 32 3 equaling equal VERB iajs-1914 32 4 these these DET iajs-1914 32 5 moments moment NOUN iajs-1914 32 6 , , PUNCT iajs-1914 32 7 that that PRON iajs-1914 32 8 is be AUX iajs-1914 32 9 𝜇 𝜇 ADP iajs-1914 32 10 ′ ′ NUM iajs-1914 32 11 𝑚′ 𝑚′ NUM iajs-1914 32 12 , , PUNCT iajs-1914 32 13 r r NOUN iajs-1914 32 14 = = SYM iajs-1914 32 15 1 1 NUM iajs-1914 32 16 , , PUNCT iajs-1914 32 17 2 2 NUM iajs-1914 32 18 , , PUNCT iajs-1914 32 19 . . PUNCT iajs-1914 32 20 . . PUNCT iajs-1914 33 1 . . PUNCT iajs-1914 34 1 , , PUNCT iajs-1914 34 2 k k PROPN iajs-1914 34 3 the the DET iajs-1914 34 4 solutions solution NOUN iajs-1914 34 5 to to ADP iajs-1914 34 6 the the DET iajs-1914 34 7 above above ADJ iajs-1914 34 8 equations equation NOUN iajs-1914 34 9 denote denote VERB iajs-1914 34 10 by by ADP iajs-1914 34 11 𝜃^ 𝜃^ PROPN iajs-1914 34 12 , , PUNCT iajs-1914 34 13 𝜃^ 𝜃^ ADJ iajs-1914 34 14 , , PUNCT iajs-1914 34 15 … … PUNCT iajs-1914 34 16 , , PUNCT iajs-1914 34 17 𝜃^ 𝜃^ ADJ iajs-1914 34 18 , , PUNCT iajs-1914 34 19 yields yield VERB iajs-1914 34 20 the the DET iajs-1914 34 21 moment moment NOUN iajs-1914 34 22 estimators estimator NOUN iajs-1914 34 23 of of ADP iajs-1914 34 24 θ1,θ2 θ1,θ2 PROPN iajs-1914 34 25 , , PUNCT iajs-1914 34 26 . . PUNCT iajs-1914 34 27 . . PUNCT iajs-1914 35 1 . . PUNCT iajs-1914 36 1 , , PUNCT iajs-1914 36 2 θk θk X iajs-1914 37 1 [ [ X iajs-1914 37 2 4 4 X iajs-1914 37 3 ] ] PUNCT iajs-1914 37 4 the the DET iajs-1914 37 5 moment moment NOUN iajs-1914 37 6 method method NOUN iajs-1914 37 7 for for ADP iajs-1914 37 8 estimating estimate VERB iajs-1914 37 9 the the DET iajs-1914 37 10 two two NUM iajs-1914 37 11 – – PUNCT iajs-1914 37 12 parameter parameter NOUN iajs-1914 37 13 gamma gamma NOUN iajs-1914 37 14 distribution distribution NOUN iajs-1914 37 15 can can AUX iajs-1914 37 16 be be AUX iajs-1914 37 17 derived derive VERB iajs-1914 37 18 as as ADP iajs-1914 37 19 𝑚 𝑚 PROPN iajs-1914 37 20 ∑ ∑ PUNCT iajs-1914 37 21 � � PROPN iajs-1914 37 22 ̅ ̅ NOUN iajs-1914 37 23 � � NOUN iajs-1914 37 24 𝑚 𝑚 NOUN iajs-1914 37 25 ∑ ∑ ADV iajs-1914 37 26 μ′ μ′ X iajs-1914 37 27 e e NOUN iajs-1914 37 28 x x X iajs-1914 37 29 α α PRON iajs-1914 37 30 β β X iajs-1914 37 31 𝜇′ 𝜇′ X iajs-1914 37 32 𝐸 𝐸 PROPN iajs-1914 37 33 𝑋 𝑋 PROPN iajs-1914 37 34 α α PROPN iajs-1914 37 35 β β X iajs-1914 37 36 α α X iajs-1914 37 37 β β NOUN iajs-1914 37 38 from from ADP iajs-1914 37 39 𝑚 𝑚 PROPN iajs-1914 37 40 𝜇′ 𝜇′ ADV iajs-1914 37 41 , , PUNCT iajs-1914 37 42 𝑚 𝑚 ADP iajs-1914 37 43 𝜇′ 𝜇′ ADV iajs-1914 37 44 , , PUNCT iajs-1914 37 45 we we PRON iajs-1914 37 46 get get VERB iajs-1914 37 47 𝛼 𝛼 PRON iajs-1914 37 48 ̅ ̅ NOUN iajs-1914 37 49 ∑ ∑ NOUN iajs-1914 37 50 ̅ ̅ X iajs-1914 37 51 ( ( PUNCT iajs-1914 37 52 2 2 NUM iajs-1914 37 53 ) ) PUNCT iajs-1914 37 54 𝛽 𝛽 NOUN iajs-1914 37 55 ̅ ̅ NOUN iajs-1914 37 56 ∑ ∑ PART iajs-1914 37 57 ̅ ̅ X iajs-1914 37 58 ( ( PUNCT iajs-1914 37 59 3 3 NUM iajs-1914 37 60 ) ) PUNCT iajs-1914 37 61 2.2 2.2 NUM iajs-1914 37 62 . . PUNCT iajs-1914 38 1 maximum maximum ADJ iajs-1914 38 2 likelihood likelihood NOUN iajs-1914 38 3 method method NOUN iajs-1914 38 4 the the DET iajs-1914 38 5 maximum maximum ADJ iajs-1914 38 6 likelihood likelihood NOUN iajs-1914 38 7 method method NOUN iajs-1914 38 8 is be AUX iajs-1914 38 9 one one NUM iajs-1914 38 10 of of ADP iajs-1914 38 11 the the DET iajs-1914 38 12 best good ADJ iajs-1914 38 13 methods method NOUN iajs-1914 38 14 of of ADP iajs-1914 38 15 obtaining obtain VERB iajs-1914 38 16 a a DET iajs-1914 38 17 point point NOUN iajs-1914 38 18 estimator estimator NOUN iajs-1914 38 19 of of ADP iajs-1914 38 20 a a DET iajs-1914 38 21 parameter parameter NOUN iajs-1914 38 22 . . PUNCT iajs-1914 39 1 this this DET iajs-1914 39 2 technique technique NOUN iajs-1914 39 3 was be AUX iajs-1914 39 4 proposed propose VERB iajs-1914 39 5 by by ADP iajs-1914 39 6 r.a r.a PROPN iajs-1914 39 7 . . PROPN iajs-1914 39 8 fisher fisher PROPN iajs-1914 39 9 ( ( PUNCT iajs-1914 39 10 1912 1912 NUM iajs-1914 39 11 ) ) PUNCT iajs-1914 39 12 , , PUNCT iajs-1914 39 13 and and CCONJ iajs-1914 39 14 he he PRON iajs-1914 39 15 developed develop VERB iajs-1914 39 16 it it PRON iajs-1914 39 17 in in ADP iajs-1914 39 18 1920s 1920 NOUN iajs-1914 39 19 [ [ X iajs-1914 39 20 5 5 NUM iajs-1914 39 21 ] ] PUNCT iajs-1914 39 22 . . PUNCT iajs-1914 39 23     SPACE iajs-1914 40 1 189 189 NUM iajs-1914 40 2         SPACE iajs-1914 40 3 ibn ibn PROPN iajs-1914 40 4 al al PROPN iajs-1914 40 5 - - PUNCT iajs-1914 40 6 haitham haitham PROPN iajs-1914 40 7 jour.for jour.for PROPN iajs-1914 40 8 pure&appl.sci pure&appl.sci PROPN iajs-1914 40 9 . . PUNCT iajs-1914 40 10 ihjpas ihjpas PROPN iajs-1914 40 11 https://doi.org/10.30526/32.1.1914 https://doi.org/10.30526/32.1.1914 X iajs-1914 40 12 vol vol NOUN iajs-1914 40 13 . . PUNCT iajs-1914 41 1 32 32 NUM iajs-1914 41 2 ( ( PUNCT iajs-1914 41 3 1 1 NUM iajs-1914 41 4 ) ) PUNCT iajs-1914 41 5 2019 2019 NUM iajs-1914 42 1 this this DET iajs-1914 42 2 method method NOUN iajs-1914 42 3 is be AUX iajs-1914 42 4 the the DET iajs-1914 42 5 most most ADV iajs-1914 42 6 popular popular ADJ iajs-1914 42 7 procedure procedure NOUN iajs-1914 42 8 in in ADP iajs-1914 42 9 estimating estimate VERB iajs-1914 42 10 the the DET iajs-1914 42 11 parameter parameter NOUN iajs-1914 42 12   X iajs-1914 42 13 which which PRON iajs-1914 42 14 specifies specify VERB iajs-1914 42 15 a a DET iajs-1914 42 16 probability probability NOUN iajs-1914 42 17 function function NOUN iajs-1914 42 18 f(x f(x PROPN iajs-1914 42 19 , , PUNCT iajs-1914 42 20 𝜃 𝜃 PRON iajs-1914 42 21 , , PUNCT iajs-1914 42 22 based base VERB iajs-1914 42 23 on on ADP iajs-1914 42 24 the the DET iajs-1914 42 25 observations observation NOUN iajs-1914 42 26 𝑥 𝑥 X iajs-1914 42 27 , , PUNCT iajs-1914 42 28 𝑥 𝑥 X iajs-1914 42 29 , , PUNCT iajs-1914 42 30 … … PUNCT iajs-1914 42 31 , , PUNCT iajs-1914 42 32 𝑥 𝑥 PRON iajs-1914 42 33 which which PRON iajs-1914 42 34 were be AUX iajs-1914 42 35 independent independent ADJ iajs-1914 42 36 sample sample NOUN iajs-1914 42 37 from from ADP iajs-1914 42 38 the the DET iajs-1914 42 39 distribution distribution NOUN iajs-1914 42 40 . . PUNCT iajs-1914 43 1 the the DET iajs-1914 43 2 maximum maximum ADJ iajs-1914 43 3 likelihood likelihood NOUN iajs-1914 43 4 estimator estimator NOUN iajs-1914 43 5 ^ ^ PUNCT iajs-1914 43 6   X iajs-1914 43 7 of of ADP iajs-1914 43 8 the the DET iajs-1914 43 9 parameter parameter NOUN iajs-1914 43 10   PROPN iajs-1914 43 11 which which PRON iajs-1914 43 12 maximizes maximize VERB iajs-1914 43 13 the the DET iajs-1914 43 14 likelihood likelihood NOUN iajs-1914 43 15 function function NOUN iajs-1914 43 16 will will AUX iajs-1914 43 17 be be AUX iajs-1914 43 18 as as SCONJ iajs-1914 43 19 follows follow VERB iajs-1914 43 20 [ [ X iajs-1914 43 21 6 6 NUM iajs-1914 43 22 ] ] X iajs-1914 43 23 𝐿 𝐿 PROPN iajs-1914 43 24 𝑥 𝑥 PROPN iajs-1914 43 25 , , PUNCT iajs-1914 43 26 𝑥 𝑥 X iajs-1914 43 27 , , PUNCT iajs-1914 43 28 … … PUNCT iajs-1914 43 29 , , PUNCT iajs-1914 43 30 𝑥 𝑥 X iajs-1914 43 31 ; ; PUNCT iajs-1914 43 32 𝜃 𝜃 X iajs-1914 43 33 𝜋 𝜋 NOUN iajs-1914 43 34 𝑓 𝑓 PRON iajs-1914 43 35 𝑥 𝑥 NOUN iajs-1914 43 36 ; ; PUNCT iajs-1914 43 37 𝜃 𝜃 X iajs-1914 43 38 the the DET iajs-1914 43 39 likelihood likelihood NOUN iajs-1914 43 40 function function NOUN iajs-1914 43 41 for for ADP iajs-1914 43 42 two two NUM iajs-1914 43 43 - - PUNCT iajs-1914 43 44 parameter parameter NOUN iajs-1914 43 45 gamma gamma NOUN iajs-1914 43 46 distribution distribution NOUN iajs-1914 43 47 is be AUX iajs-1914 43 48 𝐿 𝐿 PROPN iajs-1914 43 49 𝑥 𝑥 PROPN iajs-1914 43 50 , , PUNCT iajs-1914 43 51 𝑥 𝑥 X iajs-1914 43 52 , , PUNCT iajs-1914 43 53 … … PUNCT iajs-1914 43 54 , , PUNCT iajs-1914 43 55 𝑥 𝑥 X iajs-1914 43 56 ; ; PUNCT iajs-1914 43 57 𝛼 𝛼 X iajs-1914 43 58 , , PUNCT iajs-1914 43 59 𝛽 𝛽 NOUN iajs-1914 43 60 г г NOUN iajs-1914 43 61 𝜋 𝜋 NOUN iajs-1914 43 62 𝑥 𝑥 NOUN iajs-1914 43 63 𝑒 𝑒 PROPN iajs-1914 43 64 ∑ ∑ PUNCT iajs-1914 43 65 ( ( PUNCT iajs-1914 43 66 4 4 X iajs-1914 43 67 ) ) PUNCT iajs-1914 43 68 taking take VERB iajs-1914 43 69 the the DET iajs-1914 43 70 logarithm logarithm NOUN iajs-1914 43 71 for for ADP iajs-1914 43 72 ( ( PUNCT iajs-1914 43 73 4 4 NUM iajs-1914 43 74 ) ) PUNCT iajs-1914 43 75 , , PUNCT iajs-1914 43 76 yields yield VERB iajs-1914 43 77 ln ln PROPN iajs-1914 43 78 l l PROPN iajs-1914 43 79 nlnг nlnг NOUN iajs-1914 44 1 α α PRON iajs-1914 44 2 nαlnβ nαlnβ VERB iajs-1914 44 3 𝛼 𝛼 SYM iajs-1914 44 4 1 1 NUM iajs-1914 44 5 ∑ ∑ ADP iajs-1914 44 6 𝑙𝑛𝑥 𝑙𝑛𝑥 VERB iajs-1914 44 7 𝛽 𝛽 PROPN iajs-1914 44 8 ∑ ∑ ADP iajs-1914 44 9 𝑥 𝑥 PROPN iajs-1914 44 10 the the DET iajs-1914 44 11 parameters parameter NOUN iajs-1914 44 12 that that PRON iajs-1914 44 13 maximize maximize VERB iajs-1914 44 14 the the DET iajs-1914 44 15 likelihood likelihood NOUN iajs-1914 44 16 function function NOUN iajs-1914 44 17 are be AUX iajs-1914 44 18 the the DET iajs-1914 44 19 solution solution NOUN iajs-1914 44 20 of of ADP iajs-1914 44 21 the the DET iajs-1914 44 22 equations equation NOUN iajs-1914 44 23 = = SYM iajs-1914 44 24 𝑛𝛹 𝑛𝛹 PROPN iajs-1914 44 25 𝛼 𝛼 PROPN iajs-1914 44 26 𝑛𝑙𝑛𝛽 𝑛𝑙𝑛𝛽 NOUN iajs-1914 44 27 ∑ ∑ PUNCT iajs-1914 44 28 𝑙𝑛𝑥 𝑙𝑛𝑥 X iajs-1914 44 29 ( ( PUNCT iajs-1914 44 30 5 5 NUM iajs-1914 44 31 ) ) PUNCT iajs-1914 44 32 ∑ ∑ NOUN iajs-1914 44 33 𝑥 𝑥 PROPN iajs-1914 44 34 ( ( PUNCT iajs-1914 44 35 6 6 NUM iajs-1914 44 36 ) ) PUNCT iajs-1914 44 37 observe observe VERB iajs-1914 44 38 that that SCONJ iajs-1914 44 39 , , PUNCT iajs-1914 44 40 the the DET iajs-1914 44 41 two two NUM iajs-1914 44 42 equations equation NOUN iajs-1914 44 43 ( ( PUNCT iajs-1914 44 44 5 5 NUM iajs-1914 44 45 ) ) PUNCT iajs-1914 44 46 and and CCONJ iajs-1914 44 47 ( ( PUNCT iajs-1914 44 48 6 6 NUM iajs-1914 44 49 ) ) PUNCT iajs-1914 44 50 are be AUX iajs-1914 44 51 difficult difficult ADJ iajs-1914 44 52 and and CCONJ iajs-1914 44 53 complicated complicated ADJ iajs-1914 44 54 to to PART iajs-1914 44 55 solve solve VERB iajs-1914 44 56 , , PUNCT iajs-1914 44 57 then then ADV iajs-1914 44 58 it it PRON iajs-1914 44 59 is be AUX iajs-1914 44 60 impossible impossible ADJ iajs-1914 44 61 to to PART iajs-1914 44 62 find find VERB iajs-1914 44 63 mle mle NOUN iajs-1914 44 64 for for ADP iajs-1914 44 65   NOUN iajs-1914 44 66 and and CCONJ iajs-1914 44 67   PROPN iajs-1914 44 68 analytically analytically ADV iajs-1914 44 69 , , PUNCT iajs-1914 44 70 we we PRON iajs-1914 44 71 can can AUX iajs-1914 44 72 use use VERB iajs-1914 44 73 the the DET iajs-1914 44 74 numerical numerical ADJ iajs-1914 44 75 analysis analysis NOUN iajs-1914 44 76 ( ( PUNCT iajs-1914 44 77 numerical numerical ADJ iajs-1914 44 78 procedure procedure NOUN iajs-1914 44 79 ) ) PUNCT iajs-1914 44 80 to to PART iajs-1914 44 81 obtain obtain VERB iajs-1914 44 82 and and CCONJ iajs-1914 44 83 estimate estimate VERB iajs-1914 44 84   NOUN iajs-1914 44 85 and and CCONJ iajs-1914 44 86   NOUN iajs-1914 44 87 that that PRON iajs-1914 44 88 maximize maximize VERB iajs-1914 44 89 the the DET iajs-1914 44 90 likelihood likelihood NOUN iajs-1914 44 91 function function NOUN iajs-1914 44 92 . . PUNCT iajs-1914 45 1 one one NUM iajs-1914 45 2 of of ADP iajs-1914 45 3 these these DET iajs-1914 45 4 numerical numerical ADJ iajs-1914 45 5 procedures procedure NOUN iajs-1914 45 6 is be AUX iajs-1914 45 7 newton newton PROPN iajs-1914 45 8 - - PUNCT iajs-1914 45 9 raphson raphson NOUN iajs-1914 45 10 method method NOUN iajs-1914 45 11 and and CCONJ iajs-1914 45 12 using use VERB iajs-1914 45 13 hessian hessian ADJ iajs-1914 45 14 matrix matrix NOUN iajs-1914 45 15 , , PUNCT iajs-1914 45 16 which which PRON iajs-1914 45 17 is be AUX iajs-1914 45 18 the the DET iajs-1914 45 19 second second ADJ iajs-1914 45 20 partial partial ADJ iajs-1914 45 21 derivative derivative NOUN iajs-1914 45 22 of of ADP iajs-1914 45 23 the the DET iajs-1914 45 24 log log NOUN iajs-1914 45 25 - - PUNCT iajs-1914 45 26 likelihood likelihood NOUN iajs-1914 45 27 function function NOUN iajs-1914 45 28 . . PUNCT iajs-1914 46 1 we we PRON iajs-1914 46 2 can can AUX iajs-1914 46 3 construct construct VERB iajs-1914 46 4 hessian hessian ADJ iajs-1914 46 5 matrix matrix NOUN iajs-1914 46 6 as as SCONJ iajs-1914 46 7 follows follow VERB iajs-1914 46 8 [ [ X iajs-1914 46 9 4 4 NUM iajs-1914 46 10 ] ] X iajs-1914 46 11 𝑔 𝑔 PROPN iajs-1914 46 12 𝛼 𝛼 NOUN iajs-1914 46 13 𝑛ψ(α 𝑛ψ(α NOUN iajs-1914 46 14 ) ) PUNCT iajs-1914 46 15 𝑛𝑙𝑛𝛽 𝑛𝑙𝑛𝛽 NOUN iajs-1914 46 16 ∑ ∑ PUNCT iajs-1914 46 17 𝑙𝑛𝑥 𝑙𝑛𝑥 VERB iajs-1914 46 18 𝑔 𝑔 PROPN iajs-1914 46 19 𝛽 𝛽 PROPN iajs-1914 46 20 𝑛 𝑛 PROPN iajs-1914 46 21 � � PROPN iajs-1914 46 22 ̅ ̅ NOUN iajs-1914 46 23 � � NOUN iajs-1914 46 24 the the DET iajs-1914 46 25 partial partial ADJ iajs-1914 46 26 derivatives derivative NOUN iajs-1914 46 27 of of ADP iajs-1914 46 28 ) ) PUNCT iajs-1914 46 29 ( ( PUNCT iajs-1914 46 30 1 1 NUM iajs-1914 46 31 g g NUM iajs-1914 46 32 with with ADP iajs-1914 46 33 respect respect NOUN iajs-1914 46 34 to to ADP iajs-1914 46 35 unknown unknown ADJ iajs-1914 46 36 parameters parameter NOUN iajs-1914 46 37   X iajs-1914 46 38 and and CCONJ iajs-1914 46 39   PROPN iajs-1914 46 40 are be AUX iajs-1914 46 41 𝑛𝛹 𝑛𝛹 NOUN iajs-1914 46 42 𝛼 𝛼 NOUN iajs-1914 46 43 where where SCONJ iajs-1914 46 44 ψ ψ NOUN iajs-1914 46 45 α α PROPN iajs-1914 46 46 is be AUX iajs-1914 46 47 the the DET iajs-1914 46 48 derivative derivative NOUN iajs-1914 46 49 of of ADP iajs-1914 46 50 ψ ψ X iajs-1914 46 51 α α PRON iajs-1914 46 52 which which PRON iajs-1914 46 53 is be AUX iajs-1914 46 54 called call VERB iajs-1914 46 55 as as ADP iajs-1914 46 56 tri tri ADJ iajs-1914 46 57 - - NOUN iajs-1914 46 58 gamma gamma NOUN iajs-1914 46 59 the the DET iajs-1914 46 60 partial partial ADJ iajs-1914 46 61 derivatives derivative NOUN iajs-1914 46 62 of of ADP iajs-1914 46 63 ) ) PUNCT iajs-1914 46 64 ( ( PUNCT iajs-1914 46 65 2 2 NUM iajs-1914 46 66 g g X iajs-1914 46 67 with with ADP iajs-1914 46 68 respect respect NOUN iajs-1914 46 69 to to ADP iajs-1914 46 70 unknown unknown ADJ iajs-1914 46 71 parameters parameter NOUN iajs-1914 46 72   X iajs-1914 46 73 and and CCONJ iajs-1914 46 74   PROPN iajs-1914 46 75 are be AUX iajs-1914 46 76 hence hence ADV iajs-1914 46 77 , , PUNCT iajs-1914 47 1   ADJ iajs-1914 47 2   PROPN iajs-1914 48 1   PUNCT iajs-1914 48 2   NOUN iajs-1914 48 3   PROPN iajs-1914 48 4   PROPN iajs-1914 48 5   X iajs-1914 48 6   NOUN iajs-1914 48 7   NOUN iajs-1914 48 8   NOUN iajs-1914 48 9   NOUN iajs-1914 48 10   NOUN iajs-1914 48 11   NOUN iajs-1914 48 12   PROPN iajs-1914 48 13   ADJ iajs-1914 48 14   ADJ iajs-1914 48 15   ADJ iajs-1914 48 16   ADJ iajs-1914 48 17   ADJ iajs-1914 48 18   ADJ iajs-1914 48 19   ADJ iajs-1914 48 20   PROPN iajs-1914 48 21   PROPN iajs-1914 48 22   PROPN iajs-1914 48 23   PROPN iajs-1914 48 24   NUM iajs-1914 48 25   PROPN iajs-1914 48 26   PROPN iajs-1914 48 27   NOUN iajs-1914 48 28   X iajs-1914 48 29   X iajs-1914 48 30 ) ) PUNCT iajs-1914 48 31 ( ( PUNCT iajs-1914 48 32 ) ) PUNCT iajs-1914 48 33 ( ( PUNCT iajs-1914 48 34 ) ) PUNCT iajs-1914 48 35 ( ( PUNCT iajs-1914 48 36 ) ) PUNCT iajs-1914 48 37 ( ( PUNCT iajs-1914 48 38 22 22 NUM iajs-1914 48 39 11 11 NUM iajs-1914 48 40 gg gg NOUN iajs-1914 48 41 gg gg PROPN iajs-1914 48 42 j j PROPN iajs-1914 49 1 k k PROPN iajs-1914 50 1 = = PUNCT iajs-1914 51 1 𝑎 𝑎 PRON iajs-1914 51 2 𝑎 𝑎 X iajs-1914 51 3 𝑎 𝑎 NOUN iajs-1914 51 4 𝑎 𝑎 NOUN iajs-1914 51 5 where where SCONJ iajs-1914 51 6 , , PUNCT iajs-1914 51 7 jk jk PROPN iajs-1914 51 8 is be AUX iajs-1914 51 9 the the DET iajs-1914 51 10 jacobean jacobean ADJ iajs-1914 51 11 matrix matrix NOUN iajs-1914 51 12 and and CCONJ iajs-1914 51 13 jk jk PROPN iajs-1914 51 14 must must AUX iajs-1914 51 15 be be AUX iajs-1914 51 16 a a DET iajs-1914 51 17 non non ADJ iajs-1914 51 18 - - ADJ iajs-1914 51 19 singular singular ADJ iajs-1914 51 20 symmetric symmetric ADJ iajs-1914 51 21 matrix matrix NOUN iajs-1914 51 22 so so ADV iajs-1914 51 23 , , PUNCT iajs-1914 51 24 its its PRON iajs-1914 51 25 inverse inverse NOUN iajs-1914 51 26 can can AUX iajs-1914 51 27 be be AUX iajs-1914 51 28 found find VERB iajs-1914 51 29 as as ADP iajs-1914 51 30     SPACE iajs-1914 51 31 190 190 NUM iajs-1914 51 32         SPACE iajs-1914 51 33 ibn ibn PROPN iajs-1914 51 34 al al PROPN iajs-1914 51 35 - - PUNCT iajs-1914 51 36 haitham haitham PROPN iajs-1914 51 37 jour.for jour.for PROPN iajs-1914 51 38 pure&appl.sci pure&appl.sci PROPN iajs-1914 51 39 . . PUNCT iajs-1914 52 1 ihjpas ihjpas PROPN iajs-1914 52 2 https://doi.org/10.30526/32.1.1914 https://doi.org/10.30526/32.1.1914 X iajs-1914 52 3 vol vol NOUN iajs-1914 52 4 . . PUNCT iajs-1914 53 1 32 32 NUM iajs-1914 53 2 ( ( PUNCT iajs-1914 53 3 1 1 NUM iajs-1914 53 4 ) ) SYM iajs-1914 53 5 2019 2019 NUM iajs-1914 53 6 𝐽 𝐽 NOUN iajs-1914 54 1 | | ADV iajs-1914 54 2 | | ADV iajs-1914 54 3 𝑎 𝑎 NOUN iajs-1914 54 4 𝑎 𝑎 NOUN iajs-1914 54 5 𝑎 𝑎 NOUN iajs-1914 54 6 𝑎 𝑎 PRON iajs-1914 54 7   NOUN iajs-1914 54 8   NOUN iajs-1914 54 9   PROPN iajs-1914 54 10   PROPN iajs-1914 54 11   X iajs-1914 54 12   NOUN iajs-1914 54 13   NOUN iajs-1914 54 14   NOUN iajs-1914 54 15   VERB iajs-1914 54 16   PROPN iajs-1914 54 17   NOUN iajs-1914 54 18   PROPN iajs-1914 55 1   PROPN iajs-1914 55 2   PROPN iajs-1914 55 3   PROPN iajs-1914 55 4   X iajs-1914 55 5   NOUN iajs-1914 55 6   NOUN iajs-1914 55 7   NOUN iajs-1914 55 8   VERB iajs-1914 55 9   PROPN iajs-1914 55 10   ADJ iajs-1914 55 11   PROPN iajs-1914 55 12   PROPN iajs-1914 55 13   PROPN iajs-1914 55 14   PROPN iajs-1914 55 15   X iajs-1914 55 16   NOUN iajs-1914 55 17   NOUN iajs-1914 55 18   NOUN iajs-1914 55 19   VERB iajs-1914 55 20   PROPN iajs-1914 55 21   PROPN iajs-1914 55 22   ADV iajs-1914 55 23   PUNCT iajs-1914 55 24 ) ) PUNCT iajs-1914 55 25 ( ( PUNCT iajs-1914 55 26 ) ) PUNCT iajs-1914 55 27 ( ( PUNCT iajs-1914 55 28 2 2 NUM iajs-1914 55 29 1 1 NUM iajs-1914 55 30 1 1 NUM iajs-1914 55 31 1 1 NUM iajs-1914 55 32 1 1 NUM iajs-1914 55 33   PROPN iajs-1914 55 34   NOUN iajs-1914 55 35   PROPN iajs-1914 55 36   NOUN iajs-1914 55 37   PROPN iajs-1914 55 38   NOUN iajs-1914 55 39 g g PROPN iajs-1914 55 40 g g PROPN iajs-1914 55 41 j j PROPN iajs-1914 55 42 ik ik PROPN iajs-1914 56 1 k k PROPN iajs-1914 56 2 k k PROPN iajs-1914 57 1 k k PROPN iajs-1914 57 2 k k PROPN iajs-1914 57 3 𝛼 𝛼 VERB iajs-1914 57 4 𝛽 𝛽 NOUN iajs-1914 57 5 𝛼 𝛼 PART iajs-1914 57 6 𝛽 𝛽 NOUN iajs-1914 57 7 1 1 NUM iajs-1914 57 8 𝑎 𝑎 NOUN iajs-1914 57 9 𝑎 𝑎 NOUN iajs-1914 57 10 𝑎 𝑎 PRON iajs-1914 57 11 𝑎 𝑎 X iajs-1914 57 12 ⎣ ⎣ ADJ iajs-1914 57 13 ⎢ ⎢ NOUN iajs-1914 57 14 ⎢ ⎢ NOUN iajs-1914 57 15 ⎢ ⎢ NOUN iajs-1914 57 16 ⎢ ⎢ NOUN iajs-1914 57 17 ⎡ ⎡ NOUN iajs-1914 57 18 𝑛𝑎 𝑛𝑎 ADP iajs-1914 57 19 𝛹 𝛹 PRON iajs-1914 57 20 𝛼 𝛼 INTJ iajs-1914 57 21 𝑛𝑎 𝑛𝑎 INTJ iajs-1914 57 22 𝑙𝑛𝛽 𝑙𝑛𝛽 NOUN iajs-1914 57 23 𝑎 𝑎 X iajs-1914 57 24 𝑙𝑛𝑥 𝑙𝑛𝑥 NOUN iajs-1914 57 25 𝑛𝑎 𝑛𝑎 ADP iajs-1914 57 26 𝛼 𝛼 PROPN iajs-1914 57 27 𝛽 𝛽 PROPN iajs-1914 57 28 𝑛𝑎 𝑛𝑎 PROPN iajs-1914 57 29 � � PROPN iajs-1914 57 30 ̅ ̅ NOUN iajs-1914 57 31 � � PROPN iajs-1914 57 32 𝑛𝑎 𝑛𝑎 ADP iajs-1914 57 33 𝛹 𝛹 PROPN iajs-1914 57 34 𝛼 𝛼 INTJ iajs-1914 57 35 𝑛𝑎 𝑛𝑎 INTJ iajs-1914 57 36 𝑙𝑛𝛽 𝑙𝑛𝛽 NOUN iajs-1914 57 37 𝑎 𝑎 X iajs-1914 57 38 𝑙𝑛𝑥 𝑙𝑛𝑥 NOUN iajs-1914 57 39 𝑛𝑎 𝑛𝑎 ADP iajs-1914 57 40 𝛼 𝛼 PROPN iajs-1914 57 41 𝛽 𝛽 PROPN iajs-1914 57 42 𝑛𝑎 𝑛𝑎 PROPN iajs-1914 57 43 � � PROPN iajs-1914 57 44 ̅ ̅ NOUN iajs-1914 57 45 � � PROPN iajs-1914 57 46 ⎦ ⎦ NOUN iajs-1914 57 47 ⎥ ⎥ PROPN iajs-1914 57 48 ⎥ ⎥ PROPN iajs-1914 57 49 ⎥ ⎥ PROPN iajs-1914 57 50 ⎥ ⎥ PROPN iajs-1914 57 51 ⎤ ⎤ VERB iajs-1914 57 52 the the DET iajs-1914 57 53 absolute absolute ADJ iajs-1914 57 54 value value NOUN iajs-1914 57 55 for for ADP iajs-1914 57 56 the the DET iajs-1914 57 57 difference difference NOUN iajs-1914 57 58 between between ADP iajs-1914 57 59 the the DET iajs-1914 57 60 new new ADJ iajs-1914 57 61 value value NOUN iajs-1914 57 62 for for ADP iajs-1914 57 63   NOUN iajs-1914 57 64 and and CCONJ iajs-1914 57 65   NOUN iajs-1914 57 66 in in ADP iajs-1914 57 67 new new ADJ iajs-1914 57 68 iterative iterative NOUN iajs-1914 57 69 value value NOUN iajs-1914 57 70 with with ADP iajs-1914 57 71 previous previous ADJ iajs-1914 57 72 value value NOUN iajs-1914 57 73 for for ADP iajs-1914 57 74   NOUN iajs-1914 57 75 and and CCONJ iajs-1914 57 76   NOUN iajs-1914 57 77 in in ADP iajs-1914 57 78 last last ADJ iajs-1914 57 79 iterative iterative NOUN iajs-1914 57 80 represent represent VERB iajs-1914 57 81 the the DET iajs-1914 57 82 error error NOUN iajs-1914 57 83 term term NOUN iajs-1914 57 84 , , PUNCT iajs-1914 57 85 it it PRON iajs-1914 57 86 's be AUX iajs-1914 57 87 symbol symbol NOUN iajs-1914 57 88 is be AUX iajs-1914 57 89   PROPN iajs-1914 57 90 , , PUNCT iajs-1914 57 91 which which PRON iajs-1914 57 92 is be AUX iajs-1914 57 93 a a DET iajs-1914 57 94 very very ADV iajs-1914 57 95 small small ADJ iajs-1914 57 96 and and CCONJ iajs-1914 57 97 assumed assumed ADJ iajs-1914 57 98 value value NOUN iajs-1914 57 99 . . PUNCT iajs-1914 58 1 then then ADV iajs-1914 58 2 , , PUNCT iajs-1914 58 3 error error NOUN iajs-1914 58 4 term term NOUN iajs-1914 58 5 is be AUX iajs-1914 58 6 formulated formulate VERB iajs-1914 58 7 as as ADP iajs-1914 58 8   PROPN iajs-1914 58 9   PROPN iajs-1914 58 10   PROPN iajs-1914 58 11   PROPN iajs-1914 58 12   X iajs-1914 58 13   NOUN iajs-1914 58 14   NOUN iajs-1914 58 15   NOUN iajs-1914 58 16   VERB iajs-1914 58 17   PROPN iajs-1914 58 18   NOUN iajs-1914 58 19   PROPN iajs-1914 59 1   PROPN iajs-1914 59 2   PROPN iajs-1914 59 3   PROPN iajs-1914 59 4   X iajs-1914 59 5   NOUN iajs-1914 59 6   NOUN iajs-1914 59 7   NOUN iajs-1914 59 8   VERB iajs-1914 59 9   PROPN iajs-1914 59 10   ADJ iajs-1914 59 11   PROPN iajs-1914 59 12   PROPN iajs-1914 59 13   PROPN iajs-1914 59 14   PROPN iajs-1914 59 15   X iajs-1914 59 16   NOUN iajs-1914 59 17   NOUN iajs-1914 59 18   NOUN iajs-1914 59 19   NOUN iajs-1914 59 20   PROPN iajs-1914 