On the Bayesian analysis of 3-component mixture of Exponential distributions under different loss functions

Transcription

1 O the Bayesa aalyss of -compoet mxture of Expoetal dstrbutos uder dfferet loss fuctos Muhammad Tahr ad Muhammad Aslam ad Zawar Hussa Abstract The memory-less property of the Expoetal dstrbuto s a strog reaso of ts use for testg lfetmes of obects may lfetme modelg applcatos. Also, mxture models have extesvely bee used survval aalyss ad relablty studes. Ths artcle focuses o the Bayesa aalyss of the -compoet mxture of Expoetal dstrbutos uder type-i rght cesorg scheme. Takg dfferet oformatve ad formatve prors, Bayes estmators ad posteror rsks for the ukow parameters parameters of compoet dstrbutos ad mxg proportos are derved uder squared error loss fucto, precautoary loss fucto ad DeGroot loss fucto. The elctato of the hyperparameters s also doe usg pror predctve dstrbuto.the Bayes estmators ad posteror rsks are looked at as a fucto of the test termato tme. Some mportat propertes ad comparsos of the Bayes estmates are preseted. Smulated results ad real data example are also gve to llustrate the study. Keywords: -Compoet mxture dstrbuto, No-formatve ad formatve prors, Loss fucto, Bayes estmators, Posteror rsks, Test termato tme. 000 AMS Classfcato: AMS 6F5, 6N05.. Itroducto The Expoetal dstrbuto, because of ts memory-less property, has may real lfe applcatos testg lfetmes of obects where lfetmes do ot deped upo ther ages. There are may electroc devces whose falure rate does ot deped o ther ages, therefore, the Expoetal dstrbuto s sutable to model the lfetmes. Geerally, lfetme modelg, populato s supposed to be composed of more tha oe subpopulatos mxed by ukow mxg proportos. I our study, we take the data from a populato whch s characterzed by three dfferet members of the Expoetal famly of dstrbutos. McCullagh 994 derved some codtos uder whch quadratc ad polyomal Expoetal models ca be Departmet of Statstcs, Quad--Azam Uversty, Islamabad, Paksta Departmet of Statstcs, Govermet College Uversty, Fasalabad, Paksta Emal: tahrqaustat@yahoo.com Correspodg Author. Departmet of Basc Sceces, Rphah Iteratoal Uversty, Islamabad, Paksta, Emal: aslamsdqu@yahoo.com Departmet of Statstcs, Quad--Azam Uversty, Islamabad, Paksta, Emal: zhlagah@yahoo.com

2 geerated as mxtures of the Expoetal models. Raqab ad Ahsaullah 00 dscussed the locato ad scale parameters of geeralzed Expoetal dstrbuto based o order statstc. Hebert ad Scarao 005 compared the locato estmators for the Expoetal mxtures uder Ptma s measure of closeess. Al et al. 005 studed the Bayes estmators of the Expoetal dstrbuto ad Abu-Taleb et al. 007 preseted the Bayesa estmato of lfetme parameters of Expoetal dstrbutos whe survval tme ad cesorg tme are both expoetally dstrbuted. The use of mxture models stuatos where data are gve oly for overall mxture dstrbutos s kow as drect applcato of the mxture models. L 98 ad L ad Sedrask 98, 988 dscussed dfferet features of mxture models ad defed two types of mxture models. The mxture of the probablty desty fuctos from the same famly s kow as type-i mxture model ad type-ii mxture model s defed as a mxture of desty fuctos from several famles. I ths study, the drect applcato of mxture model wth the ukow compoet ad mxg proporto parameters of the -compoet mxture of Expoetal dstrbutos s cosdered uder type-i mxture modelg. Due to the developmet of advaced computatoal facltes, researchers are ow able to fd the Bayes estmates, fer ad predct about complex systems such as mxture models. Wth the provso of these computatoal facltes, the Bayesa techque to aalyze a -compoet mxture model has developed the terest amog may researchers. The posteror dstrbuto, whch s obtaed whe pror formato s combed wth lkelhood, s the workbech of Bayesa ferece. Thus, the pror formato, a subectve assessmet by a expert before the data are actually gathered, s very mportat ad ecessary for Bayesa ferece.i ths study, the Bayesa aalyss of a -compoet mxture of Expoetal dstrbutos usg the o-formatve uform ad Jeffreys prors ad the formatve pror IP uder squared error loss fucto SELF, precautoary loss fucto PLF ad DeGroot loss fucto DLF s cosdered. There are may felds such as egeerg, bologcal sceces, physcal sceces ad socal sceces where mxture models have bee used qute effectvely. Most of the researchers worked o the Bayesa aalyss of -compoet mxture models. For example, Sha 998 used the Bayesa couterpart of the maxmum lkelhood estmates of the -compoet mxture model cosdered by Medehall ad Hader 958. Saleem ad Aslam 008 dscussed the use of the formatve ad the o-formatve prors for Bayesa aalyss of the -compoet mxture of Raylegh dstrbutos. Saleem et al. 00 preseted the Bayesa aalyss of the -compoet mxture of Power dstrbutos usg the complete ad cesored data. Kazm et al. 0 developed the Bayesa aalyss for the -compoet mxture of Maxwell dstrbutos. I real lfe applcatos, most of the tmes, t s ot sutable to cotue the testg procedure utl falure of the last obect uder testg. I such stuatos, cesored samples are observed. Cesorg s a mportat ad valuable aspect of lfetme applcatos. A valuable accout o cesorg s gve Romeu 004, Gbels 00 ad Kalbflesch ad Pretce 0. I ths paper, a ordary type-i rght cesorg s used wth fxed lfe-test termato tme for all obects.

