IOSR Joural of Mathematcs IOSR-JM e-issn: 78-578, p-issn: 9-765X. Volume, Issue Ver. II Ja - Feb. 05, PP 4- www.osrjourals.org Bayesa Ifereces for Two Parameter Webull Dstrbuto Kpkoech W. Cheruyot, Abel Ouko, Emly Krm Maasa Mara Uversty, Keya Departmet of Mathematcs ad Physcal Sceces The East Afrca Uversty, Keya Departmet of Mathematcs 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 I. Itroducto 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. II. 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. dl 0 d 0. From. we obtaed the mamum lkelhood estmates of as. Substtutg. to., yelds a epresso terms of oly as gve by DOI: 0.9790/578-4 www.osrjourals.org 4 Page
Bayesa Ifereces for Two Parameter Webull Dstrbuto l l.4 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 III. Bayesa Estmato of Parameters of Webull Dstrbuto. Bayesa Approach. Let X, X,..., X be a radom sample from a populato wth a two parameter Webull dstrbuto gve by f /, = ep {- }. The lkelhood fucto s the gve by, ; ep{ } L The log lkelhood fucto s gve by l log L log log log DOI: 0.9790/578-4 www.osrjourals.org 5 Page Suppose that we are gorat about the parameters, so that the dffuse vague pror used s,. The jot posteror dstrbuto s the gve by g, / L /, The margal p.d.f of g, / ep{ } s gve by ep{ } 0 0 f. dd.4 Smlarly, the margal posteror p.d.f s of ad are requred order to compute the correspodg posteror epectatos of ad as
Bayesa Ifereces for Two Parameter Webull Dstrbuto E / g / L / L /.5 ad E / g / L / L / 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{ } 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 /. Laplace Appromato Sce the Bayes estmate of volve rato of two mathematcally tractable tegrals, Terey ad Kadae, 986 proposed to estmate.8 as follows L e d E /.9 L e d where ad l L log log log.0 l L log log DOI: 0.9790/578-4 www.osrjourals.org 6 Page.6.7.8
log log log Bayesa Ifereces for Two Parameter Webull Dstrbuto. The posteror mode of L s obtaed by dfferetatg L wth respect to oce ad equatg to zero, that s, dl 0 d 0 Gvg terms of as. 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 The ad mode gve by Hece Also. are equal to the mus the verse of the secod dervatve of the log posteror desty at ts " L dl d " L dl d " L.4 DOI: 0.9790/578-4 www.osrjourals.org 7 Page
Bayesa Ifereces for Two Parameter Webull Dstrbuto Thus, the Laplace appromato of.8 s gve by where ad.5 E / ep{ L L }.6 where ad E / E / L / L / e L L e d d L log log L log log log The posteror mode of L ad L are gve by ad respectvely The ad Hece of L ad L respectvely are gve by " L " L.7.8.9 DOI: 0.9790/578-4 www.osrjourals.org 8 Page
Bayesa Ifereces for Two Parameter Webull Dstrbuto Ad Hece L L " " Thus, the Laplace appromato of.7 s.0. E / ep{ L L }. Hece the posteror varace of s V / E / [ E / ].. 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 0 U U L[ U U ] L [ U U ] L 0 U U.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 U ; U U ad U U j j j j DOI: 0.9790/578-4 www.osrjourals.org 9 Page
Bayesa Ifereces for Two Parameter Webull Dstrbuto, j th elemet the verse of matr { } evaluated at,,, j, j The quattes l Lj L j ' s are the hgher order dervatves of log-lkelhood fucto gve by l L l l 0 L L l l L l 0 L l 0.5.6.7 DOI: 0.9790/578-4 www.osrjourals.org 0 Page 4 4 l l l l L l l l l L l 0.8.9.0. l L 0. l l L0. l l l L.4 Fsher s formato matr s gve by L0 L I L L0 Thus L0 L I L L 0L0 L L L0.5 where L0 L0 L L l l Net, we have log l Hece ad Sce s kow, we let U
Bayesa Ifereces for Two Parameter Webull Dstrbuto such that U ; U 0 ad U j 0 ;,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 l L 0, j th s obtaed by vertg mus secod dervatve of log lkelhood fucto wth respect to j evaluated at,, Therefore, Bayes estmate of usg.4 s gve by E / L 0.6 The E / L 0.7 Hece the posteror varace for s gve by Var / E / E /.8 IV. 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, 0, 50 ad 00, draw from a Webull dstrbuto for dfferet values of shape parameter,.5,,.5,,.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. Fgure : Graph of Estmates of agast Lambda for a Sample of =0 for affed DOI: 0.9790/578-4 www.osrjourals.org Page
Bayesa Ifereces for Two Parameter Webull Dstrbuto Fgure : Graph of Estmates of agast Lambda for a sample of =00 for affed Fgure : Graph of varace for sample sze 0 agast values of Lambda. Fgure 4: Graph of varace for sample sze 00 agast values of Lambda. DOI: 0.9790/578-4 www.osrjourals.org Page
Bayesa Ifereces for Two Parameter Webull Dstrbuto V. Dscussos 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 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. VI. Cocluso 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. VII. Recommedato 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 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. Refereces []. Cohe, A.C. 965: Mamum Lkelhood Estmato the Webull Dstrbuto Based o Complete ad Cesored Samples. Techometrcs, 7, 579-588. []. Ldley, D.V. 980. Appromate Bayesa Method, Trabajos Estadstca,, -7. []. Terey L, Kass, R.E. ad Kadae, J.B. 989: Fully epoetal Laplace appromatos to epectatos ad varaces of opostve fuctos. Joural of Amerca Statstcal Assocato, 84, 70-76. [4]. Terey L. ad Kadae, J.B. 986: Accurate Appromatos for Posteror Momets ad Margal Destes. Joural of Amerca Statstcal Assocato, 8, 8-86. DOI: 0.9790/578-4 www.osrjourals.org Page