BAYESIAN INFERENCES FOR TWO PARAMETER WEIBULL DISTRIBUTION
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1 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( BAYESIAN INFERENCES FOR TWO PARAMETER WEIBULL DISTRIBUTION Kpkoech W. Cheruyot, Abel Ouko ad Emly Krm 3 Maasa Mara Uversty, Keya, Departmet of Mathematcs ad Physcal Sceces The East Afrca Uversty, Keya, Departmet of Mathematcs 3 Techcal Uversty of Keya, Departmet of Mathematcs ad Statstcs ABSTRACT: I ths paper, Bayesa estmato usg dffuse (vague prors s carred out for the parameters of a two parameter Webull dstrbuto. Epressos for the margal posteror destes ths case are ot avalable closed form. Appromate Bayesa methods based o Ldley (980 formula ad Terey ad Kadae (986 Laplace approach are used to obta epressos for posteror destes. A comparso based o posteror ad asymptotc varaces s doe usg smulated data. The results obtaed dcate that, the posteror varaces for scale parameter obtaed by Laplace method are smaller tha both the Ldley appromato ad asymptotc varaces of ther MLE couterparts. KEYWORDS: Webull Dstrbuto, Ldley Appromato, Laplace Appromato, Mamum Lkelhood Estmates INTRODUCTION The Webull dstrbuto s oe of the most wdely used dstrbutos relablty ad survval aalyss because of varous shapes assumed by the probablty desty fuctos (p.d.f ad the hazard fucto. The Webull dstrbuto has bee used effectvely aalyzg lfetme data partcularly whe the data are cesored whch s very commo survval data ad lfe testg epermets. The Webull dstrbuto was derved from the problem of materal stregth ad t has bee wdely used as a lfetme model. Webull dstrbuto correspods to a famly of dstrbuto that covers a wde rage of dstrbutos that goes from the ormal model to the epoetal model makg t applcable dfferet areas for stace fatgue lfe, stregth of materals, geetc research, qualty cotrol ad relablty aalyss. The probablty desty fucto (p.d.f for the two parameter Webull dstrbuto s gve by ft (,, ep 0 (. Lkelhood Based Estmato of Parameters of Webull Dstrbuto. The Mamum Lkelhood Estmato Let X, X,..., X be depedet radom samples of sze from Webull dstrbuto wth the p.d.f gve by (.. Dfferetatg wth respect to ad ad equatg to zero we obta ad dl 0 d l l( ( l( 0 (. 5
2 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( dl 0 d ( 0 (. From (. we obtaed the mamum lkelhood estmates of as (.3 Substtutg (.3 to (., yelds a epresso terms of oly as gve by (.4 l l The mamum lkelhood estmate for s obtaed from (.4 wth the ad of stadard teratve procedures. Varace ad Covarace Estmates The asymptotc varace-covarace matr of ad are obtaed by vertg formato matr wth elemets that are egatves of epected values of secod order dervatves of logarthms of the lkelhood fuctos. Cohe (965 suggested that the preset stuato t s approprate to appromate the epected values by ther mamum lkelhood estmates. Accordgly, we have as the appromate varace-covarace matr wth elemets l l var( cov( l l cov( var( Whe ad are depedet, the covarace of the above matr s zero. Whe s kow the asymptotc varaces for s obtaed by ˆ var( ˆ Bayesa Estmato of Parameters of Webull Dstrbuto Bayesa Approach. 6
3 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( Let X, X,..., X be a radom sample from a populato wth a two parameter Webull dstrbuto gve by (3. f( / =, ( ep {-( } The lkelhood fucto s the gve by (, ; ep{ ( } L The log lkelhood fucto s gve by l log L log log ( log ( Suppose that we are gorat about the parameters (, so that the dffuse (vague pror used s (, (3. The jot posteror dstrbuto s the gve by g(, / L( / (, g(, / ep{ ( } (3.3 The margal p.d.f of s gve by ( ep{ ( } 0 0 f dd (3.4 Smlarly, the margal posteror p.d.f s of ad are requred order to compute the correspodg posteror epectatos of ad as E( / g( / L ( / ( L ( / ( (3.5 ad E( / g( / L ( / ( L ( / ( (3.6 7
4 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( respectvely Sce the Bayes estmate for ad volve evaluatg ratos of two mathematcally tractable tegrals, approprate Bayesa appromatos are appled. Assumg s kow, the lkelhood fucto for s gve by L ( / ep{ ( } (3.7 The log lkelhood fucto for s gve by l log L log ( Sce s kow, the pror desty for s gve by ( The posteror desty for s gve by g( / L( / ( g( / ( ep{ } Therefore the posteror epectato for s obtaed by E( / g( / d E( / L ( / ( L ( / ( (3.8 Laplace Appromato Sce the Bayes estmate of volve rato of two mathematcally tractable tegrals, Terey ad Kadae, (986 proposed to estmate (3.8 as follows E( / e L L e d d (3.9 where l L log ( 8
5 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( (3.0 log( log ( ad l L log ( log log log( log ( (3. The posteror mode of L s obtaed by dfferetatg L wth respect to oce ad equatg to zero, that s, dl d 0 ( 0 Gvg terms of as ( (3. The posteror mode (local mamum for L s obtaed by dfferetatg L wth respect to ad equatg to zero to get log log( log ( ( 0 Gvg terms of as (3.3 The ad are equal to the mus the verse of the secod dervatve of the log posteror desty at ts mode gve by 9
6 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( Hece " L ( dl ( d ( " ( L ( ( ( ( (3.4 Also dl d ( ( " L ( (3.5 Thus, the Laplace appromato of (3.8 s gve by where ad E( / ( ep{ ( L ( L( } (3.6 ( 30
7 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( E ( / L ( / ( L ( / ( (3.7 E ( / e L L e d d where L log( log ( ad L log log( log ( The posteror mode of L ad L are gve by ( (3.8 ad ( (3.9 respectvely The ad of L ad L respectvely are gve by " L ( " ( L ( ( Hece ( ( (3.0 3
8 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( ad Hece L L ( " ( ( " ( ( (3. Thus, the Laplace appromato of (3.7 s E( / ( ep{ ( L ( L( } (3. Hece the posteror varace of s V ( / E( / [ E( / ] (3.3 Ldley (980 Appromato Ldley (980 developed a multdmesoal lear Bayes estmate of a arbtrary fucto as a appromato of a asymptotc epaso of the rato of two tegrals whch caot be epressed a closed form gve by E U ( / U ( V ( ep L( d V ( ep L( d Epadg L( ad U( V( by Taylor s seres about the MLE of, Ldley (980 Bayesa appromato of two parameter case s gve by E U ( / U ( Uj U j j Ljkl jklu terms of order ad smaller. j j k l U U U U U U U U U L 30( U U L[3 U U ( ] L [3 U U( ] L 03( U U 3
9 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( (3.4 all evaluated at MLE of (see, Ldley, 980. For two parameter Webull dstrbuto we have, The MLE of (, s (, ( log V( log ( L U j L j j U ; U U j ad U U j (, j th elemet the verse of matr { } evaluated at (,,, j, j The quattes Lj L j ' s are the hgher order dervatves of log-lkelhood fucto gve by l ( L l 3 l 0 ( ( L ( ( 3 ( L 3 3 l ( l( ( L l 0 L 3 ( ( ( ( l( ( 30 (3.5 (3.6 (3.7 (3.8 (3.9 l 4 4 ( ( l( ( l( ( l( ( l( L (3.30 l l l( ( l( L 0 (3.3 33
10 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( l ( l( L 0 (3.3 l l( L03 (3.33 l 3 ( l( ( l( L (3.34 Fsher s formato matr s gve by I L L L 0 L 0 Thus L L I 0 L0 L L 0 L L L0 (3.35 where ( (l( ( l( ( ( 0 0 L L L L Net, we have Hece log ( l( ad Sce s kow, we let U such that ; 0 ad 0 ;,j=, U U U j The quattes Lj ' s are the hgher order dervatves of log-lkelhood fucto. Because s kow, the followg dervatves are used to obta Bayes estmates of 3 l ( ( 3 ( L
11 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( (, j th s obtaed by vertg mus secod dervatve of log lkelhood fucto wth j respect to evaluated at (,, Therefore, Bayes estmate of usg (3.4 s gve by (3.36 E( / L 30 The (3.37 E( / L 30 Hece the posteror varace for s gve by (3.38 Var E E ( / / ( / RESULTS Smulato epermets were carred out usg R software to compare the performace of the Bayes ad MLE estmates of the two parameter Webull dstrbuto. We assumed the shape parameter s kow. We performed smulato epermet wth dfferet sample szes (0, 30, 50 ad 00, draw from a Webull dstrbuto for dfferet values of shape parameter (,.5,,.5, 3, 3.5, 4, 4.5, 5, 5.5 ad 6. We specfed the true value of the scale parameter to be fed at oe ( for all the sample szes. We computed estmates based o two dfferet Bayesa methods, that s, Terey ad Kadae, (986 Laplace appromato method, Ldley (980 appromato ad the method of Mamum lkelhood estmato. 35
12 Estmates Estmates Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( Estmates of Alpha agast Lambda Lambda MLE LAPLACE LINDLEY Fgure : Graph of Estmates of agast Lambda ( for a Sample of =0 for affed Estmates of Alpha agast Lambda MLE LAPLACE LINDLEY Lambda Fgure : Graph of Estmates of agast Lambda ( for a sample of =00 for affed 36
13 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( Fgure 3: Graph of varace for sample sze 0 agast values of Lambda (. Fgure 4: Graph of varace for sample sze 00 agast values of Lambda (. 37
14 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( DISCUSSIONS Fgure show the estmates of uder varyg sze of, Bayes estmates obtaed by Ldley (980 ad Terey ad Kadae, (986 Laplace appromato are foud to be larger tha the MLE couter parts. Ldley (980 appromato s foud to overestmate the scale parameter ; however the three methods demostrate the tedecy for ther estmates ad varaces to perform better for larger sample sze. Ths s show Fgure whe the sample sze s creased to 00. As the sample sze creases the MLE ad Bayes estmates becomes more cosstet ad accurate. Fgure 3 show the varaces of the estmates of for varyg sze of shape parameter for a sample of 0. It s observed that posteror varaces of estmates of are smaller tha the asymptotc varaces of MLE hece more precse ad accurate. However, both varaces ted to coverge to zero as the value of get larger. For the sample of 0 Terey ad Kadae, (986 Laplace appromato s see to perform slghtly better tha Ldley (980, sce t stablzes faster. Fgure 4 show that Terey ad Kadae, (986 Laplace appromato performed better for larger sample of 00. The varace of MLE are observed to stablze faster whe the sample sze ad shape parameter crease. It s also observed that the Bayesa method geerally performed better for both small ad larger value of tha the MLE couter parts. Ldley (980 ad Laplace methods ted to perform almost smlarly for smaller. But the Terey ad Kadae, (986 Laplace appromato s foud to produce better results tha both MLE ad Ldley (980 for larger ad larger samples szes. CONCLUSION We have show Bayesa techques for estmatg the scale parameter of the two parameter Webull dstrbuto whch produces estmates wth smaller varaces tha the MLE. Terey ad Kadae, (986 Laplace appromato whch requres the secod dervatves ts computato s foud to be more accurate tha the Ldley (980 whch requres thrd dervatves ts computato. Ths s le wth Terey et al (989 fdgs, that Laplace method s more accurate tha the thrd dervatve method of Ldley (980. Eve though the two Bayesa methods are better tha the MLE couter parts, they have ther ow lmtatos. Ldley (980 appromato requres estece of MLE ts computato. Ths appears as f t s a adjustmet to the MLE to reduce varablty. O the other had, Laplace appromato requres estece of a umodal dstrbuto ts computato, hece dffcult to use cases of a mult modal dstrbuto. RECOMMENDATION I ths study, t s oted that the posteror varaces of Bayes estmates are smaller tha asymptotc varaces. Comparg the two Bayesa methods, Terey ad Kadae, (986 Laplace appromato method has smaller varace tha the Ldley (980 appromato techque hece more precse ad accurate. Laplace appromato does ot requre eplct 38
15 Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK ( thrd order dervatves ts computato whch are requred Ldley (980 appromato method hece smple to compute. We therefore recommed further work to be doe o two parameter Webull dstrbuto whe both scale ad shape parameters are ukow to vestgate accuracy of the two Bayesa methods. REFERENCES Cohe, A.C. (965: Mamum Lkelhood Estmato the Webull Dstrbuto Based o Complete ad Cesored Samples. Techometrcs, 7, Ldley, D.V. (980. Appromate Bayesa Method, Trabajos Estadstca, 3, Terey L, Kass, R.E. ad Kadae, J.B. (989: Fully epoetal Laplace appromatos to epectatos ad varaces of o-postve fuctos. Joural of Amerca Statstcal Assocato, 84, Terey L. ad Kadae, J.B. (986: Accurate Appromatos for Posteror Momets ad Margal Destes. Joural of Amerca Statstcal Assocato, 8,
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