Classical and Bayesian Approach in Estimation of Scale Parameter of Inverse Weibull Distribution

Transcription

1 Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, 5 Classal a Baysa Apoah Estmato of Sal Paramtr of Ivrs Wbull Dstrbuto Kasar Ahma*, S.P. Ahma* a A. Ahm *Dpartmt of Statsts, Uvrsty of Kashmr, Sragar, Ia. Dpartmt of Statsts a O.., Algarh Muslm Uvrsty, Algarh, Ia. Abstrat: I ths pa, Ivrs Wbull Dstrbuto s osr. Th lassal Mamum lklhoo stmator has b obta. Baysa mtho of stmato has b mploy orr to stmat th sal paramtr of Ivrs Wbull Dstrbuto by usg Jffry s a Quas s or ur thr ffrt loss futos. Also ths mthos ar ompar by usg ma squar rror wth varyg sampl szs through smulato stuy out softwar. Kywors: Ivrs Wbull Dstrbuto, Jffry s a Quas s or, loss futos a softwar.. Itrouto Th Wbull strbuto s a vry popular strbuto am aftr Walo Wbull, a Swsh physst. H appl ths strbuto 939 to aalyz th brakg strgth of matrals. S th, t has b wly us for aalyzg lftm ata rlablty grg. May ampls a b fou amog th arospa, ltros, matrals, a automotv ustrs. It a also b us as a altratv to Gamma a Log ormal strbuto rlablty grg a lf tstg. Som rt gralzatos of Wbull strbuto lug th t Wbull, mof Wbull ar suss Pham t al. 7. Zhag a X stu th haratrsts a applato of th truat Wbull strbuto whh has a bathtub shap hazar futo. Th pf of Wbull strbuto a b rasg or rasg or umol pg o th shap of th strbuto paramtrs. Du to th flblty of th Wbull pf, th Ivrs Wbull strbuto IWD has b tsvly mploy stuatos whr th mooto ata st s avalabl. Furthrmor, f th mal stus at that th Wbull pf mght b umol, th th IWD may b apoat mol. I ths pa lassal a Baysa mtho of stmato of IWD has b suss. If th raom varabl y has th Wbull Dstrbuto wth th followg pf f y;, y p y y ;, Th th raom varabl has Ivrs Wbull Dstrbuto wth th pf of th form y f ;, p ;,.

2 f Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, 5 Whr a ar th sal a th shap paramtrs rsptvly. Graphal rstato of pf Ivrs Wbull Dstrbuto alpha=.,bta=. alpha=.,bta=.5 alpha=.5,bta=. alpha =.5,bta=.5 alpha =.,bta=. alpha =.,bta= Fgur :Th pf's of Ivrs Wbull strbuto ur varous valus of alpha a bta. Matrals a Mthos: Thr ar two ma phlosophal apoahs to statsts. Th frst s all th lassal apoah whh was fou by Prof.. A. Fshr a srs of fuamtal pas rou about 93. Th altratv apoah s th Baysa apoah whh was frst sovr by vr Thomas Bays. I ths apoah, paramtrs ar trat as raom varabls a ata s trat f. A mportat -rqust ths apoah s th apoat ho of ors for th paramtrs. Vry oft, ors ar hos aorg to os subjtv kowlg a blfs. tly ths apoah has rv grat attto by most rsarhrs. ahul t al. 9 hav suss th applato of Baysa mthos. Th othr tgral part of Baysa fr s th ho of loss futo. Varous authors hav us symmtr a asymmtr loss futos for tals s; s Vara 975, Zllr 986 a S.P. Ahma a Kasar Ahma 3 a Kasar t al. 5 a, b t. Thorm.:- Lt,,..., b a raom sampl of sz havg pf., th th mamum lklhoo stmator of sal paramtr, wh th shap paramtr s kow, s gv by Proof: - Th lklhoo futo of th pf. s gv by