59 21   PROPN iajs-1914 59 22   PROPN iajs-1914 59 23   PUNCT iajs-1914 59 24   PROPN iajs-1914 59 25 k k PROPN iajs-1914 59 26 k k PROPN iajs-1914 59 27 k k PROPN iajs-1914 59 28 k k PROPN iajs-1914 59 29 k k PROPN iajs-1914 59 30 k k PROPN iajs-1914 59 31   PROPN iajs-1914 59 32   X iajs-1914 59 33   NOUN iajs-1914 59 34   NOUN iajs-1914 59 35   ADV iajs-1914 59 36   ADP iajs-1914 59 37 1 1 NUM iajs-1914 59 38 1 1 NUM iajs-1914 59 39 1 1 NUM iajs-1914 59 40 1 1 NUM iajs-1914 59 41 ) ) PUNCT iajs-1914 59 42 ( ( PUNCT iajs-1914 59 43 ) ) PUNCT iajs-1914 59 44 ( ( PUNCT iajs-1914 59 45 ( ( PUNCT iajs-1914 59 46 7 7 X iajs-1914 59 47 ) ) PUNCT iajs-1914 59 48 where where SCONJ iajs-1914 59 49 k k NOUN iajs-1914 59 50 and and CCONJ iajs-1914 59 51 k k NOUN iajs-1914 59 52 are be AUX iajs-1914 59 53 the the DET iajs-1914 59 54 initial initial ADJ iajs-1914 59 55 values value NOUN iajs-1914 59 56 for for ADP iajs-1914 59 57   NOUN iajs-1914 59 58 and and CCONJ iajs-1914 59 59   PROPN iajs-1914 59 60 respectively respectively ADV iajs-1914 59 61 , , PUNCT iajs-1914 59 62 for for ADP iajs-1914 59 63 which which PRON iajs-1914 59 64 are be AUX iajs-1914 59 65 assumed assume VERB iajs-1914 59 66 . . PUNCT iajs-1914 60 1 3 3 X iajs-1914 60 2 . . X iajs-1914 60 3 bayesian bayesian NOUN iajs-1914 60 4 estimation estimation NOUN iajs-1914 60 5 3.1 3.1 NUM iajs-1914 60 6 . . PUNCT iajs-1914 60 7 posterior posterior ADJ iajs-1914 60 8 density density NOUN iajs-1914 60 9 functions function NOUN iajs-1914 60 10 using use VERB iajs-1914 60 11 gamma gamma NOUN iajs-1914 60 12 and and CCONJ iajs-1914 60 13 exponential exponential ADJ iajs-1914 60 14 priors prior NOUN iajs-1914 60 15 to to PART iajs-1914 60 16 estimate estimate VERB iajs-1914 60 17 α α PRON iajs-1914 60 18 and and CCONJ iajs-1914 60 19 β β NOUN iajs-1914 60 20 parameters parameter NOUN iajs-1914 60 21 for for ADP iajs-1914 60 22 gamma gamma NOUN iajs-1914 60 23 distribution distribution NOUN iajs-1914 60 24 , , PUNCT iajs-1914 60 25 we we PRON iajs-1914 60 26 assume assume VERB iajs-1914 60 27 that that SCONJ iajs-1914 60 28 α α PRON iajs-1914 60 29 has have VERB iajs-1914 60 30 a a DET iajs-1914 60 31 prior prior ADJ iajs-1914 60 32 π1(ꞏ π1(ꞏ NUM iajs-1914 60 33 ) ) PUNCT iajs-1914 60 34 , , PUNCT iajs-1914 60 35 which which PRON iajs-1914 60 36 follows follow VERB iajs-1914 60 37 gamma gamma NOUN iajs-1914 60 38 ( ( PUNCT iajs-1914 60 39 a a DET iajs-1914 60 40 , , PUNCT iajs-1914 60 41 b b NOUN iajs-1914 60 42 ) ) PUNCT iajs-1914 60 43 . . PUNCT iajs-1914 61 1 at at ADP iajs-1914 61 2 this this DET iajs-1914 61 3 moment moment NOUN iajs-1914 61 4 we we PRON iajs-1914 61 5 do do AUX iajs-1914 61 6 not not PART iajs-1914 61 7 assume assume VERB iajs-1914 61 8 any any DET iajs-1914 61 9 specific specific ADJ iajs-1914 61 10 prior prior ADV iajs-1914 61 11 on on ADP iajs-1914 61 12 α α NOUN iajs-1914 61 13 . . PUNCT iajs-1914 62 1 we we PRON iajs-1914 62 2 simply simply ADV iajs-1914 62 3 assume assume VERB iajs-1914 62 4 that that SCONJ iajs-1914 62 5 the the DET iajs-1914 62 6 prior prior NOUN iajs-1914 62 7 on on ADP iajs-1914 62 8 β β X iajs-1914 62 9 is be AUX iajs-1914 62 10 π2(ꞏ π2(ꞏ NOUN iajs-1914 62 11 ) ) PUNCT iajs-1914 62 12 and and CCONJ iajs-1914 62 13 the the DET iajs-1914 62 14 density density NOUN iajs-1914 62 15 function function NOUN iajs-1914 62 16 of of ADP iajs-1914 62 17 π2(ꞏ π2(ꞏ NOUN iajs-1914 62 18 ) ) PUNCT iajs-1914 62 19 is be AUX iajs-1914 62 20 exponential exponential ADJ iajs-1914 62 21 and and CCONJ iajs-1914 62 22 it it PRON iajs-1914 62 23 is be AUX iajs-1914 62 24 independent independent ADJ iajs-1914 62 25 of of ADP iajs-1914 62 26 π1(ꞏ π1(ꞏ PROPN iajs-1914 62 27 ) ) PUNCT iajs-1914 62 28 . . PUNCT iajs-1914 63 1 𝜋 𝜋 PRON iajs-1914 63 2 𝛼 𝛼 VERB iajs-1914 63 3 г г NOUN iajs-1914 63 4 ; ; PUNCT iajs-1914 63 5 𝑎 𝑎 PROPN iajs-1914 63 6 0 0 NUM iajs-1914 63 7 , , PUNCT iajs-1914 63 8 𝑏 𝑏 PROPN iajs-1914 63 9 0 0 NUM iajs-1914 63 10 , , PUNCT iajs-1914 63 11 𝛼 𝛼 PROPN iajs-1914 63 12 0 0 NUM iajs-1914 63 13 0 0 NUM iajs-1914 63 14 0 0 NUM iajs-1914 63 15 . . PUNCT iajs-1914 64 1 𝑤 𝑤 X iajs-1914 64 2 ( ( PUNCT iajs-1914 64 3 8) 8) NUM iajs-1914 64 4 𝜋 𝜋 NOUN iajs-1914 64 5 𝛽 𝛽 NOUN iajs-1914 64 6 𝑐 𝑐 NOUN iajs-1914 64 7 𝑒 𝑒 ADV iajs-1914 64 8 ; ; PUNCT iajs-1914 64 9 𝑐 𝑐 PROPN iajs-1914 64 10 0 0 NUM iajs-1914 64 11 , , PUNCT iajs-1914 64 12 𝛽 𝛽 NOUN iajs-1914 64 13 0 0 NUM iajs-1914 64 14 0 0 NUM iajs-1914 64 15 0 0 NUM iajs-1914 64 16 . . PUNCT iajs-1914 65 1 𝑤 𝑤 X iajs-1914 65 2 ( ( PUNCT iajs-1914 65 3 9 9 X iajs-1914 65 4 ) ) PUNCT iajs-1914 65 5 the the DET iajs-1914 65 6 equations equation NOUN iajs-1914 65 7 ( ( PUNCT iajs-1914 65 8 8) 8) NUM iajs-1914 65 9 and and CCONJ iajs-1914 65 10 ( ( PUNCT iajs-1914 65 11 9 9 NUM iajs-1914 65 12 ) ) PUNCT iajs-1914 65 13 are be AUX iajs-1914 65 14 prior prior ADJ iajs-1914 65 15 distribution distribution NOUN iajs-1914 65 16 for for ADP iajs-1914 65 17   NOUN iajs-1914 65 18 and and CCONJ iajs-1914 65 19   PROPN iajs-1914 65 20 respectively respectively ADV iajs-1914 65 21 . . PUNCT iajs-1914 66 1 the the DET iajs-1914 66 2 joint joint ADJ iajs-1914 66 3 p.d.f p.d.f NOUN iajs-1914 66 4 is be AUX iajs-1914 66 5 given give VERB iajs-1914 66 6 by by ADP iajs-1914 66 7 𝐽 𝐽 PROPN iajs-1914 66 8 𝑥 𝑥 PROPN iajs-1914 66 9 , , PUNCT iajs-1914 66 10 𝑥 𝑥 X iajs-1914 66 11 , , PUNCT iajs-1914 66 12 … … PUNCT iajs-1914 66 13 , , PUNCT iajs-1914 66 14 𝑥 𝑥 X iajs-1914 66 15 ; ; PUNCT iajs-1914 66 16 𝛼 𝛼 X iajs-1914 66 17 , , PUNCT iajs-1914 66 18 𝛽 𝛽 PROPN iajs-1914 66 19 𝐿 𝐿 PROPN iajs-1914 66 20 𝑥 𝑥 PROPN iajs-1914 66 21 , , PUNCT iajs-1914 66 22 𝑥 𝑥 X iajs-1914 66 23 , , PUNCT iajs-1914 66 24 … … PUNCT iajs-1914 66 25 , , PUNCT iajs-1914 66 26 𝑥 𝑥 X iajs-1914 66 27 ; ; PUNCT iajs-1914 66 28 𝛼 𝛼 X iajs-1914 66 29 , , PUNCT iajs-1914 66 30 𝛽 𝛽 NOUN iajs-1914 66 31 𝜋 𝜋 NOUN iajs-1914 66 32 𝛼 𝛼 NOUN iajs-1914 66 33 𝜋 𝜋 NOUN iajs-1914 66 34 𝛽 𝛽 NOUN iajs-1914 66 35 г г PROPN iajs-1914 66 36 𝜋 𝜋 NOUN iajs-1914 66 37 𝑥 𝑥 NOUN iajs-1914 66 38 𝑒 𝑒 ADP iajs-1914 66 39 ∑ ∑ PUNCT iajs-1914 66 40 г г PROPN iajs-1914 66 41 𝑐 𝑐 PROPN iajs-1914 66 42 𝑒 𝑒 PROPN iajs-1914 66 43 and and CCONJ iajs-1914 66 44 the the DET iajs-1914 66 45 marginal marginal ADJ iajs-1914 66 46 p.d.f p.d.f NOUN iajs-1914 66 47 . . PUNCT iajs-1914 67 1 of of ADP iajs-1914 67 2 𝑥 𝑥 PRON iajs-1914 67 3 , , PUNCT iajs-1914 67 4 𝑥 𝑥 X iajs-1914 67 5 , , PUNCT iajs-1914 67 6 … … PUNCT iajs-1914 67 7 , , PUNCT iajs-1914 67 8 𝑥 𝑥 PROPN iajs-1914 67 9 is be AUX iajs-1914 67 10 given give VERB iajs-1914 67 11 by by ADP iajs-1914 67 12 𝑓 𝑓 PRON iajs-1914 67 13 𝑥 𝑥 PROPN iajs-1914 67 14 , , PUNCT iajs-1914 67 15 𝑥 𝑥 X iajs-1914 67 16 , , PUNCT iajs-1914 67 17 … … PUNCT iajs-1914 67 18 , , PUNCT iajs-1914 67 19 𝑥 𝑥 PROPN iajs-1914 67 20 𝐿 𝐿 PROPN iajs-1914 67 21 𝑥 𝑥 PROPN iajs-1914 67 22 , , PUNCT iajs-1914 67 23 𝑥 𝑥 X iajs-1914 67 24 , , PUNCT iajs-1914 67 25 … … PUNCT iajs-1914 67 26 , , PUNCT iajs-1914 67 27 𝑥 𝑥 X iajs-1914 67 28 ; ; PUNCT iajs-1914 67 29 𝛼 𝛼 X iajs-1914 67 30 , , PUNCT iajs-1914 67 31 𝛽 𝛽 NOUN iajs-1914 67 32 𝜋 𝜋 NOUN iajs-1914 67 33 𝛼 𝛼 NOUN iajs-1914 67 34 𝜋 𝜋 NOUN iajs-1914 67 35 𝛽 𝛽 NOUN iajs-1914 67 36 dα dα PRON iajs-1914 67 37 dβ dβ ADJ iajs-1914 67 38 ∞∞ ∞∞ NOUN iajs-1914 67 39 hence hence ADV iajs-1914 67 40 , , PUNCT iajs-1914 67 41 the the DET iajs-1914 67 42 posterior posterior ADJ iajs-1914 67 43 density density NOUN iajs-1914 67 44 functions function NOUN iajs-1914 67 45 of of ADP iajs-1914 67 46   NOUN iajs-1914 67 47 and and CCONJ iajs-1914 67 48   NOUN iajs-1914 67 49 can can AUX iajs-1914 67 50 be be AUX iajs-1914 67 51 written write VERB iajs-1914 67 52 as as SCONJ iajs-1914 67 53 follows follow VERB iajs-1914 67 54     SPACE iajs-1914 67 55 191 191 NUM iajs-1914 67 56         SPACE iajs-1914 67 57 ibn ibn PROPN iajs-1914 67 58 al al PROPN iajs-1914 67 59 - - PUNCT iajs-1914 67 60 haitham haitham PROPN iajs-1914 67 61 jour.for jour.for PROPN iajs-1914 67 62 pure&appl.sci pure&appl.sci PROPN iajs-1914 67 63 . . PUNCT iajs-1914 68 1 ihjpas ihjpas PROPN iajs-1914 68 2 https://doi.org/10.30526/32.1.1914 https://doi.org/10.30526/32.1.1914 X iajs-1914 68 3 vol vol NOUN iajs-1914 68 4 . . PUNCT iajs-1914 69 1 32 32 NUM iajs-1914 69 2 ( ( PUNCT iajs-1914 69 3 1 1 NUM iajs-1914 69 4 ) ) PUNCT iajs-1914 69 5 2019 2019 NUM iajs-1914 69 6 ℎ ℎ SYM iajs-1914 69 7 𝛼 𝛼 PROPN iajs-1914 69 8 , , PUNCT iajs-1914 69 9 𝛽|𝑥 𝛽|𝑥 PROPN iajs-1914 69 10 , , PUNCT iajs-1914 69 11 𝑥 𝑥 X iajs-1914 69 12 , , PUNCT iajs-1914 69 13 … … PUNCT iajs-1914 69 14 , , PUNCT iajs-1914 69 15 𝑥 𝑥 PROPN iajs-1914 69 16 𝐿 𝐿 PROPN iajs-1914 69 17 𝑥 𝑥 PROPN iajs-1914 69 18 , , PUNCT iajs-1914 69 19 𝑥 𝑥 X iajs-1914 69 20 , , PUNCT iajs-1914 69 21 … … PUNCT iajs-1914 69 22 , , PUNCT iajs-1914 69 23 𝑥 𝑥 X iajs-1914 69 24 ; ; PUNCT iajs-1914 69 25 𝛼 𝛼 X iajs-1914 69 26 , , PUNCT iajs-1914 69 27 𝛽 𝛽 NOUN iajs-1914 69 28 𝜋 𝜋 NOUN iajs-1914 69 29 𝛼 𝛼 NOUN iajs-1914 69 30 𝜋 𝜋 NOUN iajs-1914 69 31 𝛽 𝛽 NOUN iajs-1914 69 32 𝐿 𝐿 NOUN iajs-1914 69 33 𝑥 𝑥 PROPN iajs-1914 69 34 , , PUNCT iajs-1914 69 35 𝑥 𝑥 X iajs-1914 69 36 , , PUNCT iajs-1914 69 37 … … PUNCT iajs-1914 69 38 , , PUNCT iajs-1914 69 39 𝑥 𝑥 X iajs-1914 69 40 ; ; PUNCT iajs-1914 69 41 𝛼 𝛼 X iajs-1914 69 42 , , PUNCT iajs-1914 69 43 𝛽 𝛽 NOUN iajs-1914 69 44 𝜋 𝜋 NOUN iajs-1914 69 45 𝛼 𝛼 NOUN iajs-1914 69 46 𝜋 𝜋 NOUN iajs-1914 69 47 𝛽 𝛽 NOUN iajs-1914 69 48 dαdβ dαdβ NOUN iajs-1914 69 49 ∞∞ ∞∞ NOUN iajs-1914 69 50 = = PUNCT iajs-1914 69 51 г г PROPN iajs-1914 69 52 ∑ ∑ PROPN iajs-1914 69 53 г г PROPN iajs-1914 69 54 г г PROPN iajs-1914 69 55 ∑ ∑ PROPN iajs-1914 69 56 г г PROPN iajs-1914 69 57 ∞∞ ∞∞ NOUN iajs-1914 69 58 3.2 3.2 NUM iajs-1914 69 59 . . PUNCT iajs-1914 70 1 bayes bayes PROPN iajs-1914 70 2 estimator estimator NOUN iajs-1914 70 3 under under ADP iajs-1914 70 4 precautionary precautionary ADJ iajs-1914 70 5 loss loss NOUN iajs-1914 70 6 function function NOUN iajs-1914 70 7 norstrom norstrom ADP iajs-1914 70 8 ( ( PUNCT iajs-1914 70 9 1996 1996 NUM iajs-1914 70 10 ) ) PUNCT iajs-1914 70 11 introduced introduce VERB iajs-1914 70 12 an an DET iajs-1914 70 13 asymmetric asymmetric ADJ iajs-1914 70 14 precautionary precautionary ADJ iajs-1914 70 15 loss loss NOUN iajs-1914 70 16 function function NOUN iajs-1914 70 17 , , PUNCT iajs-1914 70 18 which which PRON iajs-1914 70 19 can can AUX iajs-1914 70 20 be be AUX iajs-1914 70 21 defined define VERB iajs-1914 70 22 as as SCONJ iajs-1914 70 23 follows follow VERB iajs-1914 70 24 [ [ X iajs-1914 70 25 7 7 NUM iajs-1914 70 26 ] ] X iajs-1914 70 27 l l NOUN iajs-1914 70 28 θ θ PROPN iajs-1914 70 29 , , PUNCT iajs-1914 70 30 θ θ PROPN iajs-1914 70 31 based base VERB iajs-1914 70 32 on on ADP iajs-1914 70 33 precautionary precautionary ADJ iajs-1914 70 34 loss loss NOUN iajs-1914 70 35 function function NOUN iajs-1914 70 36 , , PUNCT iajs-1914 70 37 risk risk NOUN iajs-1914 70 38 function function NOUN iajs-1914 70 39 r r NOUN iajs-1914 70 40 θ θ PROPN iajs-1914 70 41 , , PUNCT iajs-1914 70 42 θ θ PROPN iajs-1914 70 43 can can AUX iajs-1914 70 44 be be AUX iajs-1914 70 45 derived derive VERB iajs-1914 70 46 as as SCONJ iajs-1914 70 47 follows follow VERB iajs-1914 70 48 r r NOUN iajs-1914 70 49 θ θ PROPN iajs-1914 70 50 , , PUNCT iajs-1914 70 51 θ θ PROPN iajs-1914 70 52 e e NOUN iajs-1914 70 53 l l NOUN iajs-1914 70 54 θ θ PROPN iajs-1914 70 55 , , PUNCT iajs-1914 70 56 θ θ PROPN iajs-1914 70 57 l l NOUN iajs-1914 70 58 θ θ PROPN iajs-1914 70 59 , , PUNCT iajs-1914 71 1 θ θ PROPN iajs-1914 71 2 h h NOUN iajs-1914 72 1 θ θ NOUN iajs-1914 72 2 x x PUNCT iajs-1914 72 3 dθ dθ PROPN iajs-1914 72 4 ∞ ∞ PROPN iajs-1914 72 5 𝑅 𝑅 PROPN iajs-1914 72 6 𝜃 𝜃 PROPN iajs-1914 72 7 , , PUNCT iajs-1914 72 8 𝜃 𝜃 NOUN iajs-1914 72 9 𝜃 𝜃 NOUN iajs-1914 72 10 𝜃 𝜃 NOUN iajs-1914 72 11 𝜃 𝜃 NUM iajs-1914 72 12 ∞ ∞ NUM iajs-1914 72 13 h h NOUN iajs-1914 73 1 θ θ NOUN iajs-1914 73 2 x x PUNCT iajs-1914 74 1 dθ dθ NOUN iajs-1914 74 2 𝜃 𝜃 X iajs-1914 74 3 𝜃 𝜃 NOUN iajs-1914 74 4 h h NOUN iajs-1914 74 5 θ θ NOUN iajs-1914 74 6 x x PUNCT iajs-1914 75 1 dθ dθ PROPN iajs-1914 75 2 2𝜃h 2𝜃h ADJ iajs-1914 75 3 θ θ PROPN iajs-1914 76 1 x x PUNCT iajs-1914 77 1 dθ dθ PROPN iajs-1914 77 2 ∞ ∞ PROPN iajs-1914 77 3 𝜃 𝜃 PROPN iajs-1914 78 1 ∞∞ ∞∞ NOUN iajs-1914 78 2 h h NOUN iajs-1914 78 3 θ θ NOUN iajs-1914 78 4 x x PUNCT iajs-1914 78 5 dθ dθ PROPN iajs-1914 78 6 𝑅 𝑅 PROPN iajs-1914 78 7 𝜃 𝜃 PROPN iajs-1914 78 8 , , PUNCT iajs-1914 78 9 𝜃 𝜃 X iajs-1914 78 10 e e NOUN iajs-1914 78 11 θ θ X iajs-1914 78 12 x x SYM iajs-1914 78 13 𝜃 𝜃 NOUN iajs-1914 78 14 2e 2e NOUN iajs-1914 78 15 θ θ NOUN iajs-1914 78 16 x x PUNCT iajs-1914 78 17 𝜃 𝜃 PUNCT iajs-1914 78 18 taking take VERB iajs-1914 78 19 the the DET iajs-1914 78 20 partial partial ADJ iajs-1914 78 21 derivative derivative NOUN iajs-1914 78 22 for for ADP iajs-1914 78 23 r r PROPN iajs-1914 78 24 θ θ PROPN iajs-1914 78 25 , , PUNCT iajs-1914 78 26 θ θ PROPN iajs-1914 78 27 with with ADP iajs-1914 78 28 respect respect NOUN iajs-1914 78 29 to to ADP iajs-1914 78 30 θ θ PROPN iajs-1914 78 31 and and CCONJ iajs-1914 78 32 setting set VERB iajs-1914 78 33 it it PRON iajs-1914 78 34 equal equal ADJ iajs-1914 78 35 to to ADP iajs-1914 78 36 zero zero NUM iajs-1914 78 37 , , PUNCT iajs-1914 78 38 gives give VERB iajs-1914 78 39 𝜃 𝜃 NUM iajs-1914 78 40 e e NOUN iajs-1914 78 41 θ θ NOUN iajs-1914 78 42 x x PUNCT iajs-1914 78 43 hence hence ADV iajs-1914 78 44 , , PUNCT iajs-1914 78 45 bayes bayes PROPN iajs-1914 78 46 estimator estimator NOUN iajs-1914 78 47 relative relative ADJ iajs-1914 78 48 to to ADP iajs-1914 78 49 precautionary precautionary ADJ iajs-1914 78 50 loss loss NOUN iajs-1914 78 51 function function NOUN iajs-1914 78 52 , , PUNCT iajs-1914 78 53 denoted denote VERB iajs-1914 78 54 by by ADP iajs-1914 78 55 𝜃 𝜃 PRON iajs-1914 78 56 is be AUX iajs-1914 78 57 given give VERB iajs-1914 78 58 by by ADP iajs-1914 78 59 𝜃 𝜃 NUM iajs-1914 78 60 e e NOUN iajs-1914 78 61 θ θ NOUN iajs-1914 78 62 x x SYM iajs-1914 78 63 ( ( PUNCT iajs-1914 78 64 10 10 NUM iajs-1914 78 65 ) ) PUNCT iajs-1914 78 66 in in ADP iajs-1914 78 67 general general ADJ iajs-1914 78 68 , , PUNCT iajs-1914 78 69 𝐸 𝐸 PROPN iajs-1914 78 70 u u NOUN iajs-1914 78 71 𝛼 𝛼 NOUN iajs-1914 78 72 , , PUNCT iajs-1914 78 73 𝛽 𝛽 NOUN iajs-1914 78 74 u u X iajs-1914 78 75 𝛼 𝛼 NOUN iajs-1914 78 76 , , PUNCT iajs-1914 78 77 𝛽 𝛽 PRON iajs-1914 78 78 ∞∞ ∞∞ NOUN iajs-1914 78 79 h h NOUN iajs-1914 78 80 𝛼 𝛼 NOUN iajs-1914 78 81 , , PUNCT iajs-1914 78 82 𝛽|x 𝛽|x ADJ iajs-1914 78 83 , , PUNCT iajs-1914 78 84 … … PUNCT iajs-1914 78 85 x x SYM iajs-1914 78 86 𝑑𝛼𝑑𝛽 𝑑𝛼𝑑𝛽 NOUN iajs-1914 78 87 where where SCONJ iajs-1914 78 88 u(α u(α NOUN iajs-1914 78 89 , , PUNCT iajs-1914 78 90 β β NOUN iajs-1914 78 91 ) ) PUNCT iajs-1914 78 92 be be VERB iajs-1914 78 93 any any DET iajs-1914 78 94 function function NOUN iajs-1914 78 95 for for ADP iajs-1914 78 96 α α NOUN iajs-1914 78 97 and and CCONJ iajs-1914 78 98 β β X iajs-1914 78 99 . . PUNCT iajs-1914 79 1 therefore therefore ADV iajs-1914 79 2 , , PUNCT iajs-1914 79 3 𝐸 𝐸 PROPN iajs-1914 79 4 u u NOUN iajs-1914 79 5 𝛼 𝛼 NOUN iajs-1914 79 6 , , PUNCT iajs-1914 79 7 𝛽 𝛽 NOUN iajs-1914 79 8 , , PUNCT iajs-1914 79 9 ∞∞ ∞∞ NOUN iajs-1914 79 10 , , PUNCT iajs-1914 79 11 , , PUNCT iajs-1914 79 12 … … PUNCT iajs-1914 79 13 , , PUNCT iajs-1914 79 14 ; ; PUNCT iajs-1914 79 15 , , PUNCT iajs-1914 79 16 , , PUNCT iajs-1914 79 17 , , PUNCT iajs-1914 79 18 … … PUNCT iajs-1914 79 19 , , PUNCT iajs-1914 79 20 ; ; PUNCT iajs-1914 79 21 , , PUNCT iajs-1914 79 22 ∞∞ ∞∞ NOUN iajs-1914 79 23 i i NOUN iajs-1914 79 24 ) ) PUNCT iajs-1914 79 25 bayesian bayesian NOUN iajs-1914 79 26 estimation estimation NOUN iajs-1914 79 27 for for ADP iajs-1914 79 28 the the DET iajs-1914 79 29 shape shape NOUN iajs-1914 79 30 parameter parameter NOUN iajs-1914 79 31 α α PROPN iajs-1914 79 32 under under ADP iajs-1914 79 33 precautionary precautionary ADJ iajs-1914 79 34 loss loss NOUN iajs-1914 79 35 function function NOUN iajs-1914 79 36 to to PART iajs-1914 79 37 obtain obtain VERB iajs-1914 79 38 bayesian bayesian NOUN iajs-1914 79 39 estimation estimation NOUN iajs-1914 79 40 for for ADP iajs-1914 79 41 α α NOUN iajs-1914 79 42 , , PUNCT iajs-1914 79 43 assume assume VERB iajs-1914 79 44 that that SCONJ iajs-1914 79 45 , , PUNCT iajs-1914 79 46 𝑢 𝑢 PRON iajs-1914 79 47 𝛼 𝛼 NOUN iajs-1914 79 48 , , PUNCT iajs-1914 79 49 𝛽 𝛽 PROPN iajs-1914 79 50 𝛼 𝛼 PRON iajs-1914 79 51     SPACE iajs-1914 79 52 192 192 NUM iajs-1914 79 53         SPACE iajs-1914 79 54 ibn ibn PROPN iajs-1914 79 55 al al PROPN iajs-1914 79 56 - - PUNCT iajs-1914 79 57 haitham haitham PROPN iajs-1914 79 58 jour.for jour.for PROPN iajs-1914 79 59 pure&appl.sci pure&appl.sci PROPN iajs-1914 79 60 . . PUNCT iajs-1914 80 1 ihjpas ihjpas PROPN iajs-1914 80 2 https://doi.org/10.30526/32.1.1914 https://doi.org/10.30526/32.1.1914 X iajs-1914 80 3 vol vol NOUN iajs-1914 80 4 . . PUNCT iajs-1914 81 1 32 32 NUM iajs-1914 81 2 ( ( PUNCT iajs-1914 81 3 1 1 NUM iajs-1914 81 4 ) ) PUNCT iajs-1914 81 5 2019 2019 NUM iajs-1914 81 6 therefore therefore ADV iajs-1914 81 7 , , PUNCT iajs-1914 81 8 𝐸 𝐸 PROPN iajs-1914 81 9 𝛼 𝛼 PART iajs-1914 81 10 𝑥 𝑥 PROPN iajs-1914 81 11 ∞∞ ∞∞ NOUN iajs-1914 81 12 , , PUNCT iajs-1914 81 13 , , PUNCT iajs-1914 81 14 … … PUNCT iajs-1914 81 15 , , PUNCT iajs-1914 81 16 ; ; PUNCT iajs-1914 81 17 , , PUNCT iajs-1914 81 18 α α X iajs-1914 81 19 β β X iajs-1914 81 20 , , PUNCT iajs-1914 81 21 , , PUNCT iajs-1914 81 22 … … PUNCT iajs-1914 81 23 , , PUNCT iajs-1914 81 24 ; ; PUNCT iajs-1914 81 25 , , PUNCT iajs-1914 81 26 α α PROPN iajs-1914 81 27 β β PROPN iajs-1914 81 28 ∞∞ ∞∞ NOUN iajs-1914 81 29 notice notice VERB iajs-1914 81 30 that that SCONJ iajs-1914 81 31 , , PUNCT iajs-1914 81 32 it it PRON iajs-1914 81 33 is be AUX iajs-1914 81 34 difficult difficult ADJ iajs-1914 81 35 to to PART iajs-1914 81 36 find find VERB iajs-1914 81 37 the the DET iajs-1914 81 38 solution solution NOUN iajs-1914 81 39 of of ADP iajs-1914 81 40 the the DET iajs-1914 81 41 ratio ratio NOUN iajs-1914 81 42 of of ADP iajs-1914 81 43 two two NUM iajs-1914 81 44 integrals integral NOUN iajs-1914 81 45 . . PUNCT iajs-1914 82 1 therefore therefore ADV iajs-1914 82 2 , , PUNCT iajs-1914 82 3 lindley lindley PROPN iajs-1914 82 4 's 's PART iajs-1914 82 5 approximate approximate NOUN iajs-1914 82 6 will will AUX iajs-1914 82 7 be be AUX iajs-1914 82 8 used use VERB iajs-1914 82 9 to to PART iajs-1914 82 10 get get VERB iajs-1914 82 11 𝐸 𝐸 ADP iajs-1914 82 12 𝛼 𝛼 PART iajs-1914 82 13 𝑥 𝑥 NOUN iajs-1914 82 14 as as SCONJ iajs-1914 82 15 follows follow VERB iajs-1914 82 16 𝑢 𝑢 PRON iajs-1914 82 17 𝛼 𝛼 NOUN iajs-1914 82 18 , , PUNCT iajs-1914 82 19 𝛽 𝛽 PROPN iajs-1914 82 20 𝛼 𝛼 X iajs-1914 82 21 u u NOUN iajs-1914 82 22 α α NOUN iajs-1914 82 23 , , PUNCT iajs-1914 82 24 β β X iajs-1914 82 25 2𝛼 2𝛼 NUM iajs-1914 82 26 , , PUNCT iajs-1914 82 27 u u NOUN iajs-1914 82 28 , , PUNCT iajs-1914 82 29 2 2 NUM iajs-1914 82 30 , , PUNCT iajs-1914 82 31 u u NOUN iajs-1914 82 32 α α NOUN iajs-1914 82 33 , , PUNCT iajs-1914 82 34 β β X iajs-1914 82 35 0 0 NUM iajs-1914 82 36 , , PUNCT iajs-1914 82 37 u u NOUN iajs-1914 82 38 , , PUNCT iajs-1914 82 39 = = NOUN iajs-1914 82 40 0 0 NUM iajs-1914 82 41 𝜋 𝜋 NOUN iajs-1914 82 42 𝛼 𝛼 NOUN iajs-1914 82 43 , , PUNCT iajs-1914 82 44 𝛽 𝛽 PROPN iajs-1914 82 45 г г X iajs-1914 82 46 𝑐𝑒 𝑐𝑒 PROPN iajs-1914 82 47 𝑝 𝑝 PROPN iajs-1914 82 48 𝑙𝑛𝜋 𝑙𝑛𝜋 PROPN iajs-1914 82 49 𝛼 𝛼 PROPN iajs-1914 82 50 , , PUNCT iajs-1914 82 51 𝛽 𝛽 PROPN iajs-1914 82 52 𝑎 𝑎 PROPN iajs-1914 82 53 1 1 NUM iajs-1914 82 54 𝑙𝑛𝛼 𝑙𝑛𝛼 NOUN iajs-1914 82 55 𝑎𝑙𝑛𝑏 𝑎𝑙𝑛𝑏 VERB iajs-1914 82 56 𝑏𝛼 𝑏𝛼 NUM iajs-1914 82 57 𝑙𝑛г 𝑙𝑛г NOUN iajs-1914 82 58 𝛼 𝛼 NOUN iajs-1914 82 59 𝑙𝑛𝑐 𝑙𝑛𝑐 NOUN iajs-1914 82 60 𝑐𝛽 𝑐𝛽 ADP iajs-1914 82 61 𝑝 𝑝 PROPN iajs-1914 82 62 𝑏 𝑏 PROPN iajs-1914 82 63 , , PUNCT iajs-1914 82 64 𝑝 𝑝 PROPN iajs-1914 82 65 𝑐 𝑐 PROPN iajs-1914 82 66 recall recall VERB iajs-1914 82 67 that that PRON iajs-1914 82 68 , , PUNCT iajs-1914 82 69 ln ln ADJ iajs-1914 82 70 l l NOUN iajs-1914 82 71 ( ( PUNCT iajs-1914 82 72 x1, x1, NOUN iajs-1914 82 73 … … PUNCT iajs-1914 82 74 ,xn ,xn PUNCT iajs-1914 82 75 ; ; PUNCT iajs-1914 82 76 α α X iajs-1914 82 77 , , PUNCT iajs-1914 82 78 β β X iajs-1914 82 79 ) ) PUNCT iajs-1914 83 1 = = PUNCT iajs-1914 83 2 nα nα ADP iajs-1914 83 3 ln ln NOUN iajs-1914 83 4 β β NOUN iajs-1914 83 5 - - ADJ iajs-1914 83 6 n n CCONJ iajs-1914 83 7 ln ln ADJ iajs-1914 83 8 г(α)-β г(α)-β NOUN iajs-1914 83 9 ∑ ∑ PROPN iajs-1914 83 10 x x PROPN iajs-1914 83 11 α α PROPN iajs-1914 83 12 1 1 NUM iajs-1914 83 13 ∑ ∑ ADP iajs-1914 83 14 ln ln ADJ iajs-1914 83 15 x x NOUN iajs-1914 83 16 l12 l12 NOUN iajs-1914 83 17 = = PUNCT iajs-1914 83 18 , , PUNCT iajs-1914 83 19 β β PROPN iajs-1914 83 20 𝑙 𝑙 X iajs-1914 83 21 , , PUNCT iajs-1914 83 22 0 0 NUM iajs-1914 84 1 𝑙 𝑙 NOUN iajs-1914 84 2 , , PUNCT iajs-1914 84 3 = = SYM iajs-1914 84 4 𝑙 𝑙 X iajs-1914 84 5 , , PUNCT iajs-1914 84 6 n n CCONJ iajs-1914 84 7 ψ ψ NOUN iajs-1914 84 8 " " PUNCT iajs-1914 84 9 α α PROPN iajs-1914 84 10 σ σ PROPN iajs-1914 84 11 ψ′ ψ′ PROPN iajs-1914 84 12 α α NOUN iajs-1914 84 13 , , PUNCT iajs-1914 84 14 𝜎 𝜎 PROPN iajs-1914 84 15 𝐸 𝐸 PROPN iajs-1914 84 16 𝛼 𝛼 PART iajs-1914 84 17 𝛼 𝛼 NOUN iajs-1914 84 18 u u NOUN iajs-1914 84 19 𝜎 𝜎 PROPN iajs-1914 84 20 𝑝 𝑝 X iajs-1914 84 21 u u NOUN iajs-1914 84 22 𝜎 𝜎 NOUN iajs-1914 84 23 𝑙 𝑙 X iajs-1914 84 24 u u X iajs-1914 84 25 𝜎 𝜎 NOUN iajs-1914 84 26 𝑙 𝑙 X iajs-1914 84 27 u u X iajs-1914 84 28 𝜎 𝜎 NOUN iajs-1914 84 29 𝜎 𝜎 NOUN iajs-1914 84 30 𝛼 𝛼 ADJ iajs-1914 84 31 2 2 NUM iajs-1914 84 32 ψ′ ψ′ NUM iajs-1914 84 33 𝑏 𝑏 PROPN iajs-1914 84 34 2𝛼 2𝛼 PROPN iajs-1914 84 35 ψ′ ψ′ NUM iajs-1914 84 36 n n CCONJ iajs-1914 84 37 ψ ψ X iajs-1914 84 38 " " PUNCT iajs-1914 84 39 𝛼 𝛼 PROPN iajs-1914 84 40 ψ′ ψ′ NOUN iajs-1914 84 41 ψ′ ψ′ NUM iajs-1914 84 42 𝛼 𝛼 NOUN iajs-1914 84 43 ψ′ ψ′ NUM iajs-1914 84 44 𝑏 𝑏 PRON iajs-1914 84 45 ψ ψ NOUN