3 The rest of the paper s orgazed as follows. I Secto, the -compoet mxture of Expoetal dstrbutos s preseted. Posteror dstrbutos usg the uform pror UP, the Jeffreys pror JP ad the formatve pror IP are derved Secto. The Bayes estmators ad posteror rsks usg the UP, the JP ad the IP uder SELF, PLF ad DLF are preseted Sectos 4, 5 ad 6, respectvely. The elctato of hyperparameters s descrbed Secto 7. The lmtg expressos are derved Secto 8. A smulato study ad real data example are dscussed Sectos 9 ad 0, respectvely. Fally, the cocluso of the study s gve Secto.. -compoet mxture of expoetal dstrbutos If X s expoetally dstrbuted wth parameter θ m, ts probablty desty fucto s gve as:. f m x; θ m = θ m exp θ m x, x 0, θ m > 0, m =,,. Accordg to Barger 006 ad Stírelec ad Stehlík 0, a fte -compoet mxture of Expoetal dstrbutos wth ukow mxg proportos p ad p s defed as:. f x = p f x + p f x + p p f x, p, p 0, p + p. f x; θ, θ, θ, p, p = p θ exp θ x + p θ exp θ x + p p θ exp θ x As cumulatve dstrbuto fucto of the radom varable X s gve by:.4 F m x = exp θ m x, m =,,, the cumulatve dstrbuto fucto of -compoet mxture dstrbuto s defed as:.5 F x = p F x + p F x + p p F x.6 F x = p exp θ x p exp θ x p p exp θ x. The posteror dstrbuto usg the UP, the JP ad the IP The posteror dstrbutos of parameters gve data x are derved usg the UP, the JP ad the IP... The lkelhood fucto. Suppose uts are used a lfe testg expermet wth the -compoet mxture modelg. Let r out of uts fal before fxed test termato tme t ad the remag r uts are stll workg. Accordg to Medehall ad Hader 958, there are may practcal stuatos whch the falg obects ca be poted out easly as subset of subpopulato-, subpopulato-or subpopulato-. Out of r uts, suppose r, r ad r uts belog to subpopulato-, subpopulato- ad subpopulato-,respectvely,such that r = r + r + r. Now, defe x lk, 0 < x lk t, as the falure tme of k th k =,,, r l ut belog to l th l =,, subpopulato. Thus, the lkelhood fucto of the -compoet mxture model for the radom sample vector x s gve ascf. Evertt ad Had, 98:

4 4 L ψ x.. { r } { r } { r } p f x k p f x k p p f x k k= { F t} L ψ x θ r θr θr exp exp { θ { θ k= r t rt t + t t + r k= r k= x k } x k } exp k= p +r p +r p p +r ], { θ t + r k= x k } where ψ = θ, θ, θ, p, p ad x= x,..., x r, x,..., x r, x,..., x r... The posteror dstrbuto usg the UP. Whe o or lttle pror formato s gve, usually, the o-formatve pror s assumed to be the UP. Bayes 76, de Laplace 80 ad Gesser 984 proposed that oe may take the UP for the ukow parameters ψ = θ, θ, θ, p, p. Followg Bayes 76, de Laplace 80 ad Gesser 984, UPs over the tervals 0, ad 0, are take for the parameters θ, θ ad θ of Expoetal dstrbutos ad for the mxg proportos p ad p, respectvely. Wth these settgs, ot pror dstrbuto of the parameters θ, θ, θ, p ad p, as defed by Saleem 00, s gve by:. π ψ ; θ, θ, θ > 0, p, p 0, p + p. The ot posteror dstrbuto of parameters θ, θ, θ, p ad p gve data x, usg the UP s defed as: L ψ x π ψ.4 g ψ x = ψ L ψ x π ψ dψ.5 g ψ x = E θ A θ A θ A r exp B θ exp B θ exp B θ p A 0 p B 0 p p C 0, where A = r +, A = r +, A = r +, B = t rt t+ r k= x k, B = t t+ r k= x k, B = t+ r k= x k, A 0 = +r +, B 0 = +r +, C 0 = + r +, E = Γ A Γ A Γ A =0 =0 B A 0, B 0, C 0 B A B A B A. r

5 5.. The posteror dstrbuto usg the JP. Accordg to Jeffreys 946, 96, Berardo 979 ad Berger 985, the JP s defed as p θ m [ ] I θ m, m =,,, where I θ m = E fx θ m θ s the Fsher s formato matrx. m The pror dstrbutos of the mxg proportos p ad p are aga take to be the uform o over the terval 0,. The ot pror dstrbuto of parameters θ, θ, θ, p ad p s cf. Sha, 998 gve by:.6 π ψ θ θ θ, θ, θ, θ > 0, p, p 0, p + p The ot posteror dstrbuto of parameters θ, θ, θ, p ad p gve data x, usg the JP s: L ψ x π ψ.7 g ψ x = ψ L ψ x π ψ dψ.8 g ψ x = r exp B θ E θ A θ A θ A exp B θ exp B θ p A0 p B0 p p C0, where A = r, A = r, A = r, B = t rt t + r k= x k, B = t t + r k= x k, B = t + r k= x k, A 0 = r + r +, B 0 = + r +, C 0 = + r +, E = Γ A Γ A Γ A r B A 0, B 0, C 0 B A B A B A..4. The posteror dstrbuto usg the IP. As a formatve pror dstrbuto, we take Gamma dstrbuto for compoet parameters θ, θ, θ ad bvarate beta dstrbuto for proporto parameters p, p,.e..9 π 4 θ ; a, b = ba Γ a θa exp b θ, θ > 0, a, b > 0.0 π 5 θ ; a, b = ba Γ a θa exp b θ, θ > 0, a, b > 0. π 6 θ ; a, b = ba Γ a θa exp b θ, θ > 0, a, b > 0. π 7 p, p ; a, b, c = B a, b, c pa p b p p c, p, p 0, p + p, a, b, c > 0. So, the ot pror dstrbuto of parameters θ, θ, θ, p ad p usg the IP s. π ψ θ a exp b θ θ a exp b θ θ a exp b θ p a p b p p c

6 6 The ot posteror dstrbuto of parameters θ, θ, θ, p ad p gve data x, usg the IP s: L ψ x π ψ.4 g ψ x = ψ L ψ x π ψ dψ r exp B θ exp B θ g ψ x = E θ A θ A θ A.5 exp B θ p A 0 p B 0 p p C0, where A = r +a, A = r +a, A = r +a, B = t rt t+ r k= x k+b, B = t t+ r k= x k+b, B = t+ r k= x k+b, A 0 = +r +a, B 0 = +r +b, C 0 = +r +c, E = Γ A Γ A Γ A r B A 0, B 0, C 0 B A B A B A. 4. The Bayes estmators ad posteror rsks usg the UP, the JP ad IP uder SELF If L θ, d s a loss fucto the the expected value of the loss fucto for a gve decso wth respect to the posteror dstrbuto s posteror rsk fucto ad f ˆd s a Bayes estmator the ρ ˆd s called posteror rsk ad s gve by } ρ ˆd = E θ x {L θ, ˆd. The SELF s suggested by Legedre 806 ad s defed as: L θ, d = θ d. The Bayes estmator ad posteror rsk uder SELF are: ˆd = Eθ x θ ad ρ ˆd = E θ x θ { E θ x θ }, respectvely. So, the Bayes estmators ad posteror rsks usg the UP, the JP ad IP for parameters θ, θ, θ, p ad p uder SELF are obtaed wth ther respectve margal posteror dstrbutos as gve below: ˆθ v = Γ A v + Γ A v Γ A v 4. B Av+ v 4. B Av v 4. B Av v B Av v B Av ˆθ v = Γ A v Γ A v + Γ A v B A v+ v r v B A 0v, C 0v B B 0v, A 0v + C 0v r B Av ˆθ v = Γ A v Γ A v Γ A v + B Av v v B A 0v, C 0v B B 0v, A 0v + C 0v r B A v+ ˆp v = Γ A v Γ A v Γ A v v B A 0v, C 0v B B 0v, A 0v + C 0v r

7 7 4.4 B A v v 4.5 B A v v ˆθv B A v v B A v ˆp v = Γ A v Γ A v Γ A v ρ B A v+ v v B B 0v, C 0v B A 0v +, B 0v + C 0v r B A v v B A v v B A 0v, C 0v B B 0v +, A 0v + C 0v = Γ A v + Γ A v Γ A v r B Av v B Av v ΓA v+γa vγa v B A v+ v ρ ˆθv B A v v B A v v B A v B A 0v, C 0v B B 0v, A 0v + C 0v r v B A 0v, C 0v B B 0v, A 0v + C 0v r = Γ A v Γ A v + Γ A v B Av+ v B A v v ΓA v ΓA v +ΓA v B A v v ρ ˆθv B Av v B Av+ v B A v B A 0v, C 0v B B 0v, A 0v + C 0v r v B A 0v, C 0v B B 0v, A 0v + C 0v r = Γ A v Γ A v Γ A v + B Av v B A v+ v ΓA vγa vγa v+ B Av v B Av v B A v+ ρ ˆp v = Γ A v Γ A v Γ A v B A v v B A v v B A v v ΓA v ΓA v ΓA v B A v v B A v v B A v ρ ˆp v = Γ A v Γ A v Γ A v B A v v B A v v B A v v B A 0v, C 0v B B 0v, A 0v + C 0v r v B A 0v, C 0v B B 0v, A 0v + C 0v r B B 0v, C 0v B A 0v +, B 0v + C 0v r v B B 0v, C 0v B A 0v +, B 0v + C 0v r B A 0v, C 0v B B 0v +, A 0v + C 0v ΓA v ΓA v ΓA v r 4.0, B A v v B A v v B A v v B A 0v, C 0v B B 0v +, A 0v + C 0v where v = for the UP, v = for the JP ad v = for the IP.

8 8 5. The Bayes estmators ad posteror rsks usg the UP, the JP ad IP uder PLF Norstrom 996 dscussed a asymmetrc PLF ad a specal case of geeral class of PLFs s L θ, d = θ d d. The Bayes estmator ad posteror rsk are: ˆd = { E } θ x θ ad ρ ˆd = { E } θ x θ E θ x θ, respectvely. The respectve margal posteror dstrbutos yeld the Bayes estmators ad posteror rsks usg the UP, the JP ad the IP for parameters θ, θ, θ, p ad p uder PLF as: 5. ˆθv = 5. ˆθv = 5. ˆθv = 5.4 ˆp v = 5.5 ˆp v = ΓA v +ΓA v ΓA v B Av+ v ΓA v ΓA v +ΓA v B A v v r B A v v B A v v B A 0v, C 0v B B 0v, A 0v + C 0v B Av+ v ΓA v ΓA v ΓA v + B Av v B Av v ΓA vγa vγa v B Av v B Av v ΓA vγa vγa v B Av v B Av v r B A v v B A 0v, C 0v B B 0v, A 0v + C 0v r B Av+ v B A 0v, C 0v B B 0v, A 0v + C 0v r B Av v B B 0v, C 0v B A 0v +, B 0v + C 0v r B Av v B A 0v, C 0v B B 0v +, A 0v + C 0v 5.6 ρ ˆθv = ΓA v +ΓA v ΓA v B Av+ v B A v v ΓA v +ΓA v ΓA v B A v+ v r B A v v B A 0v, C 0v B B 0v, A 0v + C 0v r B Av v B Av v B A 0v, C 0v B B 0v, A 0v + C 0v 5.7

9 ρ ˆθv = ΓA v ΓA v +ΓA v B A v v B Av+ v B A v ΓA v ΓA v +ΓA v B A v v B Av+ v r v B A 0v, C 0v B B 0v, A 0v + C 0v r B A v v B A 0v, C 0v B B 0v, A 0v + C 0v ρ ˆθv = ΓA v ΓA v ΓA v + B Av v B Av v B Av+ ΓA vγa vγa v+ B Av v B Av v r v B A 0v, C 0v B B 0v, A 0v + C 0v r B A v+ v B A 0v, C 0v B B 0v, A 0v + C 0v 5.9 ρ ˆp v = ΓA vγa vγa v B Av v B Av v ΓA v ΓA v ΓA v B A v v r B Av v B B 0v, C 0v B A 0v +, B 0v + C 0v r B A v v B A v v B B 0v, C 0v B A 0v +, B 0v + C 0v 5.0 ρ ˆp v = ΓA v ΓA v ΓA v B A v v B A v v B A v ΓA vγa vγa v B Av v B Av v r v B A 0v, C 0v B B 0v +, A 0v + C 0v r B Av v B A 0v, C 0v B B 0v +, A 0v + C 0v 6. The Bayes estmators ad posteror rsks usg the UP, the JP ad the IP uder DLF DeGroot 005 troduced the asymmetrc loss fucto, L θ, d = θ d, d kow as DLF.The Bayes estmator ad ts posteror rsk uder DLF are: ˆd = E θ x θ E θ x θ ad ρ ˆd = {E θ x θ} E θ x θ, respectvely. The Bayes estmators ad posteror rsks usg the UP, the JP ad the IP for parameters θ, θ, θ, p ad

10 0 p uder DLF are: ˆθ v = ΓAv+ΓAvΓAv r ΓA v +ΓA v ΓA v r B A v + v B A v + v ˆθ v = ΓA vγa v +ΓA v r ΓA vγa v+γa v r ˆθ v = ΓAvΓAvΓAv+ r B A v v B A v v B A v v B A v + v ΓA v ΓA v ΓA v + r ˆp v = ΓA vγa v ΓA v r ΓA vγa vγa v r B A v v BA 0v,C 0vBB 0v,A 0v+C 0v B A v v BA 0v,C 0v BB 0v,A 0v +C 0v B A v v BA 0v,C 0v BB 0v,A 0v +C 0v B A v v B A v + v B A v v BA 0v,C 0v BB 0v,A 0v +C 0v B A v v B A v v B A v + v BA 0v,C 0vBB 0v,A 0v+C 0v B A v v B A v v B A v + v BA 0v,C 0v BB 0v,A 0v +C 0v B A v v B A v v ˆp v = ΓA vγa v ΓA v r ΓA v ΓA v ΓA v r B A v v B A v v ρ ˆθv = {ΓA v+γa v ΓA v } r r B A v v B A v v BB 0v,C 0v BA 0v +,B 0v +C 0v B A v v B A v v BB 0v,C 0vBA 0v+,B 0v+C 0v B A v v B A v v BA 0v,C 0v BB 0v +,A 0v +C 0v B A v v B A v v BA 0v,C 0v BB 0v +,A 0v +C 0v E vγa v+γa vγa v B A v + v B A v v B A v v BA 0v,C 0vBB 0v,A 0v+C 0v B A v + v B A v v ρ ˆθv = {ΓA vγa v +ΓA v } r r B A v v BA 0v,C 0v BB 0v,A 0v +C 0v ΓA v ΓA v +ΓA v B A v v B A v + v B A v v BA 0v,C 0v BB 0v,A 0v +C 0v B A v v B A v + v B A v v BA 0v,C 0vBB 0v,A 0v+C 0v