3 Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, 5 L ;,.. Th log lklhoo futo s gv by * log logl ;, log log.. Dffrtatg.. w.r.t. to a quat to zro, w gt. Loss futos us ths pa: L Loss futo:..3 For obtag th Baysa stmat of sal paramtr α w us a vry usful asymmtr l loss futo gv by L p, whr Whr a rst th tru a stmat valus of th paramtr. Prautoary Loss futo: Th autoary loss futo For tals, s Norstrom 996 s gv by l, Whr a rst th tru a stmat valus of th paramtr. Al-Bayyat s loss futo: Th Al-Bayyat s loss futo s gv by l A, ;. whh s a asymmtr loss futo, a rsts th tru a stmat valus of th paramtr. 3. Baysa Mtho of Estmato: 3. Postror sty ur Jffrys or: Lt,,, b a raom sampl of sz havg th obablty sty futo. a th lklhoo futo... Jffry s 946 o-formatv or for s of th form 3

4 Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, 5 g t I Whr s k-vtor valu paramtr a I s th Fshr s formato matr of orr k k. Thus whr k s a ostat. k g Th postror strbuto of s gv by L g Usg.. a , w gt a k 3..3 k Usg th valu of k Postror sty ur Quas or: Lt,,, b a raom sampl of sz havg th pf. a th lklhoo futo... W osr th or strbuto of to b g whr, 3.. Th postror strbuto of s gv by L g 3.. Usg.. a , w gt Whr k s pt of. k

5 Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, 5 5 a k By usg th valu of k 3..3, w gt Baysa stmato by usg Jffrys or ur ffrt Loss Futos: Thorm 4.:- Assumg th loss futo, l L, th Bays stmat of th sal paramtr, f th shap paramtr s kow, s of th form log Proof: - Th rsk futo of th stmator ur th l loss futo, l L s gv by th formula p 4.. Usg , w hav p 4.. O solvg 4.., w hav Mmzato of th rsk wth rspt to gvs us th optmal stmator

6 Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, 5 6 log 4..3 Thorm 4.:- Assumg th loss futo, L, th Bays stmat of th sal paramtr, f th shap paramtr s kow, s of th form Proof: - Th rsk futo of th stmator ur autoary loss futo, L s gv by th formula 4.. Usg , w hav 4..3 O solvg 4..3, w hav Mmzato of th rsk wth rspt to gvs us th optmal stmator 4..3 Thorm 4.3:- Assumg th loss futo, Al L, th Bays stmat of th sal paramtr, f th shap paramtr s kow, s of th form

7 Mathmatal Thory a Molg ISSN Pa Vol.5, No.3, 5 ISSN 5-5 Ol Proof: - Th rsk futo of th stmator ur Al-Bayyat s loss futo, gv by th formula Usg , w hav O solvg 4.3., w hav Mmzato of th rsk wth rspt to gvs us th optmal stmator Tabl 4a: Estmats usg Jffrys or ur ffrt loss futos ML l Al L s C=.5 C= Al Tabl 4b: MSE usg Jffry s or ur ffrt loss futos ML l Al C=.5 C= ML=Mamum Lklhoo, l=l loss futo, Pr=Prautoary loss futo, Al= Al-Bayyat s loss futo 7

8 Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, Baysa stmato by usg Quas or ur ffrt Loss Futos: Thorm 5.:- Assumg th loss futo, l L, th Bays stmat of th sal paramtr, f th shap paramtr s kow, s of th form log Proof: - Th rsk futo of th stmator ur th l loss futo, l L s gv by th formula p 5.. Usg , w hav p 5.. O solvg 5.., w hav p Mmzato of th rsk wth rspt to gvs us th optmal stmator log 5..3 mark: plag = 5..3, th sam Bays stmator s obta as 4..3 orrspog to th Jffry s or. Thorm 5.:- Assumg th loss futo, L, th Bays stmat of th sal paramtr, f th shap paramtr s kow, s of th form

9 Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, 5 9 Proof: - Th rsk futo of th stmator ur autoary loss futo, L s gv by th formula 5.. Usg , w hav 5.. O solvg 5.., w hav 3 Mmzato of th rsk wth rspt to gvs us th optmal stmator 5..3 mark: plag = 5..3, th sam Bays stmator s obta as 4..3 orrspog to th Jffry s or. Thorm 5.3:- Assumg th loss futo, Al L, th Bays stmat of th sal paramtr, f th shap paramtr s kow, s of th form Proof: - Th rsk futo of th stmator ur Al-Bayyat s loss futo, Al L s gv by th formula

