On Posterior Analysis of Mixture of Two Components of Gumbel Type II Distribution

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1 Iteratioal Joural of Probability ad Statistics 0, (4: 9-3 DOI: 0.593/j.ijps O Posterior Aalysis of Mixture of Two Compoets of Gumbel Type II Distributio Navid Feroze,*, Muhammad Aslam Departmet of Mathematics ad Statistics, Allama Iqbal Ope Uiversity, Islamabad, Paista Departmet of Statistics, Quaid-i-Azam Uiversity, Islamabad, Paista Abstract This paper describes the Bayesia aalysis of the parameters of mixture of two compoets of Gumbel type II distributio. A heterogeeous populatio has bee modeled by meas of two compoets mixture of the Gumbel type II distributio uder type I cesored data. The Bayes estimators of the said parameters have bee derived uder the assumptio of o-iformative priors o the basis of differet loss fuctios. A cesored mixture data is simulated by probabilistic mixig for the computatioal purpose. The comparisos amog the estimators have bee made i terms of correspodig posterior riss. The posterior predictive distributios ad itervals have bee derived ad evaluated uder each prior. Keywords Bayes Estimators, Posterior Riss, Mixture Models, Loss Fuctios. Itroductio The Gumbel type II distributio is used to model the extreme evets lie extreme earthquae, ra ifalls, temperature, floods etc. It has aother side of applicatios which deals with life testig experimets. Chechile[] obtaied the posterior distributio assumig that the radom sample is tae from the Gumbel distributio usig the cojugate prior. Corsii et al.[] discussed the maximum lielihood (ML algorithms ad Cramer-Rao (CR bouds for the locatio ad scale parameters of the Gumbel distributio. Mousa[3] obtaied the Bayesia estimatio for the two parameters of the Gumbel distributio based o record values. Koutsoyiais ad Baloutsos[4] described that the Gumbel distributio has bee the prevailig model for quatifyig ris associated with extreme raifall. Rasmusse ad Gautam[5] exteded the probability weighted momets (PWM to what is called the geeralized method of probability weighted momets (GPWM because there is o reaso why the PWMs provide the most efficiet estimators of Gumbel parameters ad quatile especially i hydrology. Maliowsa ad Szyal[6] obtaied the Bayes estimators for the two parameters of a Gumbel distributio based o th lower record values. Nadarajah ad Kotz[7] itroduced the beta Gumbel (BG distributio ad provided closed-form expressios for the momets, the asymptotic distributio of the extreme order statistics ad discussed the maximum lielihood estimatio procedure. Miladiovic ad * Correspodig author: avidferoz@hotmail.com (Navid Feroze Published olie at Copyright 0 Scietific & Academic Publishig. All Rights Reserved Tsoos[8] modified the classical Gumbel probability distributio i order to study the failure times of a give system. Par et al.[9] gave a ovel equatio for the scale parameter of the Gumbel distributio. Heo ad Salas[0] examied the log-gumbel distributio regardig quatile estimatio ad cofidece itervals of quatiles. Thompso et al.[] itroduced a distributioal hypothesis test for left cesored Gumbel observatios based o the probability plot correlatio coefficiet (PPCC. The mixture models have received great attetio of the aalysts i the recet era. These models iclude fiite ad ifiite umber of compoets that ca aalyze differet datasets. A fiite mixture of probability distributio is suitable to study a populatio categorized i umber of subpopulatios. A populatio of lifetimes of certai electrical elemets ca be classified ito umber of subpopulatios based o causes of failures. The aalysis of mixture models uder Bayesia framewor has developed a sigificat iterest amog the statisticias. The authors dealig with Bayesia aalysis of mixture models iclude Saleem ad Aslam[], Saleem et al.[3], Majeed ad Aslam[4] ad Kazmi et al.[5]. These cotributios to the mixture models are the great motivatios for the reset study. We cosidered two compoet mixture of Gumbel type II distributio. The populatio of certai items is assumed to be partitioed ito two subpopulatios. The radomly selected observatios from the said populatio are cosidered to be a part of oe of the above metioed subpopulatios. These subpopulatios are assumed to follow the Gumbel type II distributio. Therefore, the two compoets mixture of Gumbel type II distributios has bee proposed to model this populatio. The observatios have bee assumed to be right cesored. The iverse trasformatio techique of

2 0 Navid Feroze et al.: O Posterior Aalysis of Mixture of Two Compoets of Gumbel Type II Distributio simulatio uder a probabilistic mixig has bee used to geerate data ad to evaluate the performace of differet estimators. ad r A = x t α j j=. The Model ad Lielihood Fuctio A desity fuctio for mixture of two compoets desities with mixig weights (p,q is: f x = pf x qf x ( 0< p <, q = p The followig Gumbel type II distributio is cosidered for both mixture desities: ( αi βix α i i f ( xi; αi, βi αβ i ix = i e ( ; x > 0, α > 0, β > 0 i i i i =, With the cumulative distributio fuctio as: ai ixi F( xi; αi, βi e β = (3 The cumulative distributio fuctio for the mixture model is: F x = pf x qf x (4 Suppose items are put o a life testig experimet ad r uits failed util time T while, r uits are still worig. Now based o causes of failure, the failed items are assumed to come either from subpopulatio or from subpopulatio. Therefore it ca be observed that r ad r failed items come from first ad secod subpopulatio respectively. Where r = r r. The remaiig r items are assumed to be cesored observatios. The lielihood fuctio for above cesored data ca be obtaied as: { } ( β, β, = ( j L p x pf x r j= r { pf( x j } F ( t j= r r r r ( β, β, ββ L px p q e β β pe qe α α t βt r r xj β x j j= j= (5 (6 Expadig the last term by biomial expasio the lielihood fuctio becomes: r L px p q βa βa e e ( β, β, ββ Where r j j= r r r r A = x t α (7 3. The Posterior Aalysis uder the Assumptio of Uiform Prior Oe of the most widely used o-iformative priors, proposed by Laplace[6], is a uiform prior. It has bee applied to may problems, ad ofte the results are etirely satisfactory. This prior has bee used for the posterior estimatio. Let β Uiform β ( 0, Uiform β ( 0, p U 0,, β ad. Assumig idepedece, these priors result ito a joit prior that is proportioal to a costat. That joit prior has bee used to derive the joit posterior distributio of β, β ad p. The margial distributio for each parameter ca be obtaied by itegratig the joit posterior distributio with respect to uisace parameters. The joit posterior distributio is: r r r r r p( β, β, px = ββ p q C (8 βa β A e e Where ( ( ( θ, θ r r r Γ r Γ r C = B, A A θ = r ad θ = r Usig the posterior distributio, discussed i (8, the Bayes estimators ad posterior riss uder differet loss fuctios have bee derived ad preseted i the followig. Bayes estimators ad associated riss uder uiform prior usig squared error loss fuctio (SELF are: β r ( r ( r, SELF B (, C r r θ θ Γ Γ = ( A ( A r Γ ( r 3 Γ ( r ρ( β, SELF = ( θ, θ C 3 r r ( A ( A r Γ ( r Γ ( r ( θ, θ C r r = 0 ( A ( A β r ( r ( r, SELF B(, C r r θ θ Γ Γ = ( A ( A r Γ ( r Γ ( r 3 ρ( β, SELF = ( θ, θ C r r 3 ( A ( A r Γ ( r Γ ( r ( θ, θ C r r ( A ( A

3 Iteratioal Joural of Probability ad Statistics 0, (4: 9-3 r Γ ( r Γ ( r = ( θ, θ C ( A ( A r Γ ( r Γ ( r ( pself = (, C r r = ( A ( A pself r r ρ θ θ 0 r Γ r Γ r C A A ( ( ( θ, θ r r Bayes estimators ad ris uder uiform prior usig quadratic loss fuctio (QLF are: β ( ( θ, θ r r ( ( ( θ, θ r r r Γ r Γ r A A, QLF = r Γ r Γ r A A (, QLF ρ β β r Γ r Γ r C A A = r Γ r Γ r A A ( B ( θ, θ r r ( ( ( θ, θ r r ( ( θ, θ r r ( ( ( θ, θ r r r Γ r Γ r A A, QLF = r Γ r Γ r A A (, QLF ρ β r Γ r Γ r C A A = r Γ r Γ r A A ( B( θ, θ r r ( ( ( θ, θ r r ( ( ( θ, θ r r ( ( ( θ, θ r r r Γ r Γ r A A pqlf = r Γ r Γ r A A r Γ ( r Γ ( r B ( θ, θ C r r ( A ( A ρ ( pqlf = r Γ ( r Γ ( r ( θ, θ r r ( A ( A Bayes estimators ad ris uder uiform prior usig weighted loss fuctio (WLF are: r Γ( r Γ ( r, WLF = (, C r r ( A ( A β θ θ = (, ( ( r Γ r Γ r, WLF C r r A A ρ β θ θ r Γ( r Γ ( r ( θ, θ C r r ( A ( A β r = θ θ Γ r Γ r ( (,, WLF C r r A A = (, ( ( r Γ r Γ r, WLF C r r A A ρ β θ θ r Γ r Γ r C A A ( ( θ, θ r r = 0 r Γ r Γ r pwlf = C r r A A ( ( ( θ, θ r Γ ( r Γ ( r ( pwlf = (, C r r = ( A ( A ρ θ θ 0 r Γ r Γ r C A A ( ( ( θ, θ r r Bayes estimators ad ris uder uiform prior usig precautioary loss fuctio (PLF are: β r r 3 r, PLF ( B (, C r 3 r θ θ Γ Γ = 0 ( A ( A r Γ r 3 Γ r ρ( β, PLF = ( B ( θ, θ C r 3 r 0 ( A ( A r Γ ( r Γ ( r ( B ( θ, θ r r = C 0 A A r Γ r Γ r 3, PLF = (, C r r 3 ( A ( A β θ θ r Γ r Γ r 3 ρ( β, PLF = ( θ, θ C 3 r r 0 ( A ( A r Γ ( r Γ ( r ( θ, θ C r r ( A ( A r Γ r Γ r pplf = C r r A A ( ( ( θ, θ

4 Navid Feroze et al.: O Posterior Aalysis of Mixture of Two Compoets of Gumbel Type II Distributio ρ θ θ 0 r Γ r Γ r C A A r Γ ( r Γ ( r ( pplf = (, C r r = ( A ( A ( ( ( θ, θ r r 4. The Posterior Aalysis uder the Assumptio of Prior Aother o-iformative prior has bee suggested by [7] which is frequetly used i situatios where oe does ot have much iformatio about the parameters. This is defied as the distributio of the parameters proportioal to the square root of the determiats of the Fisher iformatio matrix i.e. { } p( β I( β Where β ( ββ, I ( β is ( Fisher iformatio matrix, defied as; I = is the vector of parameters ad ( β ( x β log f i i i = Ε βi i =, Where fi( xi β i have bee defied i ( ad U( 0,. Assumig idepedece, the joit prior is p obtaied as: h( β, β, p. (9 ββ The joit posterior distributio usig the above prior is: r r r r r p( β, β, px = β β p q C (0 βa βa e e r Γ( r Γ( r Where C = ( θ, θ r r A A The Bayes estimators ad associated posterior riss have bee derived uder differet loss fuctios usig the posterior distributio (0. Bayes estimators ad associated riss uder prior usig squared error loss fuctio (SELF are: β r ( r ( r, SELF B (, C r r θ θ Γ Γ = ( A ( A r Γ ( r Γ( r ρ( β, SELF = ( θ, θ C r r ( A ( A r Γ ( r Γ( r ( θ, θ C r r ( A ( A β θ θ ( (, ( B, r Γ r Γ r, SELF = C r r A A r Γ r Γ r (, SELF = ( C r r A A ρ β θ θ p r Γ( r Γ ( r ( θ, θ C r r ( A ( A r Γ( r Γ( r = ( θ, θ C ( A ( A r Γ( r Γ( r ( pself = (, C r r = ( A ( A SELF r r ρ θ θ 0 r Γ( r Γ( r ( θ, θ C r r ( A ( A Bayes estimators ad ris uder prior usig quadratic loss fuctio (QLF are: r Γ( r Γ( r ( θ, θ r r ( A ( A β, QLF = r Γ( r Γ( r ( θ, θ r r ( A ( A r Γ( r Γ( r ( θ, θ C r r ( A ( A ρ( β, QLF = r Γ( r Γ( r ( θ, θ r r ( A ( A r Γ( r Γ( r ( θ, θ r r ( A ( A β, QLF = r Γ( r Γ( r ( θ, θ r r ( A ( A (, QLF ρ β pqlf = ρ ( pqlf r Γ( r Γ( r B ( θ, θ C r r ( A ( A = r Γ( r Γ( r ( θ, θ r r ( A ( A r Γ( r Γ( r ( θ, θ r r ( A ( A r Γ( r Γ( r ( θ, θ r r ( A ( A r Γ r Γ r C A A = r Γ r Γ r A A B( θ, θ r r ( θ, θ r r

5 Iteratioal Joural of Probability ad Statistics 0, (4: Bayes estimators ad ris uder prior usig weighted loss fuctio (WLF are: β r ( r ( r, WLF B(, C r r θ θ Γ Γ = ( A ( A = (, ( r Γ r Γ r ρ β θ θ, WLF C r r A A r Γ r Γ r C A A ( ( θ, θ r r = 0 r Γ( r Γ( r, WLF = (, C r r ( A ( A β θ θ = (, r Γ r Γ r, WLF C r r A A ρ β θ θ r Γ( r Γ( r ( θ, θ C r r ( A ( A ( θ, θ r Γ r Γ r WLF = C r r A A p ( p = (, = r Γ r Γ r ρ θ θ WLF C r r 0 A A ( θ, θ C ( A A r Γ r Γ r r r Bayes estimators ad ris uder prior usig precautioary loss fuctio (PLF are: β r r r, PLF B (, C r r θ θ Γ Γ = ( A ( A ρ( β r = ( θ, θ Γ r Γ r r Γ r Γ r C A A (, PLF C r r A A ( ( θ, θ r r 0 β θ θ (, r Γ r Γ r, PLF = C r r A A ρ β θ θ p = (, ( θ, θ r Γ r Γ r, PLF C r r A A r Γ r Γ r B r r C A A ( θ, θ r Γ r Γ r PLF = C r r A A r Γ( r Γ r ρ( p = ( θ, θ = ( θ, θ PLF C r r 0 A A r Γ r Γ r r r C A A 6. Posterior Predictive Distributios ad Itervals The posterior predictive distributios uder uiform ad priors are: ( α r B θ, θ Γ r Γ r αy p( yx = C r 0 α = r ( A y ( A ( α B θ, θ Γ r Γ r αy r α r ( A ( A y ( α r B θ, θ Γ r Γ r αy p( yx = C r 0 α = r ( A y ( A ( α B θ, θ Γ r Γ r αy r r α ( A ( A y The posterior predictive itervals uder uiform prior ca be obtaied by solvig the followig two equatios respectively. r = C B( θ, θ Γ ( r Γ ( r r r r ( A ( A α ( A L B( θ, θ Γ ( r Γ ( r r r r ( A ( A α ( A L

6 4 Navid Feroze et al.: O Posterior Aalysis of Mixture of Two Compoets of Gumbel Type II Distributio r = C B( θ, θ Γ ( r Γ ( r r r r ( A ( A α ( A U B( θ, θ Γ ( r Γ ( r r r r ( A ( A α ( A U Where is level of sigificace The posterior predictive itervals uder uiform prior ca be obtaied by solvig the followig two equatios respectively. r = C B( θ, θ Γ( r Γ( r r r r ( A ( A α ( A L B( θ, θ Γ ( r Γ ( r r r r ( A ( A α ( A L r = C B( θ, θ Γ( r Γ( r r r r ( A ( A α ( A U B( θ, θ Γ ( r Γ ( r r r r ( A ( A α ( A U 7. Simulatio Study A simulatio study has bee coducted to assess ad compare the performace of Bayes estimators ad to aalyse the impact of sample size, mixig weight ad magitude of parametric values o the Bayes estimators. Samples of sizes = 00, 00,, ad have bee geerated by iverse trasformatio method from two compoets mixture of Gumbel type II distributio. The parametric { } values used are: (, ( 3,6,( 6,9,( 0, p ( 0.30, 0.45 β β ad. The probabilistic mixig has bee used to geerate the mixture data. For each observatio a radom umber u has bee geerated from U ( 0,. If u < p the observatio has bee radomly tae from first subpopulatio ad if u > pthe the observatio have bee tae from the secod subpopulatio. The observatios above a fixed cesorig time T have bee assumed to be right cesored. Uder each combiatio of parametric values, the choice of cesorig time has bee made so that the cesorig rate i the respective sample has bee 0%. As oe sample caot completely describe the behaviour ad properties of the Bayes estimators, the results have bee replicated 000 times ad the average of results has bee preseted i the tables below. Ta ble. Bayes estimates ad posterior riss uder uiform prior usig β = 3, β = 6 ad p = β = β = p =

7 Iteratioal Joural of Probability ad Statistics 0, (4: Ta ble. Bayes estimates ad posterior riss uder uiform prior usig β = 3, β = 6 ad p = 0.45 Ta ble 3. Bayes estimates ad posterior riss uder prior usig β = 3, β = 6 ad p = 0.30 β = 3 β = β = 6 β = p = 0.45 p =

8 6 Navid Feroze et al.: O Posterior Aalysis of Mixture of Two Compoets of Gumbel Type II Distributio Ta ble 4. Bayes estimates ad posterior riss uder prior usig β = 3, β = 6 ad p = 0.45 Ta ble 5. Bayes estimates ad posterior riss uder uiform prior usig β = 6, β = 9 ad p = 0.45 β = 3 β = β = 6 β = p = p =

9 Iteratioal Joural of Probability ad Statistics 0, (4: Ta ble 6. Bayes estimates ad posterior riss uder prior usig β = 6, β = 9 ad p = 0.45 Ta ble 7. Bayes estimates ad posterior riss uder uiform prior usig β = 6, β = 9 ad p = 0.30 β = 6 β = β = 9 β = p = 0.45 p =

10 8 Navid Feroze et al.: O Posterior Aalysis of Mixture of Two Compoets of Gumbel Type II Distributio Ta ble 8. Bayes estimates ad posterior riss uder prior usig β = 6, β = 9 ad p = 0.30 Ta ble 9. Bayes estimates ad posterior riss uder uiform prior usig β = 0, β = ad p = 0.45 β = 6 β = β = 9 β = p = 0.30 p =

11 Iteratioal Joural of Probability ad Statistics 0, (4: Ta ble 0. Bayes estimates ad posterior riss uder prior usig β = 0, β = ad p = 0.45 Table. Bayes estimates ad posterior riss uder uiform prior usig β = 0, β = ad p = 0.30 β = 0 β = β = β = p = 0.45 p =

12 30 Navid Feroze et al.: O Posterior Aalysis of Mixture of Two Compoets of Gumbel Type II Distributio Ta ble. Bayes estimates ad posterior riss uder prior usig β = 0, β = ad p = β = β = posterior riss decreases. However, the posterior riss seem to be quite large for relatively larger values of the parameters. It ca also be observed that all the parameters are over estimated with few exceptios. The exted of over estimatio is more itesive for larger values of the parameters. This idicates that the posterior distributios are positively sewed. I compariso of priors, the performace of estimates uder the prior seems better tha those uder uiform prior for each loss fuctio ad mixig weight. While, i case of loss fuctios, the estimates uder quadratic loss fuctio (QLF are associated with the miimum riss. It is also iterestig to ote that the amout of riss uder precautioary loss fuctio (PLF ad weighted loss fuctio (WLF are covergig to each other with icrease i sample size. The posterior riss uder QLF are approximately half of the riss uder WLF ad PLF. The icrease i the value of mixig proportio p imposes positive impact o the performace of estimators of β ad it egatively affects the performace of the estimators of β ad vice versa. This is because the icreasig value of p ted to icrease the value of r (the umber of observatios selected from the Gumbel type II distributio havig parameter β, wh ich will result i lesser posterior riss. The icreasig values of the mai parameters (β ad β are havig a egative effect o the behavior of the estimators of mixig proportio p. The expressios for complete samples ca simply be obtaied by icreasig the test termiatio time to uity. The riss associated with estimates uder complete samples are expected to reduce as o iformatio will be lost the. Table 3. 95% posterior predictive itervals for β = 3,β = 6, ad p = 0.30 Uiform Lower Limit Upper Limit Differece p = The simulatio study idicates that by icreasig the sample size the estimated values each parameter coverges to the true value ad the magitude of correspodig Lower Limit Upper Limit Differece

13 Iteratioal Joural of Probability ad Statistics 0, (4: From tables 3-6 it ca be assessed that posterior predictios ted to be more accurate i larger samples. Icreasig values of actual ad weight parameter imposes egative impact o the performace of the predictios. It is iterestig to ote that the posterior predictive itervals are shorter i case of prior. This simply idicates the supremacy of prior over uiform prior. Ayhow, the results uder the posterior predictive itervals are i accordace with the correspodig poit estimatio. Table 4. 95% posterior predictive itervals for β = 3, β = 6, ad p = 0.45 Uiform Lower Limit Upper Limit Differece Lower Limit Upper Limit Differece Table 6. 95% posterior predictive itervals for β = 3,β = 6, ad p = 0.30 Uiform Lower Limit Upper Limit Differece Lower Limit Upper Limit Differece Aalysis uder Real Life Data The real life data (followig Gumbel distributio regardig mothly wid speed i Camero Highlad from year preseted by Zaharimi et al.[8] is used to illustrate the applicability of the results obtaied i previous sectios. Ta ble 7. Bayes est imat es ad riss for real life dat a Table 5. 95% posterior predictive itervals for β = 6, β = 9, ad p = 0.30 Uiform Lower Limit Upper Limit Differece Lower Limit Upper Limit Differece Prior Uiform Uiform Uiform β β p =

14 3 Navid Feroze et al.: O Posterior Aalysis of Mixture of Two Compoets of Gumbel Type II Distributio Prior Uiform Uiform Uiform Ta ble 8. Bayes est imat es ad riss for real life dat a β β p = The real life data replicated the patters observed i the simulatio study. The miimum amout of riss has bee observed for the estimates uder the assumptio of prior usig quadratic loss fuctio. 9. Coclusios The purpose of the article is to fid out the appropriate combiatio of prior distributio ad loss fuctio to estimate the parameters of two-compoet mixture of Gumbel type II distributio. The parameters have bee estimated uder the assumptio of two o-iformative priors ad four loss fuctios (symmetric ad asymmetric. The posterior predictive itervals have also bee evaluated. From the fidigs of the study it ca be cocluded that i order to estimate the said parameters, the use of prior ad quadratic loss fuctio ca be preferred. REFERENCES [] A. R. Chechile, Bayesia aalysis of Gumbel distributed data, Commuicatios i Statistics - Theory ad Methods, vol.30, o.3, pp. -4, 00. [] G. Corsii, F. Gii, M. V. Gerco, et al. Cramer-Rao bouds ad estimatio of the parameters of the Gumbel distributio, IEEE, vol.3, o.3, pp. 0-04, 00. [3] A. Mousa, Bayesia estimatio, predictio ad characterizatio for the Gumbel model based o records, Statistics, vol.36, o., pp , 00. [4] D. Koutsoyiais, ad G. Baloutsos, Aalysis of a log record of aual maximum raifall i Athes, Greece, ad desig raifall ifereces, Natural Hazards, vol., o., pp. 9-48, 000. [5] Rasmusse, P. F., ad Gautam, N. Alterative PWM-estimators of the Gumbel distributio. Joural of Hydrology, vol.80, o.-4, pp. 65-7, 003. [6] I. Maliowa, ad D. Szyal, O characterizatio of certai distributios of th lower (upper record values, Applied Mathematics ad Computatio, vol.0, o., pp , 008. [7] S. Nadarajah, ad S. Kotz, The beta Gumbel distributio, Math. Probab. Eg., vol.0, 33-33, 004. [8] B. Miladiovic, ad P. C. Tsoos, Sesitivity of the Bayesia reliability estimates for the modified Gumbel failure model, Iteratioal Joural of Reliability, Quality ad Safety Egieerig (IJRQSE, vol.6(40, pp , 009. [9] Y. Par, S. Sheetli, ad L. J. Spouge, Estimatig the Gumbel scale parameter for local aligmet of radom sequeces by importace samplig with stoppig times, A. Statist., vol.37, o.6a, pp , 009. [0] H. J. Heo, ad D. J. Salas, Estimatio of quatiles ad cofidece itervals for the log-gumbel distributio, Stochastic Hydrology ad Hydraulics, vol.0, o.3, pp , 00. [] E. M. Thompso, J. B. Hewlett, ad R. M. Vogel, The Gumbel hypothesis test for left cesored observatios usig regioal earthquae records as a example, Nat. Hazards Earth Syst. Sci., vol., pp. 5-6, 0. [] M. Saleem, ad M. Aslam, "Bayesia Aalysis of the Two Compoet Mixture of the Rayleigh Dist. With the Uiform ad the Priors", J. of Applied Statistical Sciece, Vol.6, o.4, pp.05-3, 008. [3] M. Saleem, M. Aslam, ad P. Ecoomou, "O the Bayesia aalysis of the mixture of power fuctio distributio usig the complete ad the cesored sample", Joural of Applied Statistics, vol.37, o., pp. 5-40, 00. [4] M. Y. Majeed, ad M. Aslam, Bayesia aalysis of the two compoet mixture of iverted expoetial distributio uder quadratic loss fuctios, Iteratioal Joural of Physical Scieces, vol.7, o.9, pp , 0. [5] S. M. A. Kazmi, M. Aslam, ad S. Ali "O the Bayesia estimatio for two compoet mixture of maxwell distributio, Assumig Type I Cesored Data", Iteratioal Joural of Applied Sciece ad Techology (IJAST, vol., o., pp. 97-8, 0. [6] P. S. Laplace, Theorie aalytique des probabilities, Veuve Courcier Paris, 8. [7] H., Theory of Probability, 3rd ed. Oxford Uiversity Press, pp. 43, 96. [8] A. Zaharimi, S. Najid, ad A. Mahir, Aalyzig M alaysia wid speed data usig statistical distributio, Proceedigs of the 4th IASME / WSEAS Iter. Cof. o Eergy & Eviromet, Malysia, pp , 009.

Bayesian Estimation of the Parameters of Two- Component Mixture of Rayleigh Distribution under Doubly Censoring

Bayesian Estimation of the Parameters of Two- Component Mixture of Rayleigh Distribution under Doubly Censoring Joural of Moder Applied Statistical Methods Volume 3 Issue Article 4-04 Bayesia Estimatio of the Parameters of Two- Compoet Mixture of Rayleigh Distributio uder Doubly Cesorig Tahassum N. Sidhu Quaid-i-Azam