11 ρ ˆθv = {ΓA vγa v ΓA v +} r r E vγa vγa vγa v+ B A v v B A v v B A v + v BA 0v,C 0v BB 0v,A 0v +C 0v ρ ˆp v = {ΓA vγa v ΓA v } r B A v v r B A v v B A v v B A v v B A v + v BA 0v,C 0vBB 0v,A 0v+C 0v E vγa vγa vγa v B A v v B A v v BB 0v,C 0v BA 0v +,B 0v +C 0 ρ ˆp v = {ΓA vγa v ΓA v } r B A v v r B A v v B A v v B A v v BB 0v,C 0v BA 0v +,B 0v +C 0 ΓA v ΓA v ΓA v B A v v B A v v BA 0v,C 0vBB 0v+,A 0v+C 0v B A v v B A v v BA 0v,C 0v BB 0v +,A 0v +C 0v. 7. Elctato of hyperparameters Elctato s a tool used to quatfy a perso s belef ad kowledge about the parameters of terest. I Bayesa perspectve, elctato, most ofte, arses as a method for specfyg the pror dstrbuto of the radom parameters. Elctato s smply the quatfcato of pror kowledge about the radom parameters so that ths ca the be combed wth the lkelhood to obta posteror dstrbuto for further statstcal aalyss. Elctato has remaed a challegg problem for the statstca.authors who have dscussed ths problem clude Kadae et al. 980, Brch ad Bartollucc 98, Chaloer ad Duca 98, Gavasakar 988, Al-Awadh ad Gartwate 998, Aslam 00, Hah 006, Saleem ad Aslam 008 ad refereces cted there. I ths study, we adopted a method based o predctve probabltes, gve by Aslam 00. For elctg the hyperparameters, pror predctve dstrbuto PPD s used. The PPD for a radom varable X s: 7. p x = p x ψ π ψ dψ 7. p x = ψ a + b + c [ ] a a b a b + x a+ + b a b a b + x a+ + c a b a b + x a+. We choose the pror predctve probabltes, satsfyg the laws of probablty, to elct the hyperparameters of the pror desty. By followg these laws of probablty, some mor cossteces may arse whch are expected to be gorable. Usg the pror predctve dstrbuto gve 7. we cosder e tervals 0,,,,,,, 4, 4, 5, 5, 6, 6, 7, 7, 8 ad 8, 9 wth probabltes 0.57, 0.0, 0.0, 0.05, 0.0, 0.05, 0.0, ad 0.00, respectvely, gve as expert opo. The followg e equatos are derved from the gve formato

12 usg the 7. as: a + b + c a + b + c a + b + c a + b + c a + b + c a + b + c a + b + c a + b + c a + b + c [ ] a a b a b + x a + + b a b a b + x a + + c a b a b + x a dx = [ ] a a b a b + x a + + b a b a b + x a + + c a b a b + x a dx = [ ] a a b a b + x a + + b a b a b + x a + + c a b a b + x a dx = [ ] a a b a b + x a+ + b a b a b + x a+ + c a b a b + x a+ dx = 0.05 [ ] a a b a b + x a+ + b a b a b + x a+ + c a b a b + x a+ dx = 0.0 [ ] a a b a b + x a + + b a b a b + x a + + c a b a b + x a dx = [ ] a a b a b + x a + + b a b a b + x a + + c a b a b + x a dx = [ ] a a b a b + x a+ + b a b a b + x a+ + c a b a b + x a+ dx = [ ] a a b a b + x a+ + b a b a b + x a+ + c a b a b + x a+ dx = 0.00 The above e equatos are solved smultaeously by usg Mathematca software for elctg the hyperparameters a, b, a, b, a, b, a, b, c. Through ths crtera, the values of the hyperparameters are obtaed as.80,.70,.570,.60,.900,.70,.080, , The lmtg expressos Whe t teds to, r teds to ad r l teds to l, l =,,, the all the values whch are cesored become ucesored our aalyss. So, the formato cotaed the sample s creased. Cosequetly, the posteror rsks of the Bayes estmates dmsh. The effcecy of the Bayes estmates s creased because all the values are corporated our sample. The lmtg complete sample expressos for Bayes estmators ad posteror rsks usg the UP, the JP ad the IP uder SELF, PLF ad DLF are gve the Tables -6.

13 Table. Lmtg Expressos for the Bayes Estmators as t usg the UP, the JP ad the IP uder SELF Bayes Estmators Parameters UP JP IP θ + k= x k θ + k= x k θ + k= x k p + + p + + k= x k k= x k k= x k a k= x k+b +a k= x k+b +a k= x k+b +a +a+b+c +b +a+b+c Table. Lmtg Expressos for the Posteror Rsks as t usg the UP, the JP ad the IP uder SELF Posteror Rsks Parameters UP JP IP θ + k= x k +a k= x k k= x k+b θ + k= x k +a k= x k k= x k+b θ + k= x k +a k= x k k= x k+b p a + +b+c a+b+c +a+b+c+ p b + +a+c +a+b+c +a+b+c+ Table. Lmtg expressos for the Bayes estmators as usg the UP, the JP ad the IP uder PLF t Bayes Estmators Parameters UP JP IP θ + / + / / + / k= x k / k= x k / k= x k+b / +a +a + / θ + / + / + / k= x k / +a +a + / k= x k / k= x k+b / θ + / + / + / k= x k / +a +a + / k= x k / k= x k+b / p + / + / + / + / +a / +a+ / + / +4 / + / +4 / +a+b+c / +a+b+c+ / p + / + / + / +4 / + / + / + / +4 / +b / +b+ / +a+b+c / +a+b+c+ / 9. Smulato study Smulato study s a flexble methodology to llustrate the propertes of the Bayes estmates of the -compoet mxture of Expoetal dstrbutos usg the UP, the JP ad the IP uder SELF, PLF ad DLF terms of dfferet sample szes ad test termato tmes. The samples of dfferet szes = 0, 00, 00 are geerated from the -compoet mxture of Expoetal

14 4 Table 4. Lmtg expressos for the posteror rsks as t usg the UP, the JP ad the IP uder PLF Posteror Rsks Parameters { UP } { JP } { IP } θ + + / + / k= x +a k + / k= x k / +a + / k= x k+b +a { } { } { / } θ + + / + / k= x +a +a + / k + / k= x k / k= x k+b { } { } { +a / } θ + + / + / k= x +a k + / k= x k / +a + / k= x k+b +a } } / } p + + p + + { + / + / +4 / { + / + / } + / +4 / + / { + / + / +4 / { + / + / } + / +4 / + / +a +a+b+c +b +a+b+c { +a+ / +a / +a+b+c+ / { +b+ / +a+b+c / } +b / +a+b+c+ / +a+b+c / Table 5. Lmtg expressos for the Bayes estmators as usg the UP, the JP ad the IP uder DLF t Parameters θ + k= x k θ + k= x k θ + k= x k p + +4 p + +4 Bayes Estmators UP JP IP + k= x k + k= x k + k= x k a + k= x k+b +a + k= x k+b +a + k= x k+b +a+ +a+b+c+ +b+ +a+b+c+ Table 6. Lmtg expressos for the posteror rsks as t usg the UP, the JP ad the IP uder DLF Posteror Rsks Parameters UP JP IP θ + θ + θ + p p a + +a + +a + + +b+c +a++a+b+c + +a+c +b++a+b+c dstrbutos for each choce of the vector of the parameters θ, θ, θ, p, p = {4,,, 0.5, 0.,,,, 0.4, 0.4,,, 4, 0., 0.5}. The smulato s repeated 000 tmes ad the results are the averaged. Sample of szes p, p ad p p are chose radomly from frst compoet desty f x; θ, secod compoet desty f x; θ ad thrd compoet desty f x; θ, respectvely. To check the mpact of test termato tme o Bayes estmates, we

15 estmate the parameters of the -compoet mxture of Expoetal dstrbutos based o a sample cesored at fxed test termato tmes t = 0.5, 0.8. The observatos whch are greater tha test termato tme t are take as cesored. Oly falures ca be cosdered as members of subpopulato-, subpopulato- or subpopulato- of the -compoet mxture of Expoetal dstrbutos. For the sake of brevty, smulated results oly for = 0, 00, 00 ad θ, θ, θ, p, p = 4,,, 0.5, 0. are preseted the Tables 8-0 see appedx. The smulated results for θ, θ, θ, p, p = {,,, 0.4, 0.4,,, 4, 0., 0.5} are avalable wth the frst author ad ca be obtaed o demad. From Tables 8-0 see appedx, t ca be see that dffereces of Bayes estmates of compoet ad proporto parameters from assumed parameters reduce wth a crease sample sze at dfferet test termato tmes ad same s the case wth large test termato tme as compared to small test termato tme for dfferet sample szes.also, f θ > θ > θ ad p > p, frst ad secod compoet parameters ad secod proporto parameter usg the IP uder SELF, PLF ad DLF are uder-estmated but thrd compoet ad frst proporto parameters are over-estmated at dfferet sample szes ad test termato tmes wth a few exceptos.by usg the IP uder SELF, PLF ad DLF, three compoet parameters ad secod proporto parameter are uder-estmated, however,frst proporto parameter s over-estmated wth a few exceptos case of θ = θ = θ ad p = p. Also, f θ < θ < θ ad p < p, thrd compoet ad secod proporto parameters usg the IP uder SELF, PLF ad DLF are uder-estmated but there s a mxed patter over-estmato or uder-estmato for frst ad secod compoet ad frst proporto parameters usg the IP. Smlarly, the compoet parameters usg the UP ad the JP uder SELF, PLF ad DLF are over-estmated but there s a mx patter uder-estmato or over-estmato for proporto parameters usg the UP ad the JP uder SELF, PLF ad DLF at dfferet sample szes ad test termato tmes. It s, also, clear from the Tables 8-0 that for a fxed test termato tme, the posteror rsks of the Bayes estmates, usg the UP, the JP ad the IP uder SELF, PLF ad DLF, reduce wth a crease sample sze. O the other had, for all prors, loss fuctos ad sample szes cosdered ths study, posteror rsks decrease wth a crease test termato tme.the posteror rsks usg the IP are smaller tha the posteror rsks usg the UP ad the JP for dfferet sample szes ad test termato tmes.also, the posteror rsks usg the JP are smaller tha that usg the UP for dfferet sample szes ad test termato tmes. It s also observed that estmatg the compoet parameters θ, θ ad θ, posteror rsks are smaller uder DLF tha uder SELF ad PLF at dfferet sample szes ad test termato tmes cosdered ths study. However, for estmatg the mxg proportos, SELF yelds smaller posteror rsks tha SELF ad DLF, at dfferet sample szes ad test termato tmes. Thus, DLF s more sutable for estmatg compoet parameters ad SELF s a preferable choce for estmatg proporto parameters p ad p. 5

16 6 0. Real data example Davs 95 reported a mxture data o lfetmes thousad hours of may compoets used arcraft sets. To llustrate the proposed methodology, we take the data o three compoets, amely, Trasmtter Tube, Combato of Trasformers ad Combato of Relays. It s ukow that whch compoet Tubes, Trasformers ad Relays fals utl a falure of a radar set occurs at or before the test termato tme t = 0.4. The total umber of tests are coducted 70 tmes.for test termato tme t = 0.4, the data are summarzed as below. r = 70, r = 0, r = 48, r = 8, r = 69, r = 6, k= x k = 6.875, r k= x k =.90, r k= x k = 9.5. Sce r = 6, we have almost 9 percet cesored sample. Thus, ths s a type-i rght cesored data. Bayes estmates ad ther posteror rsks usg the UP, the JP ad the IP uder SELF, PLF ad DLF are showcased Table 7 gve below. Table 7. Bayes estmates BEs ad posteror rsks PRs usg the UP, the JP ad the IP uder SELF, PLF ad DLF wth Davs 95 mxture data Pror Loss Fucto ˆθ ˆθ ˆθ ˆp ˆp UP SELF BE PR PLF BE PR DLF BE PR JP SELF BE PR PLF BE PR DLF BE PR IP SELF BE PR PLF BE PR DLF BE PR From the Table 0, t s otced that results obtaed through real data are compatble wth smulato results, however, there are some exceptos whch ca be attrbuted to usg large data set. The Table 0also reveals that the performace of the IP s best. I addto, results are relatvely more precse uder the JP tha the UP.It s also observed that DLF SELF performace better tha PLF ad SELF PLF ad DLF for estmatg compoet proporto parameters.

17 . Cocluso The mportace ad applcato of the -compoet mxture models real lfe problems s udeable.a extesve smulato study s performed to compare ad hghlght some mportat ad terestg propertes of the Bayes estmates of a -compoet mxture of Expoetal dstrbutos usg the UP, the JP ad the IP uder SELF, PLF ad DLF. The smulato results revealed that a crease sample sze ad/or test termato tme produced mproved terms of closeessad relable terms of posteror rsk Bayes estmates. It s cocluded that wth a crease sample sze ad/or test termato tme, the posteror rsks decrease.to estmate compoet as well as proporto parameters, prors ca be ordered wth respect to ther performace as: IP < JP < UP. The orderg of loss fuctos depeds upo the parameters beg estmated. Specfcally, for estmatg compoet parameters, orderg of loss fuctos s: DLF < PLF < SELF, whle t chages to SELF < PLF < DLF whe proporto parameters are beg estmated. The results obtaed through real data cocde wth the smulated results. Fally,t ca be cocluded that for a Bayesa aalyss of mxture data, the IP pared wth SELF ad the IP pared wth DLF are preferable choces for estmatg proporto ad compoet parameters, respectvely. 7 Refereces [] Abu-Taleb, A. A., Smad, M. M. ad Alaweh, A. J. Bayes estmato of the lfetme parameters for the Expoetal dstrbuto, Joural of Mathematcs ad Statstcs, 06-08, 007. [] Al-Awadh, S. A. ad Gartwate, P. H. A elctato method for multvarate ormal dstrbutos, Commucato Statstcs, Part A-Theory ad Methods 7, -4, 998. [] Al, M. M., Woo, J. ad Nadaraah, S. Bayes estmators of the Expoetal dstrbuto, Joural of Statstcs ad Maagemet Systems 8, 5-58, 005. [4] Aslam, M. A Applcato of Pror Predctve Dstrbuto to Elct the Pror Desty, Joural of Statstcal Theory ad Applcatos, 70-8, 00. [5] Bayes, T. A essay towards solvg a problem the doctre of chaces, Phlosophcal Trasactos of the Royal Socety of Lodo 5, 70-48, 76. [6] Barger, K. J. Mxtures of Expoetal dstrbutos to descrbe the dstrbuto of Posso meas estmatg the umber of uobserved classes, M. Sc. Thess, Corell Uversty, 006. [7] Berger, J. O. Statstcal decso theory ad Bayesa aalyssnew York: Sprger-Verlag, 985. [8] Berardo, J. M. Referece posteror dstrbutos for Bayesa ferece, Joural of the Royal Statstcal Socety, Seres B Methodologcal, 4, -47, 979. [9] Brch, R. ad Bartolucc, A. A. Determato of the hyperparameters of a pror probablty model survval aalyss, Computer Programs Bomedce 7, 89-94, 98. [0] Chaloer, K. M. ad Duca, G. T. Assessmet of beta pror dstrbuto: PM elctato, The Statstca, 74-80, 98. [] Davs, D. J. A aalyss of some falure data, Joural of the Amerca Statstcal Assocato 47, -50, 95. [] DeGroot, M. H. Optmal statstcal decsomcgraw-hll, 005. [] de Laplace, P. S. Theore aalytque des probabltes Pars: Gauther-Vllars, 80. [4] Evertt, B. ad Had, D. J. Fte mxture dstrbuto New York: Chapma ad Hall, 98. [5] Gavasakar, U. A comparso of two elctato methods for a pror dstrbuto for a bomal parameter, Maagemet Scece 4, , 988.

18 8 [6] Gesser, S. O pror dstrbutos for bary trals, The Amerca Statstca 84, 44-47, 984. [7] Gbels, I. Cesored data, Wley Iterdscplary Revews: Computatoal Statstcs, 78-88, 00. [8] Hah, E. D. Re-examg formatve pror elctato through the les of Markovcha Mote Carlo methods, Joural of the Royal Statstcal Socety 69, Seres A, 7-48, 006. [9] Hebert, J. L. ad Scarao, S. M. Comparg locato estmators for Expoetal mxtures uder Ptma s measure of closeess, Commucatos Statstcs- Theory ad Methods, 9-46, 005. [0] Jeffreys, H. A varat form for the pror probablty estmato problems, Proceedg of the Royal Socety of Lodo, Seres A, Mathematcal ad Physcal Sceces 86007, 45-46, 946. [] Jeffreys, H. Theory of ProbabltyOxford, UK: Claredo Press, 96. [] Kadae, J. B., Dckey, J. M., Wkler, R. L., Smth, W. ad Peter, S. C. Iteractve elctato of opo for a ormal lear model, Joural of the Amerca Statstcal Assocato 75, , 980. [] Kalbflesch, J. D. ad Pretce, R. L. The Statstcal aalyss of falure tme datanew York: Joh Wley & Sos, 0. [4] Kazm, S. M. A., Aslam, M. ad Al, S. O the Bayesa estmato for two-compoet mxture of Maxwell dstrbuto assumg type-i cesored data, Iteratoal Joural of Appled Scece ad Techology, 97-8, 0. [5] Legedre, A. M. Nouvelles methodes pour la determato des orbtes des cometes: Appedce sur la Methode des Modres Carres Pars: Gauther-Vllars, 806 [6] L, L. A. Decomposto theorems, codtoal probablty, ad fte mxtures dstrbutos, Thess, State Uversty, New York, Albay, 98. [7] L, L. A. ad Sedrask, N. Iferece about the presece of a mxture, Techcal Report, State Uversty, New York, Albay, 98. [8] L, L. A. ad Sedrask, N. Mxtures of dstrbutos: A topologcal approach, The aals of Statstcs 64, 6-64, 988. [9] McCullagh, P. Expoetal mxtures ad quadratc Expoetal famles, Bometrka 84, 7-79, 994. [0] Medehall, W. ad Hader, R. J. Estmato of parameters of mxed expoetally dstrbuted falure tme dstrbutos from cesored lfe test data, Bometrka 45-4, , 958. [] Norstrom, J. G. The use of precautoary loss fucto rsk aalyss, Relablty, IEEE Trasactos o 45, , 996. [] Raqab, M. M. ad Ahsaullah, M. Estmato of the locato ad scale parameters of geeralzed Expoetal dstrbuto based o order statstc, Joural of Statstcal Computato ad Smulato 69, 09-, 00. [] Romeu, L. J. Cesored data, Strategc Arms Reducto Treaty, -8, 004. [4] Saleem. M. Bayesa aalyss of mxture dstrbutos. Ph. D. Dssertato, Dept. of Statstcs, Quad--Azam Uversty, Islamabad, Paksta, 00. [5] Saleem, M. ad Aslam, M. O pror selecto for the mxture of Ralegh dstrbuto usg predctve tervals, Paksta Joural of Statstcs 4, -5, 008. [6] Saleem, M. ad Aslam, M. Bayesa aalyss of the two-compoet mxture of the Raylegh dstrbuto assumg the uform ad the Jeffreys prors, Joural of Appled Statstcal Scece 64, 49-50, 009. [7] Saleem, M., Aslam, M. ad Ecoomus, P. O the Bayesa aalyss of the mxture of Power dstrbuto usg the complete ad cesored sample, Joural of Appled Statstcs 7, 5-40, 00. [8] Sha, S. K. Bayesa estmato New Delh: New Age Iteratoal p lmted, Publsher, 998. [9] Stírelec, L. ad Stehlík, M. O smulato of exact tests Raylegh ad Normal famles, AIP Amerca Isttute of Physcs Coferece Proceedgs 479, 0.

19 9 Appedx Table 8. Bayes Estmates BEs ad Posteror Rsks PRs of - Compoet Mxture of a Expoetal Dstrbuto usg the UP uder SELF, PLF ad DLF wth θ = 4, θ =, θ =, p = 0.5, p = 0. ad t = 0.5, 0.8 t Loss Fuctos ˆθ ˆθ ˆθ ˆp ˆp SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR

20 0 Table 9. Bayes Estmates BEs ad Posteror Rsks PRs of -Compoet Mxture of a Expoetal Dstrbuto usg the JP uder SELF, PLF ad DLF wth θ = 4, θ =, θ =, p = 0.5, p = 0. ad t = 0.5, 0.8 t Loss Fuctos ˆθ ˆθ ˆθ ˆp ˆp SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR

21 Table 0. Bayes Estmates BEs ad Posteror Rsks PRs of -Compoet Mxture of a Expoetal Dstrbuto usg the IP uder SELF, PLF ad DLF wth θ = 4, θ =, θ =, p = 0.5, p = 0. ad t = 0.5, 0.8 t Loss Fuctos ˆθ ˆθ ˆθ ˆp ˆp SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR SELF BE PR PLF BE PR DLF BE PR