10 Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, Usg , w hav O solvg 5.3., w hav Mmzato of th rsk wth rspt to gvs us th optmal stmator mark: plag = 5.3.3, th sam Bays stmator s obta as orrspog to th Jffry s or. Tabl 5a: Estmats usg Quas or ur ffrt loss futos l ML C=.5 Al C=

11 Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, 5 Tabl 5b: MSE usg Quas or ur ffrt loss futos ML l 5 5 futo C=.5 Al C= ML=Mamum Lklhoo, l=l loss futo, Pr=Prautoary loss futo, Al= Al-Bayyat s loss 6. sults a Dsusso:- W marly stu th Classal Mamum Lklhoo stmato a Baysa stmato for Ivrs Wbull Dstrbuto usg Jffrys a Quas ors ur thr ffrt loss futos. For srptv mar, w grat ffrt raom sampls of sz 5,5 a to rst small, mum a larg ata st for th Ivrs Wbull Dstrbuto Softwar, a smulato stuy was arr out 3, tms for ah pars of, whr.5,. a.,.5.th valus of tso wr =.5,.5. Th valu for th loss paramtr C =.5 a..ths was trat tms a th stmats of sal paramtr for ah mtho wr alulat. Th rsults ar st tabls 4a, 4b, 5a a 5b rsptvly. I tabl 4b, Bays stmat usg Jffry s or ur Al-Bayyat s Loss futo ovs th smallst valus most ass as ompar to othr loss futos a lassal mamum lklhoo stmator spally wh loss paramtr C s.. I tabl 5b, Bays stmat usg Quas or ur Al-Bayyat s Loss futo ovs th smallst valus most ass as ompar to othr loss futos a lassal mamum lklhoo stmator spally wh loss paramtr C s.. 7. Coluso:- From th abov rsults, w obsrv that Bays stmat ur Al-Bayyat s Loss futo has th smallst Ma Squar Error MSE valus for both or s Jffry s a Quas or as ompar to othr loss futos a lassal mamum lklhoo stmator spally wh

12 Mathmatal Thory a Molg ISSN Pa ISSN 5-5 Ol Vol.5, No.3, 5 loss paramtr C s. Thus w a olu that Bays stmat ur Al-Bayyat s Loss futo s fft. frs:. Ahma, S.P. a Kasar Ahma 3. Baysa Aalyss of Wbull Dstrbuto Usg Softwar Australa Joural of Bas a Appl Ss, 79: ISSN Hoag Pham a Ch-Dw La 7: O t Gralzatos of th Wbull Dstrbuto. IEEE Trasatos o ELIABILITY, 563: Jffrys, H A varat form for th or obablty stmato oblms; ogs of th royal Soty of Loo, Srs. A. 86, Kasar Ahma, S.P. Ahma a A. Ahm 5 a: O paramtr stmato of Erlag Dstrbuto usg Baysa mtho ur ffrt loss futos. Progs of tratoal ofr o Avas Computrs, Commuato, a Eltro Egrg hl Marh 6-8, 5, pp -6. Uvrsty of Kashmr. ISBN: Kasar Ahma, S.P. Ahma a A. Ahm 5 b: Baysa Aalyss of Gralz Gamma Dstrbuto usg Softwar. Joural of Statsts Applatos & Probablty, J. Stat. Appl. Pro. 4, No., Norstrom, J. G Th us of Prautoary Loss Futos sk Aalyss. IEEE Trasatos o rlablty, 3, ahul, G. P., Sgh a O.P. Sgh, 9: Populato ojt OF Krala usg Baysa mthoology.asa J. Appl S., : Vara, H A Baysa apoah to ral stat assssmt, Stus Baysa Eoomtrs a Statsts Hoor of Loar J. Savag s: S.E. Fbrg a A.Zllr, North-Holla, Amstram, Zllr, A., 986: Baysa stmato a to usg asymmtr loss futo. Joural of Amra Statstal Assoato potal strbuto usg smulato. Ph.D Thss, Bagha Uvrsty, Iraq. 8, Zhag T, X M. : O th up truat Wbull strbuto a ts rlablty mplatos. lablty Egrg Systm Safty; 96: