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 Uiversity, Islamabad, Paista Navid Feroze Allama Iqbal Ope Uiversity, Islamabad, Paista, avidferoz@gmail.com Muhammad Aslam Quaid-i-Azam Uiversity, Islamabad, Paista Follow this ad additioal wors at: http://digitalcommos.waye.edu/jmasm Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theory Commos Recommeded Citatio Sidhu, Tahassum N.; Feroze, Navid; ad Aslam, Muhammad (04) "Bayesia Estimatio of the Parameters of Two-Compoet Mixture of Rayleigh Distributio uder Doubly Cesorig," Joural of Moder Applied Statistical Methods: Vol. 3 : Iss., Article 4. DOI: 0.37/jmasm/4485 Available at: http://digitalcommos.waye.edu/jmasm/vol3/iss/4 This Regular Article is brought to you for free ad ope access by the Ope Access Jourals at DigitalCommos@WayeState. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of DigitalCommos@WayeState.
Joural of Moder Applied Statistical Methods November 04, Vol. 3, No., 59-86. Copyright 04 JMASM, Ic. ISSN 538 947 Bayesia Estimatio of the Parameters of Two-Compoet Mixture of Rayleigh Distributio uder Doubly Cesorig Tabassum Naz Sidhu Quaid-i-Azam Uiversity Islamabad, Paista Navid Feroze Allama Iqbal Ope Uiversity Islamabad, Paista Muhammad Aslam Quaid-i-Azam Uiversity Islamabad, Paista Recetly, the Bayesia aalysis of the two-compoet mixture of lifetime models uder sigly type I cesored samples was discussed. The Bayes estimatio of the parameters of mixture of two Rayleigh distributios (MTRD) is developed uder doubly cesorig. Differet iformative priors, uder ad -loss fuctio, have bee assumed for the posterior estimatio. The performace of differet estimators has bee compared i terms of posterior riss by aalyzig the simulated ad real life data sets. Keywords: Iverse trasformatio method, mixture model, doubly cesorig, loss fuctios, Bayes estimator Itroductio I survival aalysis, data are subject to cesorig. The most commo type of cesorig is right cesorig, i which the survival time is larger tha the observed right cesorig time. I some cases, however, data are subject to left, as well as, right, cesorig. Whe left cesorig occurs, the oly iformatio available to a aalyst is that the survival time is less tha or equal to the observed left cesorig time. A more complex cesorig scheme is foud whe both iitial ad fial times are iterval-cesored. This situatio is referred as double cesorig, or the data with both right ad left cesored observatios are ow as doubly cesored data. Aalysis of doubly cesored data for simple (sigle) distributio has bee studied by may authors. Feradez (000) ivestigated maximum lielihood predictio based o type II doubly cesored expoetial data. Feradez (006) has discussed Bayesia estimatio based o trimmed samples from Pareto populatios. Kha et al. (00) studied predictive iferece from a two-parameter Rayleigh life Mr. Feroze is i the Departmet of Statistics. Email at avidferoz@gmail.com. Mr. Sidhu ad Mr. Aslam are i the Departmet of Mathematics ad Statistics. 59
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE model give a doubly cesored sample. Kim ad Sog (00) have discussed Bayesia estimatio of the parameters of the geeralized expoetial distributio from doubly cesored samples. Kha et al. (0) studied sesitivity aalysis of predictive modelig for resposes from the three-parameter Weibull model with a follow-up doubly cesored sample of cacer patiets. Pa et al. (03) has proposed the estimatio of Rayleigh scale parameter uder doubly type-ii cesorig from imprecise data. I statistics, a mixture distributio is sigified as a covex fusio of other probability distributios. It ca be used to model a statistical populatio with subpopulatios, where costituet of mixture probability desities are the desities of the subpopulatios. Mixture distributio may appropriately be used for certai data set where the subsets of the whole data set possess differet properties that ca best be modeled separately. They ca be more mathematically maageable, because the idividual mixture compoets are dealt with more ease tha the overall mixture desity. The families of mixture distributios have a wider rage of applicatios i differet fields such as fisheries, agriculture, botay, ecoomics, medicie, psychology, electrophoresis, fiace, commuicatio theory, geology ad zoology. Solima (006) derived estimators for the fiite mixture of Rayleigh model based o progressively cesored data. Sulta, et al. (007) described the properties ad estimatio of mixture of two iverse Weibull distributios. Sulta, et al. (007) have discussed some properties of the mixture of two iverse Weibull distributios. Saleem ad Aslam (008) preseted a compariso of the Maximum Lielihood (ML) estimates with the Bayes estimates assumig the Uiform ad the Jeffreys priors for the parameters of the Rayleigh mixture. Kudu ad Howalder (00) cosidered the Bayesia iferece ad predictio of the iverse Weibull distributio for type-ii cesored data. Saleem et al. (00) cosidered the Bayesia aalysis of the mixture of Power fuctio distributio usig the complete ad the cesored sample. Shi ad Ya (00) studied the case of the two parameter expoetial distributio uder type I cesorig to get empirical Bayes estimates. Eluebaly ad Bouguila (0) have preseted a Bayesia approach to aalyze fiite geeralized Gaussia mixture models which icorporate several stadard mixtures, widely used i sigal ad image processig applicatios, such as Laplace ad Gaussia. Sulta ad Al-Moisheer (0) developed approximate Bayes estimatio of the parameters ad reliability fuctio of mixture of two iverse Weibull distributios uder Type- cesorig. 60
SINDHU ET AL The Proposed Mixture Model ad the Lielihood Fuctio The probability desity fuctio (pdf) of the Rayleigh distributio with rate parameter i is f x x x x i j () i ij iji exp ij i, 0 ij, i 0,,, ad,,..., i The cumulative distributio fuctio (CDF) of the distributio is F x x x i j () i ij exp i ij, 0 ij, i 0,,, ad,,..., i A desity fuctio for mixture of two compoets desities with mixig weights (p, - p ) is f x p f x p f x, 0 p. (3) The cumulative distributio fuctio for the mixture model is: F x p F x p F x (4) Cosider a radom sample of size from Rayleigh distributio, ad let x, x,..., x r r s be the ordered observatios that ca oly be observed. The remaiig r smallest observatios ad the s largest observatios have bee assumed to be cesored. Now based o causes of failure, the failed items are assumed to come either from subpopulatio or from subpopulatio ; so the x r,..., x s ad x r,..., x s failed items come from first ad secod subpopulatios respectively. The rest of the observatios which are less tha xr ad greater tha xs have bee assumed to be cesored from each compoet. Where max,,, ad xr mi x, r, x, r x x x s s s. Therefore, m s r ad m s r umber of failed items ca be observed from first ad secod subpopulatios respectively. The remaiig ( s r ) items are assumed to be r r r, cesored observatios, ad sr are the ucesored items. Where 6
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE s ss ad m m m. The the lielihood fuctio for the Type II doubly cesored sample x x r,..., x s, xr,..., xs of the left cesored items are idetified, ca be writte as:, assumig the causes of the failure s s x r r r r L,, p p p F x, F x, L ( ) ( ) s s s F xs,, f x( i), f x( i), r r s,, p x 0 0 0 3 ir ir p r r s 3 ( p ) s s s 3 3 x j exp x j m m exp (5) (6) where W x j ( ) = x i s å ( ) + ( - s - 3 )x ( s) + x ( r ), W x j i=r m = s - r +, ad m = s - r + ( ) = x i s å ( ) + 3 x ( s) + x ( r ), i=r Bayes Estimatio ad p i, are For the Bayesia estimatio, let us assume that the parameters i idepedet radom variables, ad the we cosider the followig priors for differet parameters: Bayesia Estimatio usig Naagami Prior The prior for the rate parameters i for i =,, is assumed to be the Naagami distributio, with the hyper-parameters a i ad b i, give by 6
SINDHU ET AL f i ai a ai i i a i i i exp, ai, bi 0 ai ai bi bi (7) by The prior for p is assumed to be the beta distributio, whose desity is give c d c d f p p p, c, d 0 p d c (8) From equatio (7)-(8), propose the followig joit prior desity of the vector,,p a ai i i c d i i g i exp p p, bi 0 p, a 0, b 0, c 0, d 0 (9) By multiplyig Equatio (9) with Equatio (6), the joit posterior desity for the vector give the data becomes r r s s3 s c p 0 0 3 0 i r r s x 3 a ( p ) exp s 3 d ( aimi) i i i ij bi x (0) Margial distributios of i uisace parameters. Bayesia Estimatio usig Chi Prior ad p i, ca be obtaied by itegratig the The prior for the rate parameters i for i=,, is assumed to be the chi distributio, with the hyperparameter e i, give by 63
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE by ei ei i i i ei e i f exp, 0 () i The prior for p is assumed to be the beta distributio, whose desity is give c d c d f p p p, c, d 0 p d c () From equatio ()-(), we propose the followig joit prior desity of the,,p vector ei i c d g i exp p p, 0 p, ei 0, c3 0, d3 0 (3) By multiplyig Equatio (3) with Equatio (6), the joit posterior desity for the vector give the data becomes r r s s3 s c p 0 0 3 0 i r r s x 3 s 3 d miei ( p) i exp i xij (4) Bayesia Estimatio usig Rayleigh Prior The prior for the rate parameters i for i=,, is assumed to be the Rayleigh distributio, with the hyperparameter v i give by f i i i i exp, v 0 i vi vi (5) by The prior for p is assumed to be the beta distributio, whose desity is give 64
SINDHU ET AL c3d3 c d f p p p, c, d 0 p d 3 c 3 3 3 3 3 (6) From equatio (5)-(6), propose the followig joit prior desity of the,, p : vector i c3 d 3 g iexp p p, 0 p, vi, c3 0, d3 0 vi (7) By multiplyig Equatio (7) with Equatio (6), the joit posterior desity for the vector give the data becomes r r s s3 s c3 p 0 0 3 0 i r r s x 3 ( p ) exp s 3 d3 ( mi ) i i ij vi x (8) Margial distributios of i ad p i, ca be obtaied by itegratig the uisace parameters. Bayes Estimatio of the Vector of Parameters The Bayesia poit estimatio is coected to a loss fuctio i geeral, sigifyig the loss iduced whe the estimate ˆ differ from true parameter. Because there is o specific rule that helps us to idetify the appropriate loss fuctio to be used, squared error loss is used i this article as it serve as stadard loss. It is well ow that uder the, the Bayes estimator of a fuctio of the parameters is the posterior mea of the fuctio ad ris is the posterior variace. It is defied as ˆ ˆ l,. It was origially used i estimatio problems whe the ubiased estimator of θ was beig cosidered. Aother reaso for its popularity is due to its relatioship to least squares theory. The use of SELF maes the calculatios simpler. 65
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE The K-Loss fuctio (KLF), defied as: l ˆ, ˆ / ˆ, was proposed by Wasa (970). It is well fitted for a measure of iaccuracy for a estimator of a R 0,. Uder K-Loss fuctio scale parameter of a distributio defied o the Bayes estimates ad posterior riss are defied as ˆ E ( x ) / E ( x ), ad ˆ ( x) ( E E x) respectively. The respective margial distributio of each parameter has bee used to derive the Bayes estimators ad posterior riss for, ad p uder the squared error loss fuctio (SELF) ad K- loss fuctios (KLF). The Bayes estimators ad posterior riss of, ad p uder (SELF) assumig Naagami prior are give as: The Bayes estimators of, ad p are:, / r r s N SELF a m / a m 0 0 3 0 r r s ˆ 3 B A A a m a m a b x j a b x j / /, / r r s ( SELF ) N a m a m / 0 0 3 0 r r s ˆ 3 B A A a m a m a b x j a b x j / /, r r s ( SELF ) N a m a m 0 0 3 0 r r s pˆ 3 B A A a m a m a b x j a b x j / / The Posterior riss of, ad p are: 66
SINDHU ET AL r r s ˆ SELF N, ( ) a ( ) m a m 0 0 3 0 r r s r r s ˆ SELF N 3 B A A a m a m a b x j a b x j / /, a b x j a b x j ˆ SELF ( ) a ( ) m a m 0 0 3 0 r r s / / 3 B A A a m a m ˆ SELF, B A A a m a m pˆ SELF N p / / r r s ( ) ( ) 0 0 3 0 r r s 3 ˆ a m a m SELF a b x j a b x j The Bayes estimators of, ad p uder KLF are: ˆ r r s 0 0 3 0 KLF r r s 0 0 3 0, / B A A a m a m r r s a / b x a / b x 3 B A, A a m / a m / r r s a / b x j a / b x 3 a m a m / j j a m a m j ˆ r r s 0 0 3 0 KLF r r s 0 0 3 0, / B A A a m a m r r s a / b x j a / b x j 3 B A, A a m a m / r r s a / b x j a / b x j 3 a m a m / / a m a m 67
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE pˆ r r s 0 0 3 0 ( KLF ) r r s 0 0 3 0, B A A a m a m r r s a / b x j a / b x j 3 B A, A a m a m / r r s a / b x j a / b x j 3 a m a m a m a m 68
SINDHU ET AL The Posterior riss of, ad p uder KLF are:, / r r s B A A a m a m N 0 0 3 0 r r s / / 3 ˆ ( KLF ) r r s B A, A a m / a m 0 0 3 0 r r / s a / b x j a / b x j 3 a m a m / a b x j a b x j a m a m, / r r s B A A a m a m N 0 0 3 0 r r s / / 3 ˆ ( KLF ) r r s B A, A a m a m / 0 0 3 0 r r s a / b x j a / b x j 3 a m a m a b x j a b x j / / a m a m, r r s B A A a m a m N 0 0 3 0 r r s / / 3 pˆ ( KLF ) r r s B A, A a m a m 0 0 3 0 r r s a / b x j a / b x j 3 Where N is formulized as N 3 a b x j a b x j a m a m a m a m, r r s B A A a m a m 0 0 3 0 r r s / / A s 3 s c ad A s 3 d a m a m a b x j a b x j Similarly, expressios for Bayes estimators ad their posterior riss uder the rest of the priors ca be obtaied with little modificatios. 69
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE Elicitatio I Bayesia aalysis the elicitatio of opiio is a crucial step. I statistical iferece, the characteristics of a certai predictive distributio proposed by a expert determie the hyper-parameters of a prior distributio. Focus o a method of elicitatio based o prior predictive distributio. The elicitatio of hyperparameter from the prior p is a difficult tas. The prior predictive distributio is used for the elicitatio of the hyper-parameters which is compared with the experts' judgmet about this distributio ad the the hyper-parameters are chose i such a way so as to mae the judgmet agree as closely as possible with the give distributio. See also Grimshaw et al. (00), O Haga et al. (006), Jeiso (005) ad Leo et al. (003). Accordig to Aslam (003), the method of elicitatio is to compare the prior predictive distributio with experts assessmet about this distributio ad the to choose the hyper-parameters that mae the assessmet agree closely with the member of the family. The prior predictive distributios uder all the priors are derived usig: p( y) p y p d Elicitatio uder Naagami distributio The prior predictive distributio usig Naagami prior is: p( y) a b a ya c a c d y a b ya d a b, y 0 a a c d y a b (9) For the elicitatio of the six hyper-parameters, six differet itervals are cosidered. From Equatio (9), the experts probabilities/assessmets are supposed to be 0.0 for each case. The six itegrals for equatio (9) are cosidered with the followig limits of the values of radom variable Y : (0, 0), (0, 0), (0, 30), (30, ), (, 50) ad (50, 60) respectively. For the elicitatio of the hyper-parameters a, a, b, b, c, ad d. These six equatios are solved simultaeously through computer program developed i SAS pacage usig the commad of PROC SYSLIN. Thus 70
SINDHU ET AL the values of hyper-parameters obtaied by applyig this methodology are: 0.0003, 0.009, 0.54, 4.9935, 0.530, ad 0.4790 respectively. Elicitatio uder Chi Prior The prior predictive distributio usig Chi prior is: e e yed / / 0.50 0.50 0.5 ye c 0.5 e c d y c d y p( y), y 0 e Now, elicit four hyper-parameters, so cosider the four itegrals. The expert probabilities are assumed to 0.5 for each itegral with the followig limits of the values of radom variable Y : (0, 5), (5, 30), (30, 45) ad (45, 60). Usig the similar id of program, as discussed above, we have the followig values of the hyper-parameters e = 0.056, e = 4.3569, c = 0.09377 ad d = 0.08749. Elicitatio uder Rayleigh Prior The prior predictive distributio usig Rayleigh prior is: v yc v yd p( y), y 0 c d y v c d y v 3 3 3 3 3 3 Agai, elicit four hyper-parameters, so cosider the four itegrals. The expert probabilities are assumed to 0.5 for each itegral with the followig limits of the values of radom variable Y : (0, 5), (5, 30), (30, 45) ad (45, 60). Usig the similar id of program, as discussed above, we attaied the followig values of the hyper-parameters v = 5.0504, v = 5.035, c 3 = 0.673 ad d 3 = 0.9035. Simulatio Study ad Comparisos A simulatio study is carried out i order to ivestigate the performace of Bayes estimators uder tefold choice of the parametric values, differet sample sizes, ad the differet values of the mixig proportio. Tae radom samples of sizes = 0,, ad from the two compoet mixture of Rayleigh distributios with tefold choice of parameters 7
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE (, ) 0., 0.,,., 0,, 0.,, 0, 0., p 0.45 ad 0.6. To geerate a mixture data we mae use of probabilistic mixig with probabilities p ad (- p ). A uiform umber u is geerated times ad if u < p the observatio is tae radomly from F (the Rayleigh distributio with parameter ) otherwise from F (from the Rayleigh distributio with parameter ). The choice of the cesorig time is made i such a way that the cesorig rate i the resultat sample is approximately 0%. To implemet cesored sampligs, we cosidered that the x,..., x ad x,..., x failed items come from first ad r s r s secod subpopulatios respectively. The rest of the observatios which are less tha xr ad greater tha xs have bee assumed to be cesored from each compoet. Where xs max x, s, x, s ad xr mi x, r, x, r bee obtaied usig followig steps:. The simulated data sets have Step : Draw samples of size from the mixture model Step : Geerate a uiform radom o. u for each observatio Step 3: If u, the tae the observatio from first subpopulatio otherwise from the secod subpopulatio Step 4: Determie the test termiatio poits o left ad right, that is, determie the values of xr ad xs Step 5: The observatios which are less tha xr ad greater tha xs have bee cosidered to be cesored from each compoet Step 6: Use the remaiig observatios from each compoet for the aalysis To avoid a extreme sample, we simulate 0, 000 data sets each of size. The Bayes estimates ad posterior riss (i parethesis) are computed usig Mathematica 8.0. The average of these estimates ad correspodig riss are reported i tables - 5. The abbreviatios used i the tables are: B.Es: Bayes estimators; P.Rs: Posterior riss; NP: Naagami prior; CP: Chi prior; RP: Rayleigh prior. 7
SINDHU ET AL Table : B.Es ad P.Rs uder NP usig,, p (0., 0., 0.45) ad (0., 0., 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 0.076 0.773 0.49845 0.09968 0.3595 0.665779 (0.000479) (0.000558) (0.039) (0.00085) (0.000904) (0.0667) 0.09947 0.65 0.486 0.0946 0.35 0.659375 (0.0003) (0.000306) (0.0073) (0.0007) (0.00043) (0.006346) 0.099036 0.57 0.47884 0.0968 0.3063 0.6648 (0.0004) (0.0006) (0.003865) (0.000057) (0.00030) (0.0038) 0.0884 0.38 0. 0.095373 0.3648 0.6553 (0.086648) (0.0690) (0.9905) (0.056005) (0.087) (0.06058) 0.0669 0.3008 0.47869 0.09664 0.38 0.645074 (0.0450) (0.037679) (0.070063) (0.07334) (0.05350) (0.0364) 0.090768 0.778 0.47094 0.0973 0.5869 0.6866 (0.0446) (0.09760) (0.034345) (0.07) (0.07) (0.06573) Table : B.Es ad P.Rs uder NP usig,, p (,., 0.45) ad (,., 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp.03790.6375 0.4988 0.970.35085 0.665995 (0.046897) (0.05460) (0.030) (0.07554) (0.08765) (0.0594).0064.5934 0.4883 0.97930.390 0.6577 (0.0787) (0.08565) (0.00734) (0.03363) (0.045048) (0.005650) 0.996073.558 0.478649 0.9893.30748 0.66586 (0.0) (0.05747) (0.003863) (0.006855) (0.0378) (0.003).0547.9750 0.484873 0.0.737600 0.4855 (0.0850) (0.0697) (0.6936) (0.08390) (0.068839) (0.9790) 0.97684.4985 0.477066 9.85076.455 0.474996 (0.04384) (0.0373) (0.068486) (0.04378) (0.037003) (0.06997) 0.99497.75 0.469378 9.9883.990 0.468498 (0.04387) (0.0374) (0.037459) (0.0678) (0.0989) (0.036650) 73
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE Table 3: B.Es ad P.Rs uder NP usig,, p (0,, 0.45) ad (0,, 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 0.8959.9605 0.497084 9.5855 3.6630 0.663355 (4.5670) (4.8388) (0.0344) (.663) (7.8546) (0.070) 0.890.886 0.47966 9.7735 3.49536 0.656693 (.3944) (.7684) (0.007305) (.887) (4.998) (0.006435) 9.6493.67 0.46094 9.8875 3.4550 0.66606 (.0503) (.58376) (0.003) (0.5877) (.30094) (0.00333) 0.0.73760 0.4855 9.848.650 0.653839 (0.08390) (0.068839) (0.979) (0.05669) (0.04837) (0.06849) 9.85076.455 0.474996 9.9449.63070 0.65788 (0.04378) (0.037003) (0.06997) (0.07645) (0.05450) (0.036) 9.9883.990 0.468498 9.958.5 0.674 (0.0678) (0.0989) (0.036650) (0.0346) (0.088) (0.05684) Table 4: B.Es ad P.Rs uder NP usig,, p (0.0,, 0.45) ad (0.0,, 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 0.09569 3.679 0.5349 0.09487 3.860 0.687435 (0.00030) (4.86890) (0.0036) (0.0000) (7.89) (0.00396) 0.090655 3.54530 0.583 0.0999 3.63860 0.6775 (0.0003) (.343030) (0.006457) (0.000096) (3.438) (0.005650) 0.090905 3.48370 0.5093 0.0960 3.59 0.670 (0.000065) (.48460) (0.003346) (0.000048) (.767790) (0.00953) 0.0945.870 0.5308 0.09344 3.860 0.6790 (0.067737) (0.05368) (0.097694) (0.0487) (0.07896) (0.04995) 0.095.7830 0.568 0.095075 3.70450 0.673099 (0.0355) (0.05796) (0.0503) (0.03389) (0.03737) (0.06303) 0.09455.3789 0.505934 0.0963.96763 0.669953 (0.05937) (0.0696) (0.0687) (0.057) (0.0868) (0.035) 74
SINDHU ET AL Table 5: B.Es ad P.Rs uder NP usig,, p (0, 0., 0.45) ad (0, 0., 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp.9930 0.064 0.33.53870 0.3048 0.585750 (4.78770) (0.00035) (0.0879) (3.6833) (0.000483) (0.07).570 0.45 0.4.465 0.53 0.5863 (.3578) (0.00055) (0.00665) (.5987) (0.0009) (0.006334).7 0.8567 0.4970.93 0.686 0.58968 (.0883) (0.000075) (0.0036) (0.756977) (0.00003) (0.00394).64350 0.076 0.47776 0.98 0.068 0.57455 (0.067737) (0.05354) (0.507) (0.0487) (0.0784) (0.078744).44770 0.874 0.43946 0.874 0.08845 0.586549 (0.0359) (0.05790) (0.0978) (0.0339) (0.037356) (0.04779).57 0.9735 0.438659 0.74670 0.0987 0.59606 (0.05944) (0.0698) (0.03) (0.058) (0.0856) (0.05) Table 6: B.Es ad P.Rs uder CP usig,, p (0., 0., 0.45) ad (0., 0., 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 0.6058 0.6908 0.47978 0.394 0.9503 0.6643 (0.00059) (0.00059) (0.0308) (0.00077) (0.000893) (0.06345) 0.3374 0.47046 0.4687 0.004 0.768 0.663936 (0.00086) (0.00039) (0.007333) (0.0006) (0.000448) (0.0065) 0. 0.399 0.448705 0.0335 0.5533 0.6673 (0.00047) (0.00096) (0.0006) (0.000058) (0.0003) (0.00396) 0.65 0.5855 0.465088 0.3577 0.98858 0.653999 (0.053) (0.043568) (0.45568) (0.0999) (0.049799) (0.06370) 0.886 0.46 0.46459 0.89 0.67765 0.65043 (0.030) (0.0399) (0.079) (0.09077) (0.036) (0.03084) 0.04790 0.3506 0.46377 0.034 0.430 0.65087 (0.03) (0.00869) (0.03839) (0.00604) (0.00665) (0.0546) 75
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE Table 7: B.Es ad P.Rs uder CP usig,, p (,., 0.45) ad (,., 0.60) ˆ 0 ˆ ˆp ˆ ˆ ˆp.5970.5775 0.47675.3536.88630 0.66357 (0.0460) (0.05958) (0.0379) (0.0788) (0.0777) (0.0650).630.48 0.469789.650.6649 0.659758 (0.06077) (0.03378) (0.0076) (0.03935) (0.04456) (0.0069).449.567 0.45906.05707.449 0.656866 (0.0475) (0.0575) (0.00446) (0.006667) (0.0669) (0.003355) -loss fuctio 0.5336.49543 0.4548.388.843 0.654349 (0.04669) (0.04585) (0.47876) (0.08990) (0.048999) (0.065537).5475.43835 0.45576.337.6307 0.6546 (0.0887) (0.0894) (0.0738) (0.09386) (0.03467) (0.0336).0677.3308 0.450778.04654.43309 0.653583 (0.08469) (0.0847) (0.036839) (0.0979) (0.099) (0.0646) Table 8: B.Es ad P.Rs uder CP usig,, p (0,, 0.45) ad (0,, 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 5.5363 5.3955 0.47034 5.58 5.00538 0.6696 (0.4336) (0.43394) (0.0375) (0.4567) (0.45878) (0.0735) 6.74 6.4 0.456349 6.679 5.85830 0.6977 (0.39050) (0.396637) (0.00799) (0.35635) (0.84) (0.006889) 7.34364 7.7637 0.450 7.6093 7.0488 0.65043 (0.339835) (0.33758) (0.0039) (0.8763) (0.38435) (0.00360) 5.5495 5.37683 0.453655 5.3 4.95789 0.6968 (0.09004) (0.030637) (0.497) (0.04786) (0.03749) (0.0737) 6.9 6.839 0.447779 6.599 5.99867 0.637 (0.0089) (0.0956) (0.077333) (0.06483) (0.054) (0.037649) 7.0543 7.7934 0.44846 7.4659 6.9630 0.643 (0.009) (0.009) (0.094) (0.00) (0.05946) (0.0936) 76
SINDHU ET AL Table 9: B.Es ad P.Rs uder CP usig,, p (0.0,, 0.45) ad (0.0,, 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 0.35859 5.5759 0.563 0.305 5.0845 0.68633 (0.00065) (0.447943) (0.0354) (0.00008) (0.46858) (0.00735) 0.5993 6.65447 0.5353 0.49 6.04774 0.674904 (0.0003) (0.904) (0.00654) (0.000099) (0.435674) (0.005747) 0.0996 8.88344 0.506878 0.098 7.33437 0.67085 (0.000059) (0.34668) (0.003368) (0.000048) (0.38437) (0.00976) 0.439 5.48345 0.53036 0.9 5.045 0.673849 (0.08685) (0.0303) (0.0496) (0.04474) (0.037048) (0.05484) 0.579 6.6053 0.506765 0.034 6.37890 0.670396 (0.09664) (0.0887) (0.05377) (0.05909) (0.04397) (0.06986) 0.05346 8.099 0.503448 0.0458 7.390 0.668463 (0.0065) (0.00753) (0.0735) (0.009303) (0.0578) (0.0479) Table 0: B.Es ad P.Rs uder CP usig,, p (0, 0., 0.45) ad (0, 0., 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 5.6845 0.46896 0.4965 5.9604 0.595 0.578365 (0.44878) (0.00038) (0.0086) (0.44476) (0.00043) (0.0083) 6.44 0.77 0.43595 6.8584 0.36448 0.57893 (0.03) (0.00053) (0.00636) (0.368546) (0.0005) (0.00649) 7.588 0.7307 0.45587 8.06977 0.0046 0.588653 (0.343598) (0.000074) (0.00387) (0.3039) (0.00004) (0.00338) 5.58490 0.475 0.5755 5.89857 0.59564 0.56665 (0.08685) (0.03033) (0.697) (0.04474 ) (0.0370) (0.083544) 6.65 0.6893 0.47657 6.844 0.3547 0.563 (0.0966) (0.08875) (0.08499) (0.05908) (0.04) (0.043066) 7.4444 0.7069 0.439330 7.7344 0.066 0.558488 (0.007) (0.00756) (0.04888) (0.009358) (0.04498) (0.087) 77
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE Table : B.Es ad P.Rs uder RP usig,, p (0., 0., 0.45) ad (0., 0., 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 0.0578 0.353 0.43 0.03079 0.38375 0.645 (0.00047) (0.00058) (0.039) (0.0008) (0.000938) (0.0038) 0.0453 0.3766 0.474736 0.09693 0.333 0.64649 (0.00033) (0.00096) (0.007004) (0.0008) (0.000453) (0.0064) 0.09683 0.934 0.46 0.09388 0.3054 0.695 (0.00003) (0.000600) (0.003756) (0.000058) (0.00034) (0.00306) 0.065 0.305 0.485036 0.093688 0.49 0.63098 (0.0687089) (0.066684) (0.9068) (0.0494) (0.088853) (0.06578) 0.0989 0.94 0.46763 0.0975664 0.9533 0.6437 (0.0436) (0.03685) (0.069873) (0.06666) (0.049849) (0.03978) 0.00056 0.875 0.46499 0.0978534 0.479 0.6373 (0.0779) (0.004) (0.037457) (0.0946) (0.03696) (0.0490) Table : B.Es ad P.Rs uder RP usig,, p (,., 0.45) ad (,., 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp.059.3334 0.43388.0393.5575 0.64457 (0.04667) (0.05606) (0.0338) (0.07869) (0.098507) (0.0345).00664.9054 0.476983 0.970745.3935 0.6496 (0.077) (0.09663) (0.007093) (0.076) (0.04543) (0.0065) 0.983895.8937 0.47585 0.98354.37069 0.645 (0.0) (0.0603) (0.003774) (0.0043) (0.0935) (0.00345).6454.873 0.468544 0.99645.0994 0.648 (0.07345) (0.0638) (0.3557) (0.06483) (0.988) (0.0709).00749.08 0.466946 0.98594.35608 0.6050 (0.04506) (0.03553) (0.069635) (0.07093) (0.0507) (0.0358) 0.95477.00 0.457543 0.99596.34535 0.68873 (0.00998) (0.097) (0.034475) (0.0399) (0.0544) (0.05959) 78
SINDHU ET AL Table 3: B.Es ad P.Rs uder RP usig,, p (0,, 0.45) ad (0,, 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 9.5834.43070 0.47949 9.58708 0.9565 0.634954 (3.69330) (3.83663) (0.0469) (.300) (5.0066) (0.087) 9.676.78300 0.4743 9.5990.933 0.63087 (.948) (.34378) (0.007076) (.989) (3.4684) (0.006453) 9.68654.3 0.470593 9.7304.86730 0.655 (0.94767) (.460) (0.003749) (0.58685) (.0879) (0.00383) 8.0.3660 0.46484 0.09989.777 0.69444 (0.0684) (0.05868) (0.3363) (0.0493049) (0.088) (0.066076) 9.3787.84870 0.467 9.474.96030 0.66346 (0.038674) (0.0343) (0.069343) (0.0743) (0.055) (0.0386) 9.450.9648 0.46099 9.576.45630 0.60353 (0.094) (0.09479) (0.036338) (0.0793) (0.05) (0.0589) Table 4: B.Es ad P.Rs uder RP usig,, p (0.0,, 0.45) ad (0.0,, 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 0.094884.89990 0.4699 0.09834.9650 0.6646 (0.00063) (3.93650) (0.054) (0.00009) (4.6595) (0.00333) 0.09558.63033 0.50987 0.098953.7857 0.656383 (0.00037) (.9308) (0.00633) (0.000098) (.7857) (0.00564) 0.0943.39790 0.49507 0.09968.49 0.646003 (0.000063) (.053) (0.003307) (0.000047) (.5893) (0.00943) 0.4775.948 0.50606 0.07666 3.307 0.65563 (0.05966) (0.045) (0.099563) (0.0439) (0.067830) (0.05989) 0.094.89850 0.503309 0.09838.75870 0.64609 (0.03053) (0.04543) (0.05498) (0.0345) (0.034796) (0.075) 0.0968.7390 0.50733 0.090458.47 0.63679 (0.05444) (0.0387) (0.06989) (0.068) (0.0763) (0.0377) 79
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE Table 5: B.Es ad P.Rs uder RP usig,, p (0, 0., 0.45) ad (0, 0., 0.60) ˆ 0 -loss fuctio 0 ˆ ˆp ˆ ˆ ˆp 0.7884 0.846 0.3775 0.50830 0.9567 0.56709 (3.33464) (0.00033) (0.00833) (.353) (0.000474) (0.0379) 0.368 0.84 0.698 0.4890 0.6756 0.567909 (.9637) (0.00053) (0.006094) (.30849) (0.000) (0.0069) 0.3087 0.963 0.47844 0.39070 0.98576 0.585739 (0.9973) (0.000074) (0.00369) (0.686735) (0.00009) (0.0036) 9.863 0.563 0.69 9.044 0.7570 0.55597 (0.05966) (0.04874) (0.50848) (0.0439) (0.067755) (0.089) 0.94770 0.75 0.498 0.648 0.357 0.55487 (0.030533) (0.045) (0.0943) (0.0345) (0.034789) (0.044) 0.57770 0.85 0.49373 0.4660 0.3356 0.55073 (0.05446) (0.0385) (0.003) (0.069) (0.0765) (0.0699) Numerical results of the simulatio study, preseted i tables -5, reveal iterestig properties of the proposed Bayes estimators. The estimated values of the parameters coverge to the true values, ad amouts of posterior riss ted to decrease for lager choice of sample size. Aother iterestig poit cocerig the posterior riss of the estimates of, is that icreasig (decreasig) the proportio of the compoet i mixture reduces (icreases) the amout of the posterior ris for the estimates of λ. I additio, whe SELF is assumed ad values of λ i are relatively smaller i.e. for (λ, λ ) = (0., 0.) ad (,.), the Bayes estimates assumig Rayleigh prior are more precise tha the rest of the iformative priors, as the averaged posterior riss of the mixture compoets are smaller as compared to those uder other priors. O the other had, for quite larger values of parameters, i.e. for (λ, λ ) = (0, ), ad for sigificatly differet values of the parameters, i.e. for (λ, λ ) = (0., ), the estimates uder chi prior (with few exceptios) perform better tha those uder Naagami ad Rayleigh priors. However, the estimates for the mixig parameter (p ), uder Rayleigh prior, are associated with the miimum amouts of posterior riss irrespective of choice of true parametric values. Whe KLF is assumed, the estimates uder chi prior are foud to be the most efficiet for all combiatios of the values of the parameters, with a exceptio i case of (λ, λ ) = (0., 0.), where the estimates uder the assumptio of Naagami prior are better tha those uder other priors. However,
SINDHU ET AL the estimates for the mixig parameter (p ) are havig mixed behavior, as for various choices of the true parametric values idicate the preferece of differet priors. The Bayes estimates of the lifetime parameters are over/uder-estimated but the size of over/uder-estimatio is greater uder. O the other had, estimates of the mixig proportio parameter have mixed behavior sometimes over-estimated ad sometimes uder-estimated, but the Bayes estimates uder Rayleigh prior are much closer to the true parametric value. I compariso of loss fuctios it has bee assessed that the magitudes of posterior riss uder are smaller tha those uder -loss fuctio for smaller choice of true parametric values i.e. for (λ, λ ) = (0., 0.) ad (,.). O the other had, for quite larger values of parameters, i.e. for (λ, λ ) = (0, ), ad for sigificatly differet values of the parameters, i.e. for (λ, λ ) = (0., ) ad (0, 0.), the -loss fuctio produces the better results. It should also be metioed here that the produces better covergece tha -loss fuctio. Real Data Aalysis I this sectio, real datasets are aalyzed to illustrate the methodology discussed i the previous sectios. I order to show the usefuless of the proposed mixture model, we applied the fidigs of the paper to the survival times (i years) of a group of patiets give chemotherapy treatmet. The data has bee reported by Beer et al. (000). We have used the Kolmogorov-Smirov ad chi square tests to see whether the data follow the Rayleigh distributio. These tests say that the data follow the Rayleigh distributio at 5% level of sigificace with p-values 0.70 ad 0.68 respectively. The data cosistig of 46 survival times (i years) for 46 patiets are: Table 6: Survival times (i years) of patiets give chemotherapy treatmet 0.047, 0.5, 0., 0.3, 0.64, 0.97, 0.03, 0.60, 0.8, 0.96, 0.334, 0.395, 0.458, 0.466, 0.50, 0.507, 0.59, 0.534, 0.5, 0.570, 0.64, 0.644, 0.696, 0.84, 0.863,.099,.9,.7,.36,.447,.485,.553,.58,.589,.78,.343,.46,.444,.85,.830, 3.578, 3.658, 3.743, 3.978, 4.003, 4.033. 8
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE Cosider the case whe the data are doubly type II cesored. Data are radomly grouped ito two sets usig probabilistic mixig for p = 0.60. Table 7: Doubly cesored mixture real life data regardig survival times of patiets give chemotherapy treatmet Populatio-I 0.97, 0.534, 0.5, 0.96, 0., 0.466, 0.59,.447, 0.863, 0.3, 0.395, 0.696,.85, 3.658, 3.978, 3.743,.343,.78, 0.5, 4.003,.553,.485,.83,.46 Populatio-II 0.60,.099, 0.50, 0.458, 0.64, 0.334, 0.570, 0.64, 0.03, 0.8, 0.047,.7,.589,.36, 0.84,.444 The followig characteristics are extracted from cesored data for the aalysis of mixture model: p = 0.6, =, r = 5, r =, r = 3, r = 9, s = 36, s =, s = 4, = 4, = 6, xr 0., x 3.978, 0.03, ad.444. s x r x s s s x ( i) x ( i) ir ir 84.6037 ad 5.833. The similar methodology has bee employed whe p = 0.45. p = 0.45, =, r = 5, r =, r = 3, r = 9, s = 36, s = 6, s = 0, = 8, =, xr 0., x 3.658, 0.64, ad 3.978, s x r x s s s x ( i) x ( i) ir ir 48.704 ad 37.999. Bayes estimates are obtaied assumig iformative priors uder squared error loss fuctio, ad -loss fuctio. 8
SINDHU ET AL Table 8: B.Es ad P.Rs i paretheses uder, ad -loss fuctio for real data set. Priors -loss fuctio p = 0.6 Naagami Prior Chi Prior Rayleigh Prior p = 0.45 Naagami Prior Chi Prior Rayleigh Prior ˆ 0.38354 (0.0068) 0.4695 (0.00678) 0.3990 (0.0068) ˆ ˆp 0.9473 (0.0657).48390 (0.05708) 0.93 (0.063) ˆ 0.37704 (0.00) 0.48646 (0.00974) 0.38867 (0.004) 0.677477 (0.00565) 0.674903 (0.005747) 0.665364 (0.00566) ˆ ˆp 0.703 (0.00757) 0.8370 (0.00888) 0.7606 (0.007504) 0.56536 (0.006537) 0.5665 (0.006693) 0.508438 (0.00638) ˆ 0.389 (0.03399) 0.46094 (0.05909) 0.39007 (0.0350) ˆ ˆp 0.9385 (0.037685).4450 (0.045) 0.9785 (0.034973) ˆ 0.373998 (0.03449) 0.479778 (0.039) 0.385574 (0.0364) 0.673064 (0.063) 0.670395 (0.06986) 0.6608996 (0.0730) ˆ ˆp 0.7649 (0.030999) 0.83775 (0.0685) 0.7357 (0.09) 0.509 (0.05386) 0.504667 (0.05585) 0.5077 (0.0535) The fidigs from the real life aalysis are i close accordace with those of simulatio study. It ca be assessed that the chi prior produces better results for parameters λ ad λ, while i case of mixig parameter the Rayleigh prior provides comparatively better results tha other priors. It should further be oted that the estimates uder are associated with smaller amouts of posterior riss. Coclusio The Bayesia iferece of the mixture of Rayleigh model uder doubly type II cesorig has bee cosidered assumig iformative priors. The simulatio study has displayed some iterestig properties of the Bayes estimates. It is oted i each case that the posterior riss of estimates of lifetime parameters are reduced as the sample size icreases. The results idicated that by usig SELF ad relatively smaller values of λ i i.e. for (λ, λ ) = (0., 0.) ad (,.), the Bayes estimates assumig Rayleigh prior are more precise tha the rest of the iformative priors. While, for quite larger values of parameters, i.e. for (λ, λ ) = (0, ), ad for sigificatly differet values of the parameters, i.e. for (λ, λ ) = (0., ) ad (0, 0.), the estimates uder chi prior perform better tha other priors. Similarly, whe KLF is cosidered, the estimates uder chi prior are foud to be the most efficiet for most of the combiatios of the values of the parameters. The performace of 83
BAYESIAN ESTIMATION OF TWO-COMPONENT MIXTURE the is better tha -loss fuctio for (λ, λ ) = (0., 0.) ad (,.). However, for (λ, λ ) = (0, ), (0., ) ad (0, 0.), the -loss fuctio produces the better results. It should also be metioed here that the produces better covergece tha -loss fuctio for almost all the cases. The real life example further stregtheed the fidigs from the simulatio study. The study ca further be exteded by cosiderig some other cesorig techiques, ad usig some more flexible probability distributio. Refereces Aslam, M. (003). A applicatio of prior predictive distributio to elicit the prior desity. Joural of Statistical Theory ad Applicatios, (), 70-83. Beer, A., Roux, J., & Mostert, P. (000). A geeralizatio of the compoud Rayleigh distributio: usig a Bayesia method o cacer survival times. Commuicatios i Statistics, Theory ad Methods, 9(7), 49-433. Blische, W. R., & Murthy, D. N. P. (000). Reliability. New Yor: Joh Wiley ad Sos. Bolstad, W. M. (004). Itroductio to Bayesia Statistics. New Yor: Joh Wiley ad Sos. Eluebaly, T., & Bougila, N. (0). Bayesia learig of fiite geeralized Gaussia mixture models o images. Sigal Processig, 9(4), -. Feradez, A. J. (000). O maximum lielihood predictio based o type II doubly cesored expoetial data. Metria, 50(3), -0. Feradez, A. J. (006). Bayesia estimatio based o trimmed samples from Pareto populatios. Computatioal statistics & data aalysis, 5(), 9-30. Ghosh, S. K., & Ebrahimi, N. (00). Bayesia aalysis of the mixig fuctio i a mixture of two expoetial distributios. