POSTERIOR ESTIMATES OF TWO PARAMETER EXPONENTIAL DISTRIBUTION USING S-PLUS SOFTWARE

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1 Joural of Rliabilit ad tatistial tudis [IN: Prit Oli] Vol. 3 Issu 00:7-34 POTERIOR ETIMATE OF TWO PARAMETER EXPONENTIAL DITRIBUTION UING -PLU OFTWARE.P. Ahmad ad Bilal Ahmad Bhat. Dartmt of tatistis Uivrsit of Kashmir riagar Idia E Mail: srvz@ahoo.om. Divisio of riultur KUAT-K halimar riagar Idia Abstrat This ar dals with th Basia stimatio of aramtrs of xotial distributio udr diffrt riors. Th umrial ad grahial illustratio of ostrior dsitis of th aramtrs of itrst has b do i -PLU oftwar. Kwords: Exotial distributio Margial ostrior dsitis Postrior stimats ad -PLU oftwar.. Itrodutio Exotial distributio is a widl usd liftim distributio whih has aard i th litratur si th arl 800s. This distributio is o of th ommol usd statistial distributios i rati. ukhatm 937 Esti ad obl Esti 954 Bartholomw 957 Mdhall 958 Johso Kotz ad Balakrisha Lawlss 003 ad othrs hav disussd this distributio with aliatios. As giv i iha 986 th df of two aramtr xotial distributio is giv b f < < < > 0 If... b iid obsrvatios from a xotial distributio th th liklihood futio is giv b Whr i i i L } i } is th first ordr statisti i th saml... suh that ad i... i. Jffr 96 ad othrs mak xtsiv us of imror rior df s. Lt us osidr a mor gral lass of riors

2 8 Joural of Rliabilit ad tatistial tudis Dmbr 00 Vol. 3 α 0 3. Postrior dsit for ad Aordig to Bas thorm w hav Postrior dsit rior dsit *liklihood i.. α L from th quatios ad 3 th ostrior dsit of ad is giv b } α K Whr K is a ormalizig ostat ad is giv b K 0 K } } d d } 4 For 0 uiform rior 4 boms For 4 boms For 4 boms } } } 3. Margial Postrior dsitis for ad Th margial ostrior dsit of is giv b

3 Postrior Estimats of Two Paramtr 9 0 } 0 } d d } 5 For 0 5 boms } For 5 boms } For 5 boms } Th margial ostrior dsit of is giv b } d d 6 For 0 6 boms For 5 boms

4 30 Joural of Rliabilit ad tatistial tudis Dmbr 00 Vol. 3 For 5 boms 4. Postrior stimats of ad Th ostrior stimat of is giv b E d d } 7 For 0 7 boms 3 E 3 E 7 boms E For 7 boms For Th Postrior stimats of is giv b 0 E d 3 For 0 8 boms 0 For 8 boms For 8 boms d E 3 E E 5. Numrial & Grahial Illustratio Grubbs F.E. 97 8

5 Postrior Estimats of Two Paramtr 3 Nit militar rsol arrirs faild i srvis for o raso or th othr at th followig milags: ad 880 mils. Numrial ad grahial illustratios ar imlmtd i -PLU oftwar for two aramtr xotial distributio. Postrior stimats of ad ar giv i Tabl I. Th grahial rrstatio for margial ostrior dsitis of ad ar show i Figurs ad rstivl. Morovr w hav dvlod th futio for stimatig aramtrs ad of two aramtr xotial distributio udr diffrt riors & is giv i Adix A. Also futios for grahial rrstatio of th margial dsitis of ad udr diffrt riors wr also dvlod i -PLU ad ar giv i Adix B & C rstivl. Prior Postrior ma of Postrior ma of Tabl I: Postrior stimats of ad udr diffrt riors usig -PLU Th ostriors of ad ar lottd i figurs & rstivl. Th ostriors ar quit robust for varig i th rior ostriors of is lss robust. α whil th

6 3 Joural of Rliabilit ad tatistial tudis Dmbr 00 Vol. 3 Postrior dsit of mu udr diffrt riors Postrior dsit of thta udr diffrt riors mu Prior Prior/thta Prior/thta^ thta Prior Prior/thta Prior/thta^ mu Figur: thta Figur: Adix A: Futio for stimatig aramtrs ad of two aramtr xotial distributio. Mu.thta<-futio <-lgth C<-0 <-mi s<-sum- stimat<--s/*c-3 stimat<-s/c-3 listmustimatthtastimat } > < > Mu.thta

7 Postrior Estimats of Two Paramtr 33 Adix B: Futio for grahial rrstatio of th margial dsit of udr diffrt riors. mu.lot<-futio <-lgth <-mi s<-sum- mu<-sq060 mu<-*-*s^-/s*-mu^- lotmumuxlab"mu"lab"mu"lim00.0 mai "Postrior dsit of mu udr diffrt riors" sub"figur: "t"l"lt mu<-*-*s^-/s*-mu^ lismumult mu<-**s^/s*-mu^ lismumult3 } > < > mu.lot > lg.ams<"prior""prior/thta""prior/thta^" > lgdloatorlg.amslt:3 Adix C: Futio for grahial rrstatio of th margial dsit of udr diffrt riors. thta.lot<-futio <-lgth <-mi s<-sum- thta<-sq50700 thta<- s^-*x-s/thta/gamma-*thta^- thta<- s^-*x-s/thta/gamma-*thta^ thta<- s^*x-s/thta/gamma*thta^ lotthtathtaxlab"thta"lab"thta"lim0.00 mai "Postrior dsit of thta udr diffrt riors" sub"figur: "t"l"lt listhtathtalt listhtathtalt3 } > < > thta.lot > lg.ams<-"prior""prior/thta""prior/thta^" > lgdloatorlg.amslt:3

8 34 Joural of Rliabilit ad tatistial tudis Dmbr 00 Vol. 3 Rfrs. Bartholomw D.J A roblm i lif tstig Joural of Amria tatistial Assoiatio Esti B Truatd lif tstig i xotial as Aals of th Mathmatial tatistis Esti B. ad obl M Lif tstig Joural of th Amria tatistial Assoiatio Esti B. ad obl M om thorms rlvat to lif tstig from a xotial distributio Aals of th Mathmatial tatistis Esti B. ad obl M qutial lif tstig i xotial as Aals of th Mathmatial tatistis Grubbs F.E. 97. Aroximat fiduial bouds o rliabilit for th two aramtr gativ xotial distributio Thomtris Vol. 3 No Jffr H. 96. Thor of Probabilit Oxford Uivrsit Prss. 8. Johso N.L. Kotz. ad Balakrisha N Cotiuous Uivariat Distributios Vol. Joh Wil & os Nw York. 9. Johso N.L. Kotz. ad Balakrisha N Cotiuous Uivariat Distributios Vol. Joh Wil & os Nw York. 0. Johso R.A Asmtoti xasios assoiatd with ostrior distributios Aals of Mathmatial tatistis Lawlss J.F tatistial Modls ad Mthods for Liftim Data. Wil Nw York.. Mdhall W A bibliograh o lif tstig ad rlatd tois. Biomtrika iha.k Rliabilit ad Lif Tstig. Wil Eastr Limitd.

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