Malaysa Joural of Mathematcal Sceces (): 97- (9) Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato Hadeel Salm Al-Kutub ad Noor Akma Ibrahm Isttute for Mathematcal Research, Uverst Putra Malaysa, 44 UPM Serdag, Selagor, Malaysa E-mal: hadeelkutub@lve.com ABSTRACT I ths paper the exteso of Jeffery pror formato wth ew loss fucto for estmatg the parameter of expoetal dstrbuto of lfe tme s preseted. Through smulato study the performace of ths estmator was compared to the stadard Bayes wth Jeffery pror formato wth respect to the mea square error (MSE) ad mea percetage error (MPE). We foud the exteso of Jeffery pror formato gves the best estmator. Keywords: Exteso of Jeffery pror formato, loss fucto, expoetal dstrbuto, Bayes method. INTRODUCTION Oe of the most useful ad wdely exploted models s the expoetal dstrbuto. Epste,(984) remarks that the expoetal dstrbuto plays as mportat role lfe expermets as that played by the ormal dstrbuto agrcultural expermets. Maxmum lkelhood estmato has bee the wdely used method to estmate the parameter of a expoetal dstrbuto. Lately Bayes method has begu to get the atteto of researchers the estmato procedure. The oly statstcal theory that combes modelg heret ucertaty ad statstcal ucertaty s Bayesa statstcs. The theorem of Bayes provdes a soluto o how to lear from data. Related to survval fucto ad by usg Bayes estmator, Ell ad Rao,(986), estmated the shape ad scale parameters of the Webull dstrbuto by assumg a weghted squared error loss fucto. They mmzed the correspodg expected loss wth respect to a gve posteror dstrbuto. Sha ad Sloa,(988), obtaed Bayes estmator of three parameters Webull dstrbuto ad compared the posteror stadard devato estmates wth the correspodg asymptotc
Hadeel Salm Al-Kutub & Noor Akma Ibrahm stadard devato of ther maxmum lkelhood couterparts ad umercal examples are gve. Elfess,() preseted some thought provokg sghts o the relatoshp betwee Bayesa ad classcal estmato usg expoetal dstrbuto. He showed how the classcal estmators ca be obtaed from varous choces made wth Bayesa framework. I, Klaus Felseste developed Bayesa procedures for vague data. These data were assumed to be vague the sese that the lkelhood s a mxture of the model dstrbuto wth error dstrbuto. I ths case the stadard updatg procedure of the model pror would fal. Al-Bayyat,() studed the problem of estmatg parameters of Webull dstrbuto ad relablty fucto stuato where there s o formato o the parameters. He proposed a method based o the prmary formato wth weghted Bayes. A exteso of Jeffery pror formato wth square error loss fucto expoetal dstrbuto was studed by Al-Kutub,(5). I ths paper, we proposed a exteso of Jeffery pror formato wth a ew loss fucto. BAYES ESTIMATOR Let t, t,..., t be the lfe tme of a radom sample of sze wth dstrbuto fucto F( t; ) ad probablty desty fucto f ( t; ).I the expoetal case (Chou,99 ad Elfess ad Reeke,), we assumed that the probablty desty fucto of the lfe tme s gve by t f ( t; ) exp. To obta Bayes estmator, the followg steps are eeded.. A umber of tems put to test ad the lfe tmes of ths radom sample are recorded wth the probablty desty fucto f ( t; ), where s real valued radom varable.. The lfe tme probablty desty fucto f ( t; ) s regarded as a codtoal probablty desty fucto f ( t ) where the margal probablty desty fucto of s gve by g( ), the Jeffery pror formato. 98 Malaysa Joural of Mathematcal Sceces
Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato. The jot probablty desty fucto s gve by: H t,..., t, f t g L t,..., t g ( ) Π ( ) ( ) ( ) 4. The margal probablty desty fucto of ( t ) (,..., ) (,..., ) p t t H t t d,...,, t s gve by: 5. The codtoal probablty desty fucto of gve the data ( t,...,, t ) s called posteror dstrbuto of, gve by ( t,..., t ) Π (,...,, ) (,..., ) H t t p t t 6. Bayes estmator of s gve by usg squared error loss fucto (,..., ) Π (..., ) E t t t t d. Jeffery Pror Iformato Cosder the oe parameter expoetal lfe tme dstrbuto t f ( t ; ) e x p ( ) We fd Jeffery pror by takg g ( ) I ( ), where I ( ) ( ) l f t, E. Takg g ( ), g ( ) k, wth k a costat. The jot probablty desty fucto f ( t, t,..., t, ) s gve by H t,..., t, f t, g ( ) ( ) ( ) Π L ( t t ) g,...,, Malaysa Joural of Mathematcal Sceces 99
where Hadeel Salm Al-Kutub & Noor Akma Ibrahm (,..., ) Π ( ) e x p L t t f t t tt k k H ( t,..., t, ) exp exp. + The margal probablty desty fucto of gve the data ( t, t,..., t ) s t p t,..., t H ( t,..., t, ) d t k k e x p d + t!, t ( )! where e d + ( t ) The codtoal probablty desty fucto of gve the data t t t s gve by ( t,..., t ) (,,..., ) Π H ( t,..., t, ) p t t (,..., ) t t k exp exp + t + k ( )! ( )! t Malaysa Joural of Mathematcal Sceces
Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato By usg squared error loss fucto the Rsk fucto, such that R ( ) l (, ) Π( t,..., t ) d c l ( ) t t c c d + ζ! exp ( ) c ( t ) where ζ ( ). ( ) ( ) R, Let, the the Bayes estmator s, we ca obta t t t B exp d ( )! (). New Exteso Of Jeffery Pror formato The ew exteso of Jeffery pror (Al-Kutub 5) s g ( ) I ( ) c, + R a costat. The lkelhood fucto s c. Wth g ( ) c (,..., ) ( ) L t t f t c, k s, the g k c Malaysa Joural of Mathematcal Sceces
Hadeel Salm Al-Kutub & Noor Akma Ibrahm The jot probablty desty fucto s gve by C (,...,, ) Π ( ) ( ) e x p. + C H t t f t g The margal probablty desty fucto of ( tt..., t ) s gve by c k ( + c )! (,..., ) (,...,, ). + c ( t ) p t t H t t d The codtoal probablty desty fucto of gve the data ( t t..., t ) s c t t k c exp exp + c Π,...,. ( t t ) The Rsk fucto, k c! c k + c + c t c t ( + ) t! t c ex p ˆ ˆ t ( + c )! R (, ) c ( ) d c R, Let, the ˆ B c t + () Malaysa Joural of Mathematcal Sceces
Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato If we let c, t whch s Jeffrey estmator ad t s a B specal case of our proposed method.. New Loss Fucto Al-Bayyat,() troduced a ew loss fucto usg Webull dstrbuto that s c l,. Here we employ ths loss fucto wth expoetal dstrbuto. (a) Jeffery Pror Iformato Through the Rsk fucto, the Bayes estmator s R (, ) (, ) R c Π ( t,..., t ) d, ad by, the, c + Π(,..., ) B c Π(,..., ) t t d t t d t t c + c + Π t,..., t d exp d ( )! ( ) ( ) c + t c! ( ) ( )! t t c c Π t,..., t d exp d ( )! Malaysa Joural of Mathematcal Sceces
Hadeel Salm Al-Kutub & Noor Akma Ibrahm Fally, c t c! ( ) ( )!. B c + t ( c )! t ( )! c ( )! c t ( c )! () (b) Exteso of Jeffery Pror Iformato Takg the posteror dstrbuto as the exteso of Jeffery pror, such t + c that t e ( t,..., t ) the Rsk fucto s gve + c ( + c )! c by R ˆ, ) ˆ ( ) ( ( t,.., t ) d c c + t,..., t (,..., ) d t t d ζ Π Π + c + ( t ) ( + c c ϕ)! where ζ ( ). ( + c )! R( ˆ, ) Let ˆ, the the Bayes estmator s c + Π,..., B4 c Π (,..., ) t t d, t t d 4 Malaysa Joural of Mathematcal Sceces
Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato wth t c exp c + c + ( t,..., t ) d Π d c t ( + c )! ad c + t ( + c c )!! ( + c ) (,..., ) c Π t t d c t y c e x p d c t ( + c )! c t ( + c c )!,! ( + c ) ˆ c + t ( + c c _ )! ( + c )! t ( + c c )! B4 c ( + c )! (4) ( + c c ) t Malaysa Joural of Mathematcal Sceces 5
Hadeel Salm Al-Kutub & Noor Akma Ibrahm (c) New Loss Fucto wth Jeffery Pror Iformato By lettg c, the Rsk fucto s, ( ˆ, ) ( ˆ ) Π (..., ). R t t d Lettg R (, ), Π,..., B5 Π (,..., ) t t d, t t d t t Π t,..., t exp d ( )! wth ( ) t ( ) ( ) 4!,! t t Π t,..., t exp d ( )! ad ( ) t ( ) ( )!! 6 Malaysa Joural of Mathematcal Sceces
Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato The ew Bayes estmator wth ew loss fucto ad Jeffery pror formato s ˆ B5 t ( 4 )! t ( )! ( )! t ( )! ( ) (5) (d) New Loss Fucto wth Exteso of Jeffery Pror Iformato By lettg c, the Rsk fucto s, ( ˆ, ) ( ˆ ) Π (..., ) R t t d t,..., t (,..., ) d t t d ζ Π Π + 4 ( t ) ( + c ϕ)! where ζ ( ). ( + c )! R, Lettg wth (,..., ), the Π,..., B6 Π(,..., ) Π c t t d, t t d t c exp t t d d t ( + c )! t ( + c 5 )!! ( + c ) Malaysa Joural of Mathematcal Sceces 7
ad (,..., ) Hadeel Salm Al-Kutub & Noor Akma Ibrahm ty c exp t t d d t ( + c )! Π c t ( + c 4 )!! ( + c ) The the ew Bayes estmator wth ew loss fucto ad exteso of Jeffery pror formato s ˆ B6 t ( c 5 )! t ( c )! ( c )! c 4 t ( c 4 )!. (6) SIMULATION AND RESULTS I ths smulato study, we have chose 5,5, to represet small, moderate ad large sample sze, several values of parameter.5,,.5, four values of Jeffery exteso, c.4,.4 ad four values of ew loss fucto c,.. The umber of replcato used was R. The smulato program was wrtte by usg Matlab program. After the parameter was estmated, mea square error (MSE) ad mea percetage error (MPE) were calculated to compare the methods of estmato, where 8 Malaysa Joural of Mathematcal Sceces
Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato MSE ( ) R ad MPE R The results of the smulato study are summarzed ad tabulated Table ad Table for the MSE ad the MPE of the sx estmators for all sample szes ad values respectvely. It s obvous from these tables, Bayes estmator wth the exteso of Jeffery pror formato, s the best ˆB estmator. I most of the cases, t s apparet that Bayes estmator wth ew loss fucto ˆB 5, s the ext best estmator. Bayes wth Jeffery pror formato most case has the largest MSE ad MPE. ˆB The effect of sample sze o the MSE ad MPE of all sx estmators are depcted Fgure ad Fgure. I these two fgures, the detty umber of scearo ( x axs ) dcates the creased of sample sze from 5 to (cosderg all factors). I Fgure, for sample sze 5 (dcated by detty umber of scearo from to 6), the tred s the same for 5 ad. But overall, the MSE decreases as creases. For the MPE of the estmators, they also decreases as creases (Fgure ). As show Fgure, the varablty of MPE wth the sample sets are smaller as sample creases. TABLE : MSE of estmated parameter of expoetal dstrbuto Sze Theta C C ˆB ˆB ˆB ˆB4 ˆB5 ˆB6 5.5.4.4.4.4.5.4.4........49.4.4.479.586.456.586.59.4.5.4..8.95.86.48.468.77.9.4.986.87.87..57..8.494.66.44.4. 5.85.99..57..8.476.597.48.588.59.4.98.9.6.9...54.5.47.46.44.84.8.856....4.494.5.498.499..4.9.8 Malaysa Joural of Mathematcal Sceces 9