iajs-1914 84 46 " " PUNCT iajs-1914 84 47 ψ′ ψ′ PUNCT iajs-1914 84 48 ( ( PUNCT iajs-1914 84 49 11 11 NUM iajs-1914 84 50 ) ) PUNCT iajs-1914 84 51 now now ADV iajs-1914 84 52 , , PUNCT iajs-1914 84 53 substituting substitute VERB iajs-1914 84 54 ( ( PUNCT iajs-1914 84 55 11 11 NUM iajs-1914 84 56 ) ) PUNCT iajs-1914 84 57 into into ADP iajs-1914 84 58 ( ( PUNCT iajs-1914 84 59 10 10 NUM iajs-1914 84 60 ) ) PUNCT iajs-1914 84 61 yields yield NOUN iajs-1914 84 62 , , PUNCT iajs-1914 84 63 𝛼 𝛼 VERB iajs-1914 84 64 𝛼 𝛼 NOUN iajs-1914 84 65 ψ′ ψ′ VERB iajs-1914 84 66 𝑏 𝑏 DET iajs-1914 84 67 ψ ψ NOUN iajs-1914 84 68 " " PUNCT iajs-1914 84 69 ψ′ ψ′ PROPN iajs-1914 84 70 ii ii PROPN iajs-1914 84 71 ) ) PUNCT iajs-1914 84 72 bayesian bayesian NOUN iajs-1914 84 73 estimation estimation NOUN iajs-1914 84 74 for for ADP iajs-1914 84 75 the the DET iajs-1914 84 76 scale scale NOUN iajs-1914 84 77 parameter parameter NOUN iajs-1914 84 78 β β PROPN iajs-1914 84 79 under under ADP iajs-1914 84 80 precautionary precautionary ADJ iajs-1914 84 81 loss loss NOUN iajs-1914 84 82 function function NOUN iajs-1914 84 83 assume assume VERB iajs-1914 84 84 that that SCONJ iajs-1914 84 85 , , PUNCT iajs-1914 84 86 u u NOUN iajs-1914 84 87 α α NOUN iajs-1914 84 88 , , PUNCT iajs-1914 84 89 β β X iajs-1914 84 90 β β X iajs-1914 84 91 then then ADV iajs-1914 84 92 , , PUNCT iajs-1914 84 93     SPACE iajs-1914 84 94 193 193 NUM iajs-1914 84 95         SPACE iajs-1914 84 96 ibn ibn PROPN iajs-1914 84 97 al al PROPN iajs-1914 84 98 - - PUNCT iajs-1914 84 99 haitham haitham PROPN iajs-1914 84 100 jour.for jour.for PROPN iajs-1914 84 101 pure&appl.sci pure&appl.sci PROPN iajs-1914 84 102 . . PUNCT iajs-1914 85 1 ihjpas ihjpas PROPN iajs-1914 85 2 https://doi.org/10.30526/32.1.1914 https://doi.org/10.30526/32.1.1914 X iajs-1914 85 3 vol vol NOUN iajs-1914 85 4 . . PUNCT iajs-1914 86 1 32 32 NUM iajs-1914 86 2 ( ( PUNCT iajs-1914 86 3 1 1 NUM iajs-1914 86 4 ) ) PUNCT iajs-1914 86 5 2019 2019 NUM iajs-1914 86 6 u u NOUN iajs-1914 86 7 α α NOUN iajs-1914 86 8 , , PUNCT iajs-1914 86 9 β β X iajs-1914 86 10 0 0 NUM iajs-1914 86 11 , , PUNCT iajs-1914 86 12 u u NOUN iajs-1914 86 13 , , PUNCT iajs-1914 86 14 = = NOUN iajs-1914 86 15 0 0 NUM iajs-1914 86 16 , , PUNCT iajs-1914 86 17 u u NOUN iajs-1914 86 18 α α NOUN iajs-1914 86 19 , , PUNCT iajs-1914 86 20 β β PROPN iajs-1914 86 21 2𝛽 2𝛽 NOUN iajs-1914 86 22 , , PUNCT iajs-1914 86 23 u u NOUN iajs-1914 86 24 , , PUNCT iajs-1914 86 25 2 2 NUM iajs-1914 86 26 thus thus ADV iajs-1914 86 27 , , PUNCT iajs-1914 86 28 𝐸 𝐸 PROPN iajs-1914 86 29 𝛽 𝛽 PROPN iajs-1914 86 30 𝛽 𝛽 NOUN iajs-1914 86 31 u u X iajs-1914 86 32 𝜎 𝜎 PROPN iajs-1914 86 33 𝑝 𝑝 X iajs-1914 86 34 u u NOUN iajs-1914 86 35 𝜎 𝜎 NOUN iajs-1914 86 36 𝑙 𝑙 X iajs-1914 86 37 u u X iajs-1914 86 38 𝜎 𝜎 PROPN iajs-1914 86 39 𝛽 𝛽 PROPN iajs-1914 86 40 𝑐 𝑐 PROPN iajs-1914 86 41 𝛽 𝛽 PROPN iajs-1914 86 42 ( ( PUNCT iajs-1914 86 43 12 12 NUM iajs-1914 86 44 ) ) PUNCT iajs-1914 86 45 after after ADP iajs-1914 86 46 substituting substitute VERB iajs-1914 86 47 ( ( PUNCT iajs-1914 86 48 12 12 NUM iajs-1914 86 49 ) ) PUNCT iajs-1914 86 50 into into ADP iajs-1914 86 51 ( ( PUNCT iajs-1914 86 52 10 10 NUM iajs-1914 86 53 ) ) PUNCT iajs-1914 86 54 yields yield NOUN iajs-1914 86 55 , , PUNCT iajs-1914 86 56 𝛽 𝛽 NOUN iajs-1914 86 57 𝛽 𝛽 NOUN iajs-1914 86 58 where where SCONJ iajs-1914 86 59 𝛼 𝛼 X iajs-1914 86 60 , , PUNCT iajs-1914 86 61 𝛽 𝛽 NOUN iajs-1914 86 62 are be AUX iajs-1914 86 63 the the DET iajs-1914 86 64 maximum maximum ADJ iajs-1914 86 65 likelihood likelihood NOUN iajs-1914 86 66 estimators estimator NOUN iajs-1914 86 67 for for ADP iajs-1914 86 68 α α NOUN iajs-1914 86 69 , , PUNCT iajs-1914 86 70 β β X iajs-1914 86 71 respectively respectively ADV iajs-1914 86 72 . . PUNCT iajs-1914 87 1 4 4 X iajs-1914 87 2 . . X iajs-1914 87 3 simulation simulation NOUN iajs-1914 87 4 study study NOUN iajs-1914 87 5 in in ADP iajs-1914 87 6 this this DET iajs-1914 87 7 section section NOUN iajs-1914 87 8 , , PUNCT iajs-1914 87 9 monte monte PROPN iajs-1914 87 10 – – PUNCT iajs-1914 87 11 carlo carlo PROPN iajs-1914 87 12 simulation simulation NOUN iajs-1914 87 13 is be AUX iajs-1914 87 14 employed employ VERB iajs-1914 87 15 to to PART iajs-1914 87 16 compare compare VERB iajs-1914 87 17 the the DET iajs-1914 87 18 performance performance NOUN iajs-1914 87 19 of of ADP iajs-1914 87 20 three three NUM iajs-1914 87 21 estimates estimate NOUN iajs-1914 87 22 ( ( PUNCT iajs-1914 87 23 moment moment NOUN iajs-1914 87 24 , , PUNCT iajs-1914 87 25 maximum maximum ADJ iajs-1914 87 26 likelihood likelihood NOUN iajs-1914 87 27 and and CCONJ iajs-1914 87 28 bayes bayes NOUN iajs-1914 87 29 estimators estimator NOUN iajs-1914 87 30 under under ADP iajs-1914 87 31 precautionary precautionary ADJ iajs-1914 87 32 loss loss NOUN iajs-1914 87 33 function function NOUN iajs-1914 87 34 ) ) PUNCT iajs-1914 87 35 for for ADP iajs-1914 87 36 unknown unknown ADJ iajs-1914 87 37 shape shape NOUN iajs-1914 87 38 and and CCONJ iajs-1914 87 39 scale scale NOUN iajs-1914 87 40 parameters parameter NOUN iajs-1914 87 41 based base VERB iajs-1914 87 42 on on ADP iajs-1914 87 43 the the DET iajs-1914 87 44 mean mean ADJ iajs-1914 87 45 squared square VERB iajs-1914 87 46 errors error NOUN iajs-1914 87 47 ( ( PUNCT iajs-1914 87 48 mse mse PROPN iajs-1914 87 49 ’s ’s PART iajs-1914 87 50 ) ) PUNCT iajs-1914 87 51 as as SCONJ iajs-1914 87 52 follows follow VERB iajs-1914 87 53 mse(𝜃 mse(𝜃 PROPN iajs-1914 87 54 = = PUNCT iajs-1914 87 55 ∑ ∑ PROPN iajs-1914 87 56 where where SCONJ iajs-1914 87 57 , , PUNCT iajs-1914 87 58 i i PRON iajs-1914 87 59 is be AUX iajs-1914 87 60 the the DET iajs-1914 87 61 number number NOUN iajs-1914 87 62 of of ADP iajs-1914 87 63 replications replication NOUN iajs-1914 87 64 . . PUNCT iajs-1914 88 1 we we PRON iajs-1914 88 2 generated generate VERB iajs-1914 88 3 i i NOUN iajs-1914 88 4 = = NOUN iajs-1914 88 5 3000 3000 NUM iajs-1914 88 6 samples sample NOUN iajs-1914 88 7 of of ADP iajs-1914 88 8 size size NOUN iajs-1914 88 9 n n NOUN iajs-1914 88 10 = = SYM iajs-1914 88 11 20 20 NUM iajs-1914 88 12 , , PUNCT iajs-1914 88 13 30 30 NUM iajs-1914 88 14 , , PUNCT iajs-1914 88 15 50 50 NUM iajs-1914 88 16 , , PUNCT iajs-1914 88 17 and and CCONJ iajs-1914 88 18 100 100 NUM iajs-1914 88 19 to to PART iajs-1914 88 20 represent represent VERB iajs-1914 88 21 small small ADJ iajs-1914 88 22 , , PUNCT iajs-1914 88 23 moderate moderate ADJ iajs-1914 88 24 and and CCONJ iajs-1914 88 25 large large ADJ iajs-1914 88 26 sample sample NOUN iajs-1914 88 27 sizes size NOUN iajs-1914 88 28 from from ADP iajs-1914 88 29 gamma gamma NOUN iajs-1914 88 30 distribution distribution NOUN iajs-1914 88 31 with with ADP iajs-1914 88 32 α α NOUN iajs-1914 88 33 = = SYM iajs-1914 88 34 2 2 NUM iajs-1914 88 35 , , PUNCT iajs-1914 88 36 3 3 NUM iajs-1914 88 37 and and CCONJ iajs-1914 88 38 β β X iajs-1914 88 39 = = SYM iajs-1914 88 40 0.5 0.5 NUM iajs-1914 88 41 , , PUNCT iajs-1914 88 42 1 1 NUM iajs-1914 88 43 . . PUNCT iajs-1914 89 1 the the DET iajs-1914 89 2 values value NOUN iajs-1914 89 3 of of ADP iajs-1914 89 4 α α PROPN iajs-1914 89 5 's 's PART iajs-1914 89 6 prior prior ADJ iajs-1914 89 7 parameters parameter NOUN iajs-1914 89 8 are be AUX iajs-1914 89 9 chosen choose VERB iajs-1914 89 10 as as ADP iajs-1914 89 11 a a DET iajs-1914 89 12 = = SYM iajs-1914 89 13 3 3 NUM iajs-1914 89 14 , , PUNCT iajs-1914 89 15 b b X iajs-1914 89 16 = = SYM iajs-1914 89 17 3 3 NUM iajs-1914 89 18 and and CCONJ iajs-1914 89 19 for for ADP iajs-1914 89 20 β β PROPN iajs-1914 89 21 's 's PART iajs-1914 89 22 prior prior ADJ iajs-1914 89 23 parameter parameter NOUN iajs-1914 89 24 , , PUNCT iajs-1914 89 25 c c NOUN iajs-1914 89 26 = = SYM iajs-1914 89 27 4 4 NUM iajs-1914 89 28 . . NOUN iajs-1914 89 29 5 5 NUM iajs-1914 89 30 . . X iajs-1914 89 31 discussion discussion NOUN iajs-1914 89 32 and and CCONJ iajs-1914 89 33 conclusion conclusion NOUN iajs-1914 89 34 the the DET iajs-1914 89 35 expected expect VERB iajs-1914 89 36 values value NOUN iajs-1914 89 37 and and CCONJ iajs-1914 89 38 ( ( PUNCT iajs-1914 89 39 mse mse NOUN iajs-1914 89 40 's 's PART iajs-1914 89 41 ) ) PUNCT iajs-1914 89 42 for for ADP iajs-1914 89 43 estimating estimate VERB iajs-1914 89 44 α α PROPN iajs-1914 89 45 and and CCONJ iajs-1914 89 46 β β X iajs-1914 89 47 are be AUX iajs-1914 89 48 tabulated tabulate VERB iajs-1914 89 49 in in ADP iajs-1914 89 50 tables table NOUN iajs-1914 89 51 ( ( PUNCT iajs-1914 89 52 1 1 NUM iajs-1914 89 53 - - SYM iajs-1914 89 54 8) 8) NUM iajs-1914 89 55 . . PUNCT iajs-1914 90 1 the the DET iajs-1914 90 2 results result NOUN iajs-1914 90 3 of of ADP iajs-1914 90 4 the the DET iajs-1914 90 5 tables table NOUN iajs-1914 90 6 can can AUX iajs-1914 90 7 be be AUX iajs-1914 90 8 summarized summarize VERB iajs-1914 90 9 by by ADP iajs-1914 90 10 the the DET iajs-1914 90 11 following follow VERB iajs-1914 90 12 points point NOUN iajs-1914 90 13 1 1 NUM iajs-1914 90 14 . . PUNCT iajs-1914 91 1 the the DET iajs-1914 91 2 performance performance NOUN iajs-1914 91 3 of of ADP iajs-1914 91 4 bayes bayes PROPN iajs-1914 91 5 estimates estimate NOUN iajs-1914 91 6 under under ADP iajs-1914 91 7 precautionary precautionary ADJ iajs-1914 91 8 loss loss NOUN iajs-1914 91 9 function function NOUN iajs-1914 91 10 for for ADP iajs-1914 91 11 two two NUM iajs-1914 91 12 parameters parameter NOUN iajs-1914 91 13 α α NOUN iajs-1914 91 14 and and CCONJ iajs-1914 91 15 β β X iajs-1914 91 16 are be AUX iajs-1914 91 17 the the DET iajs-1914 91 18 best good ADJ iajs-1914 91 19 , , PUNCT iajs-1914 91 20 since since SCONJ iajs-1914 91 21 they they PRON iajs-1914 91 22 give give VERB iajs-1914 91 23 smallest small ADJ iajs-1914 91 24 mean mean ADJ iajs-1914 91 25 square square NOUN iajs-1914 91 26 error error NOUN iajs-1914 91 27 , , PUNCT iajs-1914 91 28 as as SCONJ iajs-1914 91 29 indicated indicate VERB iajs-1914 91 30 for for ADP iajs-1914 91 31 all all DET iajs-1914 91 32 combinations combination NOUN iajs-1914 91 33 of of ADP iajs-1914 91 34 initial initial ADJ iajs-1914 91 35 values value NOUN iajs-1914 91 36 of of ADP iajs-1914 91 37 parameters parameter NOUN iajs-1914 91 38 . . PUNCT iajs-1914 92 1 followed follow VERB iajs-1914 92 2 by by ADP iajs-1914 92 3 maximum maximum ADJ iajs-1914 92 4 - - PUNCT iajs-1914 92 5 likelihood likelihood NOUN iajs-1914 92 6 estimates estimate NOUN iajs-1914 92 7 , , PUNCT iajs-1914 92 8 for for ADP iajs-1914 92 9 all all DET iajs-1914 92 10 cases case NOUN iajs-1914 92 11 2 2 NUM iajs-1914 92 12 . . PUNCT iajs-1914 93 1 it it PRON iajs-1914 93 2 is be AUX iajs-1914 93 3 clear clear ADJ iajs-1914 93 4 that that SCONJ iajs-1914 93 5 , , PUNCT iajs-1914 93 6 the the DET iajs-1914 93 7 result result NOUN iajs-1914 93 8 for for ADP iajs-1914 93 9 α α PROPN iajs-1914 93 10 ( ( PUNCT iajs-1914 93 11 expected expect VERB iajs-1914 93 12 values value NOUN iajs-1914 93 13 and and CCONJ iajs-1914 93 14 mse mse NOUN iajs-1914 93 15 's 's PART iajs-1914 93 16 ) ) PUNCT iajs-1914 93 17 at at ADP iajs-1914 93 18 β β X iajs-1914 93 19 = = SYM iajs-1914 93 20 0.5 0.5 NUM iajs-1914 93 21 are be AUX iajs-1914 93 22 the the DET iajs-1914 93 23 same same ADJ iajs-1914 93 24 as as ADP iajs-1914 93 25 the the DET iajs-1914 93 26 corresponding corresponding ADJ iajs-1914 93 27 result result NOUN iajs-1914 93 28 when when SCONJ iajs-1914 93 29 β β X iajs-1914 93 30 = = SYM iajs-1914 93 31 1 1 NUM iajs-1914 93 32 , , PUNCT iajs-1914 93 33 the the DET iajs-1914 93 34 reason reason NOUN iajs-1914 93 35 can can AUX iajs-1914 93 36 be be AUX iajs-1914 93 37 clarified clarify VERB iajs-1914 93 38 easily easily ADV iajs-1914 93 39 , , PUNCT iajs-1914 93 40 as as SCONJ iajs-1914 93 41 follows follow VERB iajs-1914 93 42 according accord VERB iajs-1914 93 43 to to ADP iajs-1914 93 44 moment moment NOUN iajs-1914 93 45 method method NOUN iajs-1914 93 46 we we PRON iajs-1914 93 47 have have VERB iajs-1914 93 48 𝛼 𝛼 NUM iajs-1914 93 49 ̅ ̅ NOUN iajs-1914 93 50 ∑ ∑ NOUN iajs-1914 93 51 ̅ ̅ NOUN iajs-1914 93 52 = = SYM iajs-1914 93 53 ∑ ∑ PUNCT iajs-1914 93 54 𝑥 𝑥 PROPN iajs-1914 93 55 ( ( PUNCT iajs-1914 93 56 13 13 NUM iajs-1914 93 57 ) ) PUNCT iajs-1914 93 58 note note VERB iajs-1914 93 59 that that SCONJ iajs-1914 93 60 , , PUNCT iajs-1914 93 61 𝑥 𝑥 X iajs-1914 93 62 , , PUNCT iajs-1914 93 63 𝑥 𝑥 X iajs-1914 93 64 , , PUNCT iajs-1914 93 65 … … PUNCT iajs-1914 93 66 , , PUNCT iajs-1914 93 67 𝑥 𝑥 PROPN iajs-1914 93 68 is be AUX iajs-1914 93 69 a a DET iajs-1914 93 70 random random ADJ iajs-1914 93 71 sample sample NOUN iajs-1914 93 72 from from ADP iajs-1914 93 73 a a DET iajs-1914 93 74 gamma gamma NOUN iajs-1914 93 75 distribution distribution NOUN iajs-1914 93 76 defined define VERB iajs-1914 93 77 by by ADP iajs-1914 93 78 ( ( PUNCT iajs-1914 93 79 1 1 NUM iajs-1914 93 80 ) ) PUNCT iajs-1914 93 81 , , PUNCT iajs-1914 93 82 where where SCONJ iajs-1914 93 83 each each DET iajs-1914 93 84 observation observation NOUN iajs-1914 93 85 say say VERB iajs-1914 93 86 𝑥 𝑥 NOUN iajs-1914 93 87 is be AUX iajs-1914 93 88 generated generate VERB iajs-1914 93 89 independently independently ADV iajs-1914 93 90 and and CCONJ iajs-1914 93 91 identically identically ADV iajs-1914 93 92 by by ADP iajs-1914 93 93 the the DET iajs-1914 93 94 following follow VERB iajs-1914 93 95 equation equation NOUN iajs-1914 93 96     SPACE iajs-1914 93 97 194 194 NUM iajs-1914 93 98         SPACE iajs-1914 93 99 ibn ibn PROPN iajs-1914 93 100 al al PROPN iajs-1914 93 101 - - PUNCT iajs-1914 93 102 haitham haitham PROPN iajs-1914 93 103 jour.for jour.for PROPN iajs-1914 93 104 pure&appl.sci pure&appl.sci PROPN iajs-1914 93 105 . . PUNCT iajs-1914 94 1 ihjpas ihjpas PROPN iajs-1914 94 2 https://doi.org/10.30526/32.1.1914 https://doi.org/10.30526/32.1.1914 X iajs-1914 94 3 vol vol NOUN iajs-1914 94 4 . . PUNCT iajs-1914 95 1 32 32 NUM iajs-1914 95 2 ( ( PUNCT iajs-1914 95 3 1 1 NUM iajs-1914 95 4 ) ) PUNCT iajs-1914 95 5 2019 2019 NUM iajs-1914 95 6 𝑥 𝑥 PRON iajs-1914 95 7 ∑ ∑ ADV iajs-1914 95 8 log log VERB iajs-1914 95 9 𝑢 𝑢 PRON iajs-1914 95 10 , , PUNCT iajs-1914 95 11 i i PRON iajs-1914 95 12 = = NOUN iajs-1914 95 13 1 1 NUM iajs-1914 95 14 , , PUNCT iajs-1914 95 15 2 2 NUM iajs-1914 95 16 , , PUNCT iajs-1914 95 17 … … PUNCT iajs-1914 95 18 , , PUNCT iajs-1914 95 19 n n CCONJ iajs-1914 95 20 ( ( PUNCT iajs-1914 95 21 14 14 NUM iajs-1914 95 22 ) ) PUNCT iajs-1914 95 23 where where SCONJ iajs-1914 95 24 , , PUNCT iajs-1914 95 25 𝑢 𝑢 PRON iajs-1914 95 26 is be AUX iajs-1914 95 27 a a DET iajs-1914 95 28 random random ADJ iajs-1914 95 29 number number NOUN iajs-1914 95 30 followed follow VERB iajs-1914 95 31 uniform uniform ADJ iajs-1914 95 32 distribution distribution NOUN iajs-1914 95 33 with with ADP iajs-1914 95 34 ( ( PUNCT iajs-1914 95 35 0,1 0,1 NUM iajs-1914 95 36 ) ) PUNCT iajs-1914 95 37 , , PUNCT iajs-1914 95 38 i.e. i.e. X iajs-1914 95 39 , , PUNCT iajs-1914 95 40 𝑢 𝑢 X iajs-1914 95 41 ~ ~ PUNCT iajs-1914 95 42 𝑈 𝑈 PROPN iajs-1914 95 43 0 0 NUM iajs-1914 95 44 , , PUNCT iajs-1914 95 45 1 1 NUM iajs-1914 95 46 after after ADP iajs-1914 95 47 substituting substitute VERB iajs-1914 95 48 ( ( PUNCT iajs-1914 95 49 14 14 NUM iajs-1914 95 50 ) ) PUNCT iajs-1914 95 51 into into ADP iajs-1914 95 52 ( ( PUNCT iajs-1914 95 53 13 13 NUM iajs-1914 95 54 ) ) PUNCT iajs-1914 95 55 yields yield NOUN iajs-1914 95 56 , , PUNCT iajs-1914 95 57 𝛼 𝛼 PROPN iajs-1914 95 58 𝛽 𝛽 NOUN iajs-1914 95 59 𝑛 𝑛 PROPN iajs-1914 95 60 1 1 NUM iajs-1914 95 61 𝛽 𝛽 NOUN iajs-1914 95 62 log log NOUN iajs-1914 95 63 𝑢 𝑢 PRON iajs-1914 95 64 therefore therefore ADV iajs-1914 95 65 , , PUNCT iajs-1914 95 66 𝛽 𝛽 NOUN iajs-1914 95 67 will will AUX iajs-1914 95 68 be be AUX iajs-1914 95 69 canceled cancel VERB iajs-1914 95 70 from from ADP iajs-1914 95 71 moment moment NOUN iajs-1914 95 72 estimation estimation NOUN iajs-1914 95 73 for for ADP iajs-1914 95 74 𝛼. 𝛼. NOUN iajs-1914 95 75 recall recall PROPN iajs-1914 95 76 that that SCONJ iajs-1914 95 77 , , PUNCT iajs-1914 95 78 the the DET iajs-1914 95 79 moment moment NOUN iajs-1914 95 80 is be AUX iajs-1914 95 81 the the DET iajs-1914 95 82 initial initial ADJ iajs-1914 95 83 value value NOUN iajs-1914 95 84 for for ADP iajs-1914 95 85 mle mle PROPN iajs-1914 95 86 . . PUNCT iajs-1914 96 1 also also ADV iajs-1914 96 2 bayesian bayesian NOUN iajs-1914 96 3 estimator estimator NOUN iajs-1914 96 4 are be AUX iajs-1914 96 5 depending depend VERB iajs-1914 96 6 on on ADP iajs-1914 96 7 mle mle PROPN iajs-1914 96 8 , , PUNCT iajs-1914 96 9 so so ADV iajs-1914 96 10 the the DET iajs-1914 96 11 result result NOUN iajs-1914 96 12 for for ADP iajs-1914 96 13 expected expect VERB iajs-1914 96 14 values value NOUN iajs-1914 96 15 and and CCONJ iajs-1914 96 16 mse mse NOUN iajs-1914 96 17 for for ADP iajs-1914 96 18 𝛼 𝛼 NOUN iajs-1914 96 19 are be AUX iajs-1914 96 20 the the DET iajs-1914 96 21 same same ADJ iajs-1914 96 22 as as ADP iajs-1914 96 23 the the DET iajs-1914 96 24 corresponding corresponding ADJ iajs-1914 96 25 value value NOUN iajs-1914 96 26 of of ADP iajs-1914 96 27 𝛼 𝛼 PRON iajs-1914 96 28 for for ADP iajs-1914 96 29 different different ADJ iajs-1914 96 30 values value NOUN iajs-1914 96 31 of of ADP iajs-1914 96 32 𝛽. 𝛽. NOUN iajs-1914 96 33 3 3 NUM iajs-1914 96 34 . . PUNCT iajs-1914 97 1 it it PRON iajs-1914 97 2 is be AUX iajs-1914 97 3 observed observe VERB iajs-1914 97 4 that that SCONJ iajs-1914 97 5 , , PUNCT iajs-1914 97 6 mse mse PROPN iajs-1914 97 7 's 's PART iajs-1914 97 8 of of ADP iajs-1914 97 9 all all DET iajs-1914 97 10 estimators estimator NOUN iajs-1914 97 11 of of ADP iajs-1914 97 12 shape shape NOUN iajs-1914 97 13 parameter parameter NOUN iajs-1914 97 14 is be AUX iajs-1914 97 15 increasing increase VERB iajs-1914 97 16 with with ADP iajs-1914 97 17 the the DET iajs-1914 97 18 increase increase NOUN iajs-1914 97 19 of of ADP iajs-1914 97 20 the the DET iajs-1914 97 21 value value NOUN iajs-1914 97 22 of of ADP iajs-1914 97 23 the the DET iajs-1914 97 24 shape shape NOUN iajs-1914 97 25 parameter parameter NOUN iajs-1914 97 26 . . PUNCT iajs-1914 98 1 also also ADV iajs-1914 98 2 , , PUNCT iajs-1914 98 3 mse mse NOUN iajs-1914 98 4 values value NOUN iajs-1914 98 5 for for ADP iajs-1914 98 6 all all DET iajs-1914 98 7 estimates estimate NOUN iajs-1914 98 8 are be AUX iajs-1914 98 9 increasing increase VERB iajs-1914 98 10 with with ADP iajs-1914 98 11 the the DET iajs-1914 98 12 increase increase NOUN iajs-1914 98 13 of of ADP iajs-1914 98 14 the the DET iajs-1914 98 15 scale scale NOUN iajs-1914 98 16 parameter parameter NOUN iajs-1914 98 17 value value NOUN iajs-1914 98 18 in in ADP iajs-1914 98 19 all all DET iajs-1914 98 20 cases case NOUN iajs-1914 98 21 . . PUNCT iajs-1914 99 1 table table NOUN iajs-1914 99 2 1 1 NUM iajs-1914 99 3 . . PUNCT iajs-1914 100 1 the the DET iajs-1914 100 2 expected expect VERB iajs-1914 100 3 values value NOUN iajs-1914 100 4 for for ADP iajs-1914 100 5 different different ADJ iajs-1914 100 6 estimators estimator NOUN iajs-1914 100 7 for for ADP iajs-1914 100 8 unknown unknown ADJ iajs-1914 100 9 shape shape NOUN iajs-1914 100 10 parameter parameter NOUN iajs-1914 100 11 α α PROPN iajs-1914 100 12 of of ADP iajs-1914 100 13 gamma gamma NOUN iajs-1914 100 14 distribution distribution NOUN iajs-1914 100 15 when when SCONJ iajs-1914 100 16 α α NOUN iajs-1914 100 17 = = SYM iajs-1914 100 18 2 2 NUM iajs-1914 100 19 method method NOUN iajs-1914 100 20 n n NOUN iajs-1914 100 21 𝛼 𝛼 NOUN iajs-1914 100 22 𝛼 𝛼 NOUN iajs-1914 100 23 𝛼 𝛼 NOUN iajs-1914 100 24 β=0.5 β=0.5 NOUN iajs-1914 100 25 β=1 β=1 PUNCT iajs-1914 100 26 β=0.5 β=0.5 NOUN iajs-1914 100 27 β=1 β=1 PRON iajs-1914 100 28 β=0.5 β=0.5 NOUN iajs-1914 100 29 β=1 β=1 SYM iajs-1914 100 30 20 20 NUM iajs-1914 100 31 2.486393 2.486393 NUM iajs-1914 100 32 2.486393 2.486393 NUM iajs-1914 100 33 2.33479 2.33479 NUM iajs-1914 100 34 2.334791 2.334791 NUM iajs-1914 100 35 2.14737 2.14737 NUM iajs-1914 100 36 2.147371 2.147371 NUM iajs-1914 100 37 30 30 NUM iajs-1914 100 38 2.298321 2.298321 NUM iajs-1914 100 39 2.298321 2.298321 NUM iajs-1914 100 40 2.194657 2.194657 NUM iajs-1914 100 41 2.194658 2.194658 NUM iajs-1914 100 42 2.085778 2.085778 NUM iajs-1914 100 43 2.085778 2.085778 NUM iajs-1914 100 44 50 50 NUM iajs-1914 100 45 2.183145 2.183145 NUM iajs-1914 100 46 2.183145 2.183145 NUM iajs-1914 100 47 2.118412 2.118412 NUM iajs-1914 100 48 2.118412 2.118412 NUM iajs-1914 100 49 2.058392 2.058392 NUM iajs-1914 100 50 2.058392 2.058392 NUM iajs-1914 100 51 100 100 NUM iajs-1914 100 52 2.090724 2.090724 NUM iajs-1914 100 53 2.090724 2.090724 NUM iajs-1914 100 54 2.055311 2.055311 NUM iajs-1914 100 55 2.055311 2.055311 NUM iajs-1914 100 56 2.027358 2.027358 NUM iajs-1914 100 57 2.027357 2.027357 NUM iajs-1914 100 58 table table NOUN iajs-1914 100 59 2 2 NUM iajs-1914 100 60 . . PUNCT iajs-1914 101 1 the the DET iajs-1914 101 2 expected expect VERB iajs-1914 101 3 values value NOUN iajs-1914 101 4 for for ADP iajs-1914 101 5 different different ADJ iajs-1914 101 6 estimators estimator NOUN iajs-1914 101 7 for for ADP iajs-1914 101 8 unknown unknown ADJ iajs-1914 101 9 shape shape NOUN iajs-1914 101 10 parameter parameter NOUN iajs-1914 101 11 α α PROPN iajs-1914 101 12 of of ADP iajs-1914 101 13 gamma gamma NOUN iajs-1914 101 14 distribution distribution NOUN iajs-1914 101 15 when when SCONJ iajs-1914 101 16 α α PROPN iajs-1914 101 17 = = SYM iajs-1914 101 18 3 3 NUM iajs-1914 101 19 method method NOUN iajs-1914 101 20 n n X iajs-1914 101 21 𝛼 𝛼 NOUN iajs-1914 101 22 𝛼 𝛼 NOUN iajs-1914 101 23 𝛼 𝛼 NOUN iajs-1914 101 24 β=0.5 β=0.5 NOUN iajs-1914 101 25 β=1 β=1 PUNCT iajs-1914 101 26 β=0.5 β=0.5 NOUN iajs-1914 101 27 β=1 β=1 PRON iajs-1914 101 28 β=0.5 β=0.5 NOUN iajs-1914 101 29 β=1 β=1 SYM iajs-1914 101 30 20 20 NUM iajs-1914 101 31 3.600494 3.600494 NUM iajs-1914 101 32 3.600494 3.600494 NUM iajs-1914 101 33 3.447432 3.447432 NUM iajs-1914 101 34 3.447433 3.447433 NUM iajs-1914 101 35 3.091556 3.091556 NUM iajs-1914 101 36 3.091556 3.091556 NUM iajs-1914 101 37 30 30 NUM iajs-1914 101 38 3.405721 3.405721 NUM iajs-1914 101 39 3.405721 3.405721 NUM iajs-1914 101 40 3.299321 3.299321 NUM iajs-1914 101 41 3.299319 3.299319 NUM iajs-1914 101 42 3.082031 3.082031 NUM iajs-1914 101 43 3.08203 3.08203 NUM iajs-1914 101 44 50 50 NUM iajs-1914 101 45 3.255532 3.255532 NUM iajs-1914 101 46 3.405721 3.405721 NUM iajs-1914 101 47 3.18809 3.18809 NUM iajs-1914 101 48 3.299319 3.299319 NUM iajs-1914 101 49 3.066335 3.066335 NUM iajs-1914 101 50 3.08203 3.08203 NUM iajs-1914 101 51 100 100 NUM iajs-1914 101 52 3.126059 3.126059 NUM iajs-1914 101 53 3.126059 3.126059 NUM iajs-1914 101 54 3.089527 3.089527 NUM iajs-1914 101 55 3.089528 3.089528 NUM iajs-1914 101 56 3.032201 3.032201 NUM iajs-1914 101 57 3.032202 3.032202 NUM iajs-1914 101 58 table table NOUN iajs-1914 101 59 3 3 NUM iajs-1914 101 60 . . PUNCT iajs-1914 102 1 the the DET iajs-1914 102 2 mse mse PROPN iajs-1914 102 3 values value NOUN iajs-1914 102 4 for for ADP iajs-1914 102 5 different different ADJ iajs-1914 102 6 estimators estimator NOUN iajs-1914 102 7 for for ADP iajs-1914 102 8 unknown unknown ADJ iajs-1914 102 9 shape shape NOUN iajs-1914 102 10 parameter parameter NOUN iajs-1914 102 11 α α PROPN iajs-1914 102 12 of of ADP iajs-1914 102 13 gamma gamma NOUN iajs-1914 102 14 distribution distribution NOUN iajs-1914 102 15 when when SCONJ iajs-1914 102 16 α α NOUN iajs-1914 102 17 = = SYM iajs-1914 102 18 2 2 NUM iajs-1914 102 19 method method NOUN iajs-1914 102 20 n n NOUN iajs-1914 102 21 𝛼 𝛼 NOUN iajs-1914 102 22 𝛼 𝛼 NOUN iajs-1914 102 23 𝛼 𝛼 NOUN iajs-1914 102 24 β=0.5 β=0.5 NOUN iajs-1914 102 25 β=1 β=1 PUNCT iajs-1914 102 26 β=0.5 β=0.5 NOUN iajs-1914 102 27 β=1 β=1 PRON iajs-1914 102 28 β=0.5 β=0.5 NOUN iajs-1914 102 29 β=1 β=1 SYM iajs-1914 102 30 20 20 NUM iajs-1914 102 31 1.13161 1.13161 NUM iajs-1914 102 32 1.13161 1.13161 NUM iajs-1914 102 33 0.80765 0.80765 NUM iajs-1914 102 34 0.80765 0.80765 NUM iajs-1914 102 35 0.52387 0.52387 NUM iajs-1914 102 36 0.52387 0.52387 NUM iajs-1914 102 37 30 30 NUM iajs-1914 102 38 0.58915 0.58915 NUM iajs-1914 102 39 0.58915 0.58915 NUM iajs-1914 102 40 0.38833 0.38833 NUM iajs-1914 102 41 0.38833 0.38833 NUM iajs-1914 102 42 0.29354 0.29354 NUM iajs-1914 102 43 0.29354 0.29354 NUM iajs-1914 102 44 50 50 NUM iajs-1914 102 45 0.29714 0.29714 NUM iajs-1914 102 46 0.29714 0.29714 NUM iajs-1914 102 47 0.18510 0.18510 NUM iajs-1914 102 48 0.18510 0.18510 NUM iajs-1914 102 49 0.15579 0.15579 NUM iajs-1914 102 50 0.15579 0.15579 NUM iajs-1914 102 51 100 100 NUM iajs-1914 102 52 0.13609 0.13609 NUM iajs-1914 102 53 0.13609 0.13609 NUM iajs-1914 102 54 0.08313 0.08313 NUM iajs-1914 102 55 0.08313 0.08313 NUM iajs-1914 102 56 0.07647 0.07647 NUM iajs-1914 102 57 0.07647 0.07647 NUM iajs-1914 102 58     SPACE iajs-1914 102 59 195 195 NUM iajs-1914 102 60         SPACE iajs-1914 102 61 ibn ibn PROPN iajs-1914 102 62 al al PROPN iajs-1914 102 63 - - PUNCT iajs-1914 102 64 haitham haitham PROPN iajs-1914 102 65 jour.for jour.for PROPN iajs-1914 102 66 pure&appl.sci pure&appl.sci PROPN iajs-1914 102 67 . . PUNCT iajs-1914 102 68 ihjpas ihjpas PROPN iajs-1914 102 69 https://doi.org/10.30526/32.1.1914 https://doi.org/10.30526/32.1.1914 X iajs-1914 102 70 vol vol NOUN iajs-1914 102 71 . . PUNCT iajs-1914 103 1 32 32 NUM iajs-1914 103 2 ( ( PUNCT iajs-1914 103 3 1 1 NUM iajs-1914 103 4 ) ) SYM iajs-1914 103 5 2019 2019 NUM iajs-1914 103 6 table table NOUN iajs-1914 103 7 4 4 NUM iajs-1914 103 8 . . PUNCT iajs-1914 104 1 the the DET iajs-1914 104 2 mse mse PROPN iajs-1914 104 3 values value NOUN iajs-1914 104 4 for for ADP iajs-1914 104 5 different different ADJ iajs-1914 104 6 estimators estimator NOUN iajs-1914 104 7 for for ADP iajs-1914 104 8 unknown unknown ADJ iajs-1914 104 9 shape shape NOUN iajs-1914 104 10 parameter parameter NOUN iajs-1914 104 11 α α PROPN iajs-1914 104 12 of of ADP iajs-1914 104 13 gamma gamma NOUN iajs-1914 104 14 distribution distribution NOUN iajs-1914 104 15 when when SCONJ iajs-1914 104 16 α α PROPN iajs-1914 104 17 = = SYM iajs-1914 104 18 3 3 NUM iajs-1914 104 19 method method NOUN iajs-1914 104 20 n n X iajs-1914 104 21 𝛼 𝛼 NOUN iajs-1914 104 22 𝛼 𝛼 NOUN iajs-1914 104 23 𝛼 𝛼 NOUN iajs-1914 104 24 β=0.5 β=0.5 NOUN iajs-1914 104 25 β=1 β=1 PUNCT iajs-1914 104 26 β=0.5 β=0.5 NOUN iajs-1914 104 27 β=1 β=1 PRON iajs-1914 104 28 β=0.5 β=0.5 NOUN iajs-1914 104 29 β=1 β=1 SYM iajs-1914 104 30 20 20 NUM iajs-1914 104 31 2.06203 2.06203 NUM iajs-1914 104 32 2.06203 2.06203 NUM iajs-1914 104 33 1.62539 1.62539 NUM iajs-1914 104 34 1.62540 1.62540 NUM iajs-1914 104 35 1.02131 1.02131 NUM iajs-1914 104 36 1.02132 1.02132 NUM iajs-1914 104 37 30 30 NUM iajs-1914 104 38 1.11338 1.11338 NUM iajs-1914 104 39 1.11338 1.11338 NUM iajs-1914 104 40 0.84883 0.84883 NUM iajs-1914 104 41 0.84883 0.84883 NUM iajs-1914 104 42 0.62012 0.62012 NUM iajs-1914 104 43 0.62012 0.62012 NUM iajs-1914 104 44 50 50 NUM iajs-1914 104 45 0.61598 0.61598 NUM iajs-1914 104 46 1.11338 1.11338 NUM iajs-1914 104 47 0.44923 0.44923 NUM iajs-1914 104 48 0.84883 0.84883 NUM iajs-1914 104 49 0.37063 0.37063 NUM iajs-1914 104 50 0.62012 0.62012 NUM iajs-1914 104 51 100 100 NUM iajs-1914 104 52 0.25924 0.25924 NUM iajs-1914 104 53 0.25924 0.25924 NUM iajs-1914 104 54 0.18543 0.18543 NUM iajs-1914 104 55 0.18543 0.18543 NUM iajs-1914 104 56 0.16827 0.16827 NUM iajs-1914 104 57 0.16827 0.16827 NUM iajs-1914 104 58 table table NOUN iajs-1914 104 59 5 5 NUM iajs-1914 104 60 . . PUNCT iajs-1914 105 1 the the DET iajs-1914 105 2 expected expect VERB iajs-1914 105 3 values value NOUN iajs-1914 105 4 for for ADP iajs-1914 105 5 different different ADJ iajs-1914 105 6 estimators estimator NOUN iajs-1914 105 7 for for ADP iajs-1914 105 8 unknown unknown ADJ iajs-1914 105 9 scale scale NOUN iajs-1914 105 10 parameter parameter NOUN iajs-1914 105 11 β β X iajs-1914 105 12 of of ADP iajs-1914 105 13 gamma gamma NOUN iajs-1914 105 14 distribution distribution NOUN iajs-1914 105 15 when when SCONJ iajs-1914 105 16 β β X iajs-1914 105 17 = = SYM iajs-1914 105 18 0.5 0.5 NUM iajs-1914 105 19 method method NOUN iajs-1914 105 20 n n X iajs-1914 105 21 𝛼 𝛼 NOUN iajs-1914 105 22 𝛼 𝛼 NOUN iajs-1914 105 23 𝛼 𝛼 NOUN iajs-1914 105 24 β=0.5 β=0.5 NOUN iajs-1914 105 25 β=1 β=1 PUNCT iajs-1914 105 26 β=0.5 β=0.5 NOUN iajs-1914 105 27 β=1 β=1 PRON iajs-1914 105 28 β=0.5 β=0.5 NOUN iajs-1914 105 29 β=1 β=1 SYM iajs-1914 105 30 20 20 NUM iajs-1914 105 31 0.63802 0.63802 NUM iajs-1914 105 32 0.61058 0.61058 NUM iajs-1914 105 33 0.59870 0.59870 NUM iajs-1914 105 34 0.58464 0.58464 NUM iajs-1914 105 35 0.58620 0.58620 NUM iajs-1914 105 36 0.57713 0.57713 NUM iajs-1914 105 37 30 30 NUM iajs-1914 105 38 0.58456 0.58456 NUM iajs-1914 105 39 0.57368 0.57368 NUM iajs-1914 105 40 0.55831 0.55831 NUM iajs-1914 105 41 0.55562 0.55562 NUM iajs-1914 105 42 0.55171 0.55171 NUM iajs-1914 105 43 0.55141 0.55141 NUM iajs-1914 105 44 50 50 NUM iajs-1914 105 45 0.55128 0.55128 NUM iajs-1914 105 46 0.54540 0.54540 NUM iajs-1914 105 47 0.53514 0.53514 NUM iajs-1914 105 48 0.53420 0.53420 NUM iajs-1914 105 49 0.53178 0.53178 NUM iajs-1914 105 50 0.53201 0.53201 NUM iajs-1914 105 51 100 100 NUM iajs-1914 105 52 0.52472 0.52472 NUM iajs-1914 105 53 0.52256 0.52256 NUM iajs-1914 105 54 0.51588 0.51588 NUM iajs-1914 105 55 0.51658 0.51658 NUM iajs-1914 105 56 0.51444 0.51444 NUM iajs-1914 105 57 0.51562 0.51562 NUM iajs-1914 105 58 table table NOUN iajs-1914 105 59 6 6 NUM iajs-1914 105 60 . . PUNCT iajs-1914 106 1 the the DET iajs-1914 106 2 expected expect VERB iajs-1914 106 3 values value NOUN iajs-1914 106 4 for for ADP iajs-1914 106 5 different different ADJ iajs-1914 106 6 estimators estimator NOUN iajs-1914 106 7 for for ADP iajs-1914 106 8 unknown unknown ADJ iajs-1914 106 9 scale scale NOUN iajs-1914 106 10 parameter parameter NOUN iajs-1914 106 11 β β X iajs-1914 106 12 of of ADP iajs-1914 106 13 gamma gamma NOUN iajs-1914 106 14 distribution distribution NOUN iajs-1914 106 15 when when SCONJ iajs-1914 106 16 β β X iajs-1914 106 17 = = SYM iajs-1914 106 18 1 1 NUM iajs-1914 106 19 method method NOUN iajs-1914 106 20 n n NOUN iajs-1914 106 21 𝛼 𝛼 NOUN iajs-1914 106 22 𝛼 𝛼 NOUN iajs-1914 106 23 𝛼 𝛼 NOUN iajs-1914 106 24 β=0.5 β=0.5 NOUN iajs-1914 106 25 β=1 β=1 PUNCT iajs-1914 106 26 β=0.5 β=0.5 NOUN iajs-1914 106 27 β=1 β=1 PRON iajs-1914 106 28 β=0.5 β=0.5 NOUN iajs-1914 106 29 β=1 β=1 SYM iajs-1914 106 30 20 20 NUM iajs-1914 106 31 1.27605 1.27605 NUM iajs-1914 106 32 1.22115 1.22115 NUM iajs-1914 106 33 1.19739 1.19739 NUM iajs-1914 106 34 1.16928 1.16928 NUM iajs-1914 106 35 1.10595 1.10595 NUM iajs-1914 106 36 1.11258 1.11258 NUM iajs-1914 106 37 30 30 NUM iajs-1914 106 38 1.16913 1.16913 NUM iajs-1914 106 39 1.14735 1.14735 NUM iajs-1914 106 40 1.11661 1.11661 NUM iajs-1914 106 41 1.11124 1.11124 NUM iajs-1914 106 42 1.06368 1.06368 NUM iajs-1914 106 43 1.07708 1.07708 NUM iajs-1914 106 44 50 50 NUM iajs-1914 106 45 1.10256 1.10256 NUM iajs-1914 106 46 1.14735 1.14735 NUM iajs-1914 106 47 1.07027 1.07027 NUM iajs-1914 106 48 1.11124 1.11124 NUM iajs-1914 106 49 1.04133 1.04133 NUM iajs-1914 106 50 1.07708 1.07708 NUM iajs-1914 106 51 100 100 NUM iajs-1914 106 52 1.04945 1.04945 NUM iajs-1914 106 53 1.04511 1.04511 NUM iajs-1914 106 54 1.03176 1.03176 NUM iajs-1914 106 55 1.03317 1.03317 NUM iajs-1914 106 56 1.01838 1.01838 NUM iajs-1914 106 57 1.02428 1.02428 NUM iajs-1914 106 58 table table NOUN iajs-1914 106 59 7 7 NUM iajs-1914 106 60 . . PUNCT iajs-1914 107 1 the the DET iajs-1914 107 2 mse mse PROPN iajs-1914 107 3 values value NOUN iajs-1914 107 4 for for ADP iajs-1914 107 5 different different ADJ iajs-1914 107 6 estimators estimator NOUN iajs-1914 107 7 for for ADP iajs-1914 107 8 unknown unknown ADJ iajs-1914 107 9 scale scale NOUN iajs-1914 107 10 parameter parameter NOUN iajs-1914 107 11 β β X iajs-1914 107 12 of of ADP iajs-1914 107 13 gamma gamma NOUN iajs-1914 107 14 distribution distribution NOUN iajs-1914 107 15 when when SCONJ iajs-1914 107 16 β β X iajs-1914 107 17 = = SYM iajs-1914 107 18 0.5 0.5 NUM iajs-1914 107 19 method method NOUN iajs-1914 107 20 n n X iajs-1914 107 21 𝛼 𝛼 NOUN iajs-1914 107 22 𝛼 𝛼 NOUN iajs-1914 107 23 𝛼 𝛼 NOUN iajs-1914 107 24 β=0.5 β=0.5 NOUN iajs-1914 107 25 β=1 β=1 PUNCT iajs-1914 107 26 β=0.5 β=0.5 NOUN iajs-1914 107 27 β=1 β=1 PRON iajs-1914 107 28 β=0.5 β=0.5 NOUN iajs-1914 107 29 β=1 β=1 SYM iajs-1914 107 30 20 20 NUM iajs-1914 107 31 0.09335 0.09335 NUM iajs-1914 107 32 0.06904 0.06904 NUM iajs-1914 107 33 0.06875 0.06875 NUM iajs-1914 107 34 0.05533 0.05533 NUM iajs-1914 107 35 0.05957 0.05957 NUM iajs-1914 107 36 0.05055 0.05055 NUM iajs-1914 107 37 30 30 NUM iajs-1914 107 38 0.04509 0.04509 NUM iajs-1914 107 39 0.03596 0.03596 NUM iajs-1914 107 40 0.03113 0.03113 NUM iajs-1914 107 41 0.02778 0.02778 NUM iajs-1914 107 42 0.02835 0.02835 NUM iajs-1914 107 43 0.02615 0.02615 NUM iajs-1914 107 44 50 50 NUM iajs-1914 107 45 0.02224 0.02224 NUM iajs-1914 107 46 0.01936 0.01936 NUM iajs-1914 107 47 0.01490 0.01490 NUM iajs-1914 107 48 0.01465 0.01465 NUM iajs-1914 107 49 0.01408 0.01408 NUM iajs-1914 107 50 0.01412 0.01412 NUM iajs-1914 107 51 100 100 NUM iajs-1914 107 52 0.01023 0.01023 NUM iajs-1914 107 53 0.00824 0.00824 NUM iajs-1914 107 54 0.00681 0.00681 NUM iajs-1914 107 55 0.00627 0.00627 NUM iajs-1914 107 56 0.00662 0.00662 NUM iajs-1914 107 57 0.00615 0.00615 NUM iajs-1914 107 58 table table NOUN iajs-1914 107 59 8 8 NUM iajs-1914 107 60 . . PUNCT iajs-1914 108 1 the the DET iajs-1914 108 2 mse mse PROPN iajs-1914 108 3 values value NOUN iajs-1914 108 4 for for ADP iajs-1914 108 5 different different ADJ iajs-1914 108 6 estimators estimator NOUN iajs-1914 108 7 for for ADP iajs-1914 108 8 unknown unknown ADJ iajs-1914 108 9 scale scale NOUN iajs-1914 108 10 parameter parameter NOUN iajs-1914 108 11 β β X iajs-1914 108 12 of of ADP iajs-1914 108 13 gamma gamma NOUN iajs-1914 108 14 distribution distribution NOUN iajs-1914 108 15 when when SCONJ iajs-1914 108 16 β β X iajs-1914 108 17 = = SYM iajs-1914 108 18 1 1 NUM iajs-1914 108 19 method method NOUN iajs-1914 108 20 n n NOUN iajs-1914 108 21 𝛼 𝛼 NOUN iajs-1914 108 22 𝛼 𝛼 NOUN iajs-1914 108 23 𝛼 𝛼 NOUN iajs-1914 108 24 β=0.5 β=0.5 NOUN iajs-1914 108 25 β=1 β=1 PUNCT iajs-1914 108 26 β=0.5 β=0.5 NOUN iajs-1914 108 27 β=1 β=1 PRON iajs-1914 108 28 β=0.5 β=0.5 NOUN iajs-1914 108 29 β=1 β=1 SYM iajs-1914 108 30 20 20 NUM iajs-1914 108 31 0.37339 0.37339 NUM iajs-1914 108 32 0.27617 0.27617 NUM iajs-1914 108 33 0.27500 0.27500 NUM iajs-1914 108 34 0.22131 0.22131 NUM iajs-1914 108 35 0.18993 0.18993 NUM iajs-1914 108 36 0.17587 0.17587 NUM iajs-1914 108 37 30 30 NUM iajs-1914 108 38 0.18037 0.18037 NUM iajs-1914 108 39 0.14383 0.14383 NUM iajs-1914 108 40 0.12452 0.12452 NUM iajs-1914 108 41 0.11114 0.11114 NUM iajs-1914 108 42 0.09764 0.09764 NUM iajs-1914 108 43 0.09486 0.09486 NUM iajs-1914 108 44 50 50 NUM iajs-1914 108 45 0.08896 0.08896 NUM iajs-1914 108 46 0.14383 0.14383 NUM iajs-1914 108 47 0.05961 0.05961 NUM iajs-1914 108 48 0.11114 0.11114 NUM iajs-1914 108 49 0.05129 0.05129 NUM iajs-1914 108 50 0.09490 0.09490 NUM iajs-1914 108 51 100 100 NUM iajs-1914 108 52 0.04091 0.04091 NUM iajs-1914 108 53 0.03295 0.03295 NUM iajs-1914 108 54 0.02723 0.02723 NUM iajs-1914 108 55 0.02508 0.02508 NUM iajs-1914 108 56 0.02535 0.02535 NUM iajs-1914 108 57 0.02386 0.02386 NUM iajs-1914 108 58     SPACE iajs-1914 108 59 196 196 NUM iajs-1914 108 60         SPACE iajs-1914 108 61 ibn ibn PROPN iajs-1914 108 62 al al PROPN iajs-1914 108 63 - - PUNCT iajs-1914 108 64 haitham haitham PROPN iajs-1914 108 65 jour.for jour.for PROPN iajs-1914 108 66 pure&appl.sci pure&appl.sci PROPN iajs-1914 108 67 . . PUNCT iajs-1914 108 68 ihjpas ihjpas PROPN iajs-1914 108 69 https://doi.org/10.30526/32.1.1914 https://doi.org/10.30526/32.1.1914 X iajs-1914 108 70 vol vol NOUN iajs-1914 108 71 . . PUNCT iajs-1914 109 1 32 32 NUM iajs-1914 109 2 ( ( PUNCT iajs-1914 109 3 1 1 NUM iajs-1914 109 4 ) ) PUNCT iajs-1914 109 5 2019 2019 NUM iajs-1914 109 6 references reference NOUN iajs-1914 109 7 1 1 NUM iajs-1914 109 8 . . PUNCT iajs-1914 109 9 hogg hogg PROPN iajs-1914 109 10 , , PUNCT iajs-1914 109 11 r.v r.v PROPN iajs-1914 109 12 . . PROPN iajs-1914 109 13 ; ; PUNCT iajs-1914 109 14 mckean mckean PROPN iajs-1914 109 15 , , PUNCT iajs-1914 109 16 j.w j.w PROPN iajs-1914 109 17 . . PROPN iajs-1914 109 18 ; ; PUNCT iajs-1914 109 19 craig craig PROPN iajs-1914 109 20 , , PUNCT iajs-1914 109 21 a.t a.t PROPN iajs-1914 109 22 . . PROPN iajs-1914 109 23 introduction introduction NOUN iajs-1914 109 24 to to ADP iajs-1914 109 25 mathematical mathematical ADJ iajs-1914 109 26 statistics statistic NOUN iajs-1914 109 27 , , PUNCT iajs-1914 109 28 7th 7th ADJ iajs-1914 109 29 ed ed NOUN iajs-1914 109 30 . . PROPN iajs-1914 109 31 , , PUNCT iajs-1914 109 32 pearson pearson PROPN iajs-1914 109 33 education education PROPN iajs-1914 109 34 publication publication NOUN iajs-1914 109 35 , , PUNCT iajs-1914 109 36 2014 2014 NUM iajs-1914 109 37 . . PUNCT iajs-1914 110 1 2 2 NUM iajs-1914 110 2 . . X iajs-1914 110 3 apolloni apolloni PROPN iajs-1914 110 4 , , PUNCT iajs-1914 110 5 b. b. PROPN iajs-1914 110 6 ; ; PUNCT iajs-1914 110 7 bassis bassis PROPN iajs-1914 110 8 , , PUNCT iajs-1914 110 9 s. s. PROPN iajs-1914 110 10 algorithmic algorithmic ADJ iajs-1914 110 11 inference inference NOUN iajs-1914 110 12 of of ADP iajs-1914 110 13 two two NUM iajs-1914 110 14 - - PUNCT iajs-1914 110 15 parameter parameter NOUN iajs-1914 110 16 gamma gamma NOUN iajs-1914 110 17 distribution distribution NOUN iajs-1914 110 18 . . PUNCT iajs-1914 111 1 communications communication NOUN iajs-1914 111 2 in in ADP iajs-1914 111 3 statistics statistic NOUN iajs-1914 111 4 simulation simulation NOUN iajs-1914 111 5 and and CCONJ iajs-1914 111 6 computation computation NOUN iajs-1914 111 7 . . PUNCT iajs-1914 112 1 2009 2009 NUM iajs-1914 112 2 , , PUNCT iajs-1914 112 3 38 38 NUM iajs-1914 112 4 , , PUNCT iajs-1914 112 5 9 9 NUM iajs-1914 112 6 , , PUNCT iajs-1914 112 7 1950 1950 NUM iajs-1914 112 8 – – PUNCT iajs-1914 112 9 1968 1968 NUM iajs-1914 112 10 . . PUNCT iajs-1914 113 1 3 3 X iajs-1914 113 2 . . X iajs-1914 113 3 douglas douglas PROPN iajs-1914 113 4 , , PUNCT iajs-1914 113 5 c.m c.m PROPN iajs-1914 113 6 . . PROPN iajs-1914 113 7 ; ; PUNCT iajs-1914 113 8 george george PROPN iajs-1914 113 9 , , PUNCT iajs-1914 113 10 c.r c.r PROPN iajs-1914 113 11 . . PROPN iajs-1914 113 12 applied apply VERB iajs-1914 113 13 statistics statistic NOUN iajs-1914 113 14 and and CCONJ iajs-1914 113 15 probability probability NOUN iajs-1914 113 16 for for ADP iajs-1914 113 17 engineering engineering NOUN iajs-1914 113 18 , , PUNCT iajs-1914 113 19 3rd 3rd ADJ iajs-1914 113 20 edition edition NOUN iajs-1914 113 21 , , PUNCT iajs-1914 113 22 john john PROPN iajs-1914 113 23 wiley wiley PROPN iajs-1914 113 24 and and CCONJ iajs-1914 113 25 sons son NOUN iajs-1914 113 26 , , PUNCT iajs-1914 113 27 inc inc PROPN iajs-1914 113 28 . . PROPN iajs-1914 113 29 2003 2003 NUM iajs-1914 113 30 . . PUNCT iajs-1914 114 1 4 4 X iajs-1914 114 2 . . X iajs-1914 114 3 sahoo sahoo PROPN iajs-1914 114 4 , , PUNCT iajs-1914 114 5 s.a s.a PROPN iajs-1914 114 6 . . PROPN iajs-1914 114 7 study study NOUN iajs-1914 114 8 on on ADP iajs-1914 114 9 bayesian bayesian NOUN iajs-1914 114 10 estimation estimation NOUN iajs-1914 114 11 of of ADP iajs-1914 114 12 parameters parameter NOUN iajs-1914 114 13 of of ADP iajs-1914 114 14 some some DET iajs-1914 114 15 well well ADV iajs-1914 114 16 known know VERB iajs-1914 114 17 distribution distribution NOUN iajs-1914 114 18 functions function NOUN iajs-1914 114 19 ; ; PUNCT iajs-1914 114 20 master master NOUN iajs-1914 114 21 thesis thesis NOUN iajs-1914 114 22 , , PUNCT iajs-1914 114 23 department department NOUN iajs-1914 114 24 of of ADP iajs-1914 114 25 mathematics mathematics PROPN iajs-1914 114 26 , , PUNCT iajs-1914 114 27 national national PROPN iajs-1914 114 28 institute institute PROPN iajs-1914 114 29 of of ADP iajs-1914 114 30 technology technology PROPN iajs-1914 114 31 rourkela-769008 rourkela-769008 NOUN iajs-1914 114 32 , , PUNCT iajs-1914 114 33 india india PROPN iajs-1914 114 34 , , PUNCT iajs-1914 114 35 2014 2014 NUM iajs-1914 114 36 . . PUNCT iajs-1914 115 1 5 5 NUM iajs-1914 115 2 . . X iajs-1914 115 3 he he PRON iajs-1914 115 4 , , PUNCT iajs-1914 115 5 h. h. PROPN iajs-1914 115 6 ; ; PUNCT iajs-1914 115 7 zhou zhou PROPN iajs-1914 115 8 , , PUNCT iajs-1914 115 9 n. n. PROPN iajs-1914 115 10 ; ; PUNCT iajs-1914 115 11 zhang zhang PROPN iajs-1914 115 12 , , PUNCT iajs-1914 115 13 r. r. PROPN iajs-1914 115 14 on on ADP iajs-1914 115 15 estimation estimation NOUN iajs-1914 115 16 for for ADP iajs-1914 115 17 the the DET iajs-1914 115 18 pareto pareto ADJ iajs-1914 115 19 distribution distribution NOUN iajs-1914 115 20 . . PUNCT iajs-1914 116 1 statistical statistical ADJ iajs-1914 116 2 methodology methodology NOUN iajs-1914 116 3 . . PUNCT iajs-1914 117 1 2014 2014 NUM iajs-1914 117 2 , , PUNCT iajs-1914 117 3 21 21 NUM iajs-1914 117 4 , , PUNCT iajs-1914 117 5 49–58 49–58 NUM iajs-1914 117 6 . . PUNCT iajs-1914 118 1 6 6 NUM iajs-1914 118 2 . . X iajs-1914 118 3 alabadee alabadee NOUN iajs-1914 118 4 , , PUNCT iajs-1914 118 5 f.a f.a PROPN iajs-1914 118 6 . . PROPN iajs-1914 118 7 ; ; PUNCT iajs-1914 119 1 al al PROPN iajs-1914 119 2 - - PUNCT iajs-1914 119 3 mayali mayali PROPN iajs-1914 119 4 , , PUNCT iajs-1914 119 5 y.m y.m PROPN iajs-1914 119 6 . . PROPN iajs-1914 119 7 ; ; PUNCT iajs-1914 119 8 neama neama NOUN iajs-1914 119 9 , , PUNCT iajs-1914 119 10 i.a i.a PROPN iajs-1914 119 11 . . PROPN iajs-1914 119 12 a a DET iajs-1914 119 13 comparison comparison NOUN iajs-1914 119 14 between between ADP iajs-1914 119 15 the the DET iajs-1914 119 16 bayesian bayesian NOUN iajs-1914 119 17 and and CCONJ iajs-1914 119 18 the the DET iajs-1914 119 19 classical classical ADJ iajs-1914 119 20 estimators estimator NOUN iajs-1914 119 21 of of ADP iajs-1914 119 22 weibull weibull NOUN iajs-1914 119 23 distribution distribution NOUN iajs-1914 119 24 . . PUNCT iajs-1914 120 1 j. j. PROPN iajs-1914 120 2 kufa kufa PROPN iajs-1914 120 3 math math PROPN iajs-1914 120 4 . . PUNCT iajs-1914 121 1 comput comput NOUN iajs-1914 121 2 . . PUNCT iajs-1914 122 1 2013,1 2013,1 NOUN iajs-1914 122 2 , , PUNCT iajs-1914 122 3 8 8 NUM iajs-1914 122 4 , , PUNCT iajs-1914 122 5 21 21 NUM iajs-1914 122 6 - - SYM iajs-1914 122 7 28 28 NUM iajs-1914 122 8 . . PUNCT iajs-1914 123 1 7 7 X iajs-1914 123 2 . . X iajs-1914 123 3 li li PROPN iajs-1914 123 4 , , PUNCT iajs-1914 123 5 j. j. PROPN iajs-1914 123 6 ; ; PUNCT iajs-1914 123 7 ren ren PROPN iajs-1914 123 8 , , PUNCT iajs-1914 123 9 h. h. PROPN iajs-1914 123 10 estimation estimation NOUN iajs-1914 123 11 of of ADP iajs-1914 123 12 one one NUM iajs-1914 123 13 parameter parameter NOUN iajs-1914 123 14 exponential exponential ADJ iajs-1914 123 15 family family NOUN iajs-1914 123 16 under under ADP iajs-1914 123 17 a a DET iajs-1914 123 18 precautionary precautionary ADJ iajs-1914 123 19 loss loss NOUN iajs-1914 123 20 function function NOUN iajs-1914 123 21 based base VERB iajs-1914 123 22 on on ADP iajs-1914 123 23 record record NOUN iajs-1914 123 24 values value NOUN iajs-1914 123 25 . . PUNCT iajs-1914 124 1 international international ADJ iajs-1914 124 2 journal journal NOUN iajs-1914 124 3 of of ADP iajs-1914 124 4 engineering engineering NOUN iajs-1914 124 5 and and CCONJ iajs-1914 124 6 manufacturing manufacturing NOUN iajs-1914 124 7 . . PUNCT iajs-1914 125 1 2012 2012 NUM iajs-1914 125 2 . . PUNCT iajs-1914 126 1 2 2 NUM iajs-1914 126 2 , , PUNCT iajs-1914 126 3 3 3 NUM iajs-1914 126 4 , , PUNCT iajs-1914 126 5 75 75 NUM iajs-1914 126 6 - - SYM iajs-1914 126 7 81 81 NUM iajs-1914 126 8 . . PUNCT