Bayesian Analysis of Weibull Distribution Using R Software

Transcription

1 Ausrl Jourl of Bs Ale Sees 79: ISSN Byes Alyss of Webull Dsrbuo Us R Sofwre S.P. Ahm Ksr Ahm Derme of Sss Uversy of Kshmr Srr Jmmu Kshmr I-96 Absr: The Webull srbuo s erhs he mos wely use lfe me srbuo moel. Webull srbuo s wely emloye moel lyz lfeme. Is lo oeo wh lfemes of my yes of mufure ems hs bee wely voe hs bee use s moel wh verse yes of ems suh s vuum ubes bll bers elerl sulo. I s lso wely use bomel los for emle sues of he me o he ourree of umors hum oulo or lborory mls my oher suos. I Byes rmeer feree he hoe of fe ror srbuos loss fuos hs re more. The rese er osers he esmo of he sle rmeer of wo rmeer Webull srbuo wh ow she. Byes esmor s obe us Jeffrey s eeso of Jeffrey s ror uer ler eoel loss fuo symmer loss fuo. Mmum lelhoo esmo s susse. These mehos re omre us me squre error hrouh smulo suy wh vry smle szes by us R sofwre. Key wors: Webull srbuo Prors Byes esmo loss fuos R sofwre. INTRODUCTION The Webull srbuo s oe of he mos wely use srbuos for lyz lfeme. I s fou o be useful verse fels r from eeer o mel sees see wless Mrz Wller 98. The Webull fmly s eerlzo of he eoel fmly moel ehb moooe hzr re behvor.e. ommoe hree yes of flure res mely res eres os. The robbly esy fuo of he Webull srbuo s ve by: f ; e where > > re he sle she rmeers of he srbuo. I Webull lfeme lyss s freque se h he vlue of she rmeer s ow. For emle he eoel Ryleh srbuos re obe whe resevely. Sol 968 ves jusfo for hs suo. The Webull srbuo ws sue by Webull 95 oeo wh he sreh of merls; eble Zele 956; Ko 959 oser lo relbly Pe 966 los mee. Fr 988 hve sse me erreos for he Webull srbuo. Hll 993 hs reely rove eelle revew of he Webull srbuo by rese hsorl fs he my ffere forms of hs srbuo s use by roers ossble ofusos errors h rse ue o hs o-uqueess. Mmum elhoo Esmo hs bee he mos wely use meho for esm he rmeers of he Webull srbuo. Reely Byes esmo roh hs reeve re eo by mos reserhers mo hem re Ahme e l.. They osere Byes Survvl Esmor for Webull srbuo wh esore whle Al-Abou 9 sue Byes esmo for he ereme vlue srbuo us roressve esore symmer loss. Byes esmor for eoel srbuo wh eeso of Jeffreys ror formo ws osere by Al-Kuub 9. Ohers lu Pey e l. Al-Ar Hoss Zmmer 3 some omrve sues o he esmo of Webull rmeers us omlee esore smles ye e l. 993 eerme Byes esmo of he Ereme-Vlue Relbly Fuo. The er s rre s follows: Seo susses he Mmum elhoo esmor Byes mehooloy us Jeffrey s ror eeso of Jeffrey s ror formo uer INEX squre error loss fuos for esmo of he sle rmeer of Webull srbuo wh ow she. Seo 3 fouses he smulo suy resuls o omre he esmors flly seo 4 s he oluso of he er. MATERIA AND METHODS. Correso Auhor: S.P. Ahm Derme of Sss Uversy of Kshmr Srr Jmmu Kshmr I-96 E-ml: srvz@yhoo.om 56

2 Aus. J. Bs & Al. S. 79: Mmum elhoo Esmo: Mmum lelhoo esmo of he rmeers of Webull srbuo s well susse lerure see Cohe 965 M e l e be rom smle of sze hv he robbly esy fuo s e ; f The lelhoo fuo s ve by e lo lo lo As he she rmeer s ssume o be ow he M esmor of s obe by solv he lo. Byes Esmo of Webull Dsrbuo Uer Jeffrey s Pror by Us Dffere oss Fuos: Pror Dsrbuo Que ofe he ervo of he ror srbuo bse o formo oher h he urre s mossble or rher fful. Moreover he ss my be requre o emloy s lle subjeve us s ossble so h he oluso my er solely bse o sml moel he urre. : Jeffrey s 946 roose forml rule for ob o-formve ror s e θ θ I Where θ s -veor vlue rmeer Iθ s he Fsher s formo mr of orer. I rulr f θ s slr rmeer Jeffrey s o- formve ror for θ s e θ I Thus our roblem we oser he ror srbuo of o be e I where s os. The oseror srbuo of s ve by e e Where s eee of.

3 Aus. J. Bs & Al. S. 79: e Hee he oseror srbuo of s ve by e Esmo Uer INEX oss Fuo: For eerm he Byes esme of sle rmeer we wll roue very useful symmer le loss fuo ve by e Where To ob he Byes esmor we mmze he oseror eee loss ve by [ ] E { } e e e e Where e e e e e e e

4 Aus. J. Bs & Al. S. 79: Now solv we ob he Byes esmor s e e e e e e B e. Esmo Uer Squre Error oss Fuo: The squre error loss fuo SEF ws roose by eere 85 Guss o evelo les squre heory. er ws use esmo roblems whe ubse esmos of θ were evlue erms of he rs fuo Rθ whh beomes oh bu he vre of he esmor. I our roblem SEF s ve by l By us he squre error loss fuo l The Rs fuo s ve by R El l E e e e Where e Now solv we ob he Byes esmor s