Hadeel Salm Al-Kutub & Noor Akma Ibrahm TABLE (cotued): MSE of estmated parameter of expoetal dstrbuto Sze Theta C C ˆB ˆB ˆB ˆB4 ˆB5 ˆB6 5.5.4.4.4.4.5.4.4.5.4.4.4.4.5.4.4.............57.6.55.6..46..5.5.547.478.566.7.7.5.6...6..7.8.58.4.55.55.48.49.8.8.95.7.498.49.44.4.7.5.4..9.95..96.6.4.4.5.6.5.47.5.8.48..4.5.54.45.44.5.7.5..99..6.99.45.4.8.8.9.5.5.6.5.4.8.4.5.59.46.55.5.6.7.5.98..4.8.4.9..4.9.9.46.5..5..98.5.49.47.46.6.5.5..99.7.6.97.48..9.6..54.5.56.8....5.48.469.475.4..7.4.99.6.6..45...5 TABLE : MPE of estmated parameter of expoetal dstrbuto Sze Theta C C ˆB ˆB ˆB ˆB4 ˆB5 ˆB6 5.5.4.4...65.89.74.897.67.687.589.554.775.868.67.6.745.85.749.848.87.78.6.598.775.689.78.75.4.4.5.4.4.....7.96.65.8.69.96.79.99.78.74.54.57.679.7.586.576.755.96.6.6.78.8.56.69.7.97.74.889.7.787.66.89.79.8.6.598.758.67.57.59.755.767.75.74.78.64.655.75 Malaysa Joural of Mathematcal Sceces
Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato TABLE (cotued): MPE of estmated parameter of expoetal dstrbuto Sze Theta C C ˆB ˆB ˆB ˆB4 ˆB5 ˆB6 5.5.4.4.4.4.5.4.4.5.4.4.4.4.5.4.4.............9.6.7.7.95.5.49.98.5..69..8.788.847.8.8.84.8.89.78.89.78.848.86.4.8.94.9.59...6.9 4.79.77.89.77.88.86.79.8.779.788.784.8.78.5.89.4.9.9.77.6.75.98.794.86.8.769.786.84.89.79.8.85.796.774.68.5.4.7.5.9..69.9.7.94.79.86.84.769.78.8.857.8.87.8.8.798.89.96.9..49.7 4..86.5.7.84.795.86.84.769.79.89.8.784.85.8.796.77.78.85.5.68.4.6..54.77.4.8.4.794.8.84.776.78.87.86.8.8.88.86.784.5 MES of theta hat..5..5 Theta hat Theta hat Theta hat Theta hat 4 Theta hat 5 Theta hat 6 4 5 6 7 8 9 4 5 6 7 8 9 detty umber of scearo Fgure : Comparso estmators dstrbuted dfferet sample sze usg MSE Malaysa Joural of Mathematcal Sceces
Hadeel Salm Al-Kutub & Noor Akma Ibrahm.5 MPE of Theta hat..5..5 Theta hat Theta hat Theta hat Theta hat 4 Theta hat 5 Theta hat 6 4 7 4 5 66 79 9 5 8 44 57 7 8 96 detty umber of scearo Fgure : Comparso estmators dstrbuted dfferet sample sze usg MPE CONCLUSION The ew estmator wth exteso of Jeffery pror formato s ˆB the best estmator whe compared to stadard Bayes ad other estmators. We ca easly coclude that MSE ad MPE of Bayes estmators decrease wth a creased of sample sze. REFERENCES Al-Bayyat, H.N.. Comparg methods of estmatg Webull falure models usg smulato. Ph.D. Thess, College of Admstrato ad Ecoomcs, Baghdad Uversty, Iraq. Al-Kutub, H.S. 5. O Comparso estmato procedures for parameter ad survval fucto expoetal dstrbuto usg smulato.ph.d. Thess, College of Ib Al-Hatham, Baghdad Uversty, Iraq. Chou, P. 99. Emprcal Bayes shrkage estmato of relablty the expoetal dstrbuto. Comm. Statstc, (5): 48-494. Elfess, A. ad Reeke, D. M.. Bayesa look at classcal estmato: the expoetal dstrbuto. Joural of Statstcs Educato, 9(). www.amstat.org/publcatos/jse/jse_/v9l/elfess.html. Malaysa Joural of Mathematcal Sceces
Bayes Estmator for Expoetal Dstrbuto wth Exteso of Jeffery Pror Iformato Epste, B. ad Sobcl, M. 984. Some theorems to lfe restg from a expoetal dstrbuto, Aals of Mathematcal Statstcs, 5. Ells, W. C. ad Rao, T. V. 986. Mmum expected loss estmators of the shape ad scale parameters of Webull dstrbuto. IEE Trasacto o Relablty, 5(): -5. Felseste, K.. Bayesa ferece for questoable data. Austra Joural of Statstc, (&):-4. Kumar, A.. Bayes estmator for oe parameter expoetal dstrbuto uder multply type II cesorg scheme. Iteratoal Coferece o Statstcs, Combatorcs ad Related Area, Uversty of Souther Mae, Portlad, ME, USA. Sha, S. K. ad Sloa, J. A. 988. Bayes estmato of the parameters ad relablty fucto of the -parameters Webull dstrbuto. IEEE Trasacto o Relablty, 7(4):64-69. Malaysa Joural of Mathematcal Sceces