5 Aus. J. Bs & Al. S. 79: Byes esmo of Webull srbuo uer Eeso of Jeffrey s ror by us ffere oss Fuos: e be rom smle of sze hv he robbly esy fuo s e ; f The lelhoo fuo s ve by e Thus our roblem we oser he ror srbuo of o be [ ] R I e where s os. The oseror srbuo of s ve by e e Where s eee of. e e

6 Aus. J. Bs & Al. S. 79: Afer us he vlue of we e he oseror srbuo s e Esmo Uer INEX oss Fuo: For eerm he Byes esme of sle rmeer we wll roue very useful symmer le loss fuo ve by e Where To ob he Byes esmor we mmze he oseror eee loss ve by [ ] E { } e e e e Where e e e e e e Now solv we ob he Byes esmor s e e e e e e B e

7 Aus. J. Bs & Al. S. 79: B e.4 Whh s he requre Byes esme of sle rmeer uer eeso of Jeffrey s ror. Esmo Uer Squre Error oss Fuo: I our roblem SEF s ve by l By us he squre error loss fuo l The Rs fuo s ve by R El l E e e e ψ Where e ψ ψ Now solv we ob he Byes esmor s ψ.5 Whh s requre Byes esmor of uer eeso of Jeffrey s ror. Smulo Suy: I our smulo suy we hose smle sze of 5 5 o rerese smll meum lre se. The sle rmeer s esme for Webull srbuo wh Mmum elhoo Byes us Jeffrey s & eeso of Jeffrey s ror mehos. For he sle rmeer we hve osere.5.5. The She rmeer hs bee fe.8.. The vlues of Jeffrey s eeso were.4.4. The vlue for he loss rmeer ±.6 ±.6. Ths ws ere mes he sle rmeer for eh meho ws lule. A smulo suy ws oue R-sofwre o eme omre he erforme of he esmes for ffere smle szes wh ffere vlues for he Eeso of Jeffreys ror he loss fuos. The resuls re resee bles for ffere seleos of he rmeers eeso of Jeffrey s ror. RESUTS AND DISCUSSION Tble : Me Squre Error for uer Jeffrey s ror. Α M B B.6 B B

8 Aus. J. Bs & Al. S. 79: M Mmum elhoo B INEX oss Fuo Squre Error oss Fuo Tble : Me Squre Error for uer eeso of Jeffrey s ror. M B.6 B.6 B.6 B M Mmum elhoo B INEX oss Fuo Squre Error oss Fuo I ble Byes esmo wh INEX oss fuo uer Jeffrey s ror roves he smlles vlues mos ses eselly whe loss rmeer s.6. Smlrly ble Byes esmo wh INEX oss fuo uer eeso of Jeffrey s ror roves he smlles vlues mos ses eselly whe loss rmeer s.6 wheher he eeso of Jeffrey s ror s.4 or.4. Coluso: I hs er we hve resse he roblem of Byes esmo for he Webull srbuo uer symmer symmer loss fuos h of Mmum elhoo Esmo. From he resuls we observe h mos ses Byes Esmor uer er Eoel oss fuo INEX hs he smlles Me Squre Error vlues for boh ror s.e Jeffrey s eeso of Jeffrey s ror formo. REFERENCES Ahme A.O.M. N.A. Ibrhm J. Ars M.B. Am. Eeso of Jeffrey s Pror Esme for Webull Cesore us ley s romo. Ausrl Jourl of Bs Ale Sees 5: Al-Abou F.M. 9. Byes esmos for he ereme vlue srbuo us roressve esore symmer loss. Ierol Mheml Forum 433: Al-Ahr F.M.. Prmeer esmo for he ouble-reo srbuo. Jourl of Mhems Sss 74:

9 Aus. J. Bs & Al. S. 79: Al-Kuub H.S. N.A. Ibrhm 9. Byes esmor for eoel srbuo wh eeso of Jeffery ror formo. Mlys Jourl of Mheml Sees 3: Cohe A.C Mmum lelhoo esmo Webull srbuo bse o omlee o esore smles. Tehomers 7: Fr J.R A smle elo of he Webull srbuo s los. Relbly Revew 83: Hll A.J A revew of he Webull srbuo. Jourl of Quly Teholoy 5: Hoss A. W. Zmmer 3.Comrso of esmo mehos for Webull rmeers: omlee esore smles. Jourl of Ssl Comuo Smulo 73: Jefferys H A vr form for he ror robbly esmo roblems; roees of he royl Soey of oo Seres. A. 86: Ko J.H.K A rhl esmo of me Webull rmeers lfe es of elero ubes. Tehomers : wless J.F.. Ssl Moels Mehos for feme D. e. Wley New Yor. eere A. 85. Nouelles Mehos Pour Deermo Des Orbes Ds Comees Couer Prs. eble J. M. Zele 956. Ssl veso of he fue lfe of ee roove bll ber. J. Res. N. Bur. S. 57: ye.m. K.P. Hurhh S. Ry 993. Byes esmo of he ereme-vlue relbly. IEEE Trsos o Relbly 44: M N.R. R.E. Shffer N.D. Surwl 974. Mehos for Ssl Alyss of Relbly fe D. Joh Wley & Sos New Yor. Mrz H.F. R.A. Wller 98. Byes Relbly Alyss. Wley New Yor. Pey B.N. N. Dwv B. Pulsy. Comrso bewee Byes mmum lelhoo esmo of he sle rmeer Webull srbuo wh ow she uer le loss fuo. Jourl of Sef Reserh 55: Pe M.C A meho of lyss of er lsses of eermes roeess. Bomers : 4-6. Sol R.M Byes lyss of he Webull roess wh uow sle rmeer s lo o ee sml. IEEE Trsos o Relbly 7: Webull W. 95. A ssl srbuo fuo of we lbly. Al. Meh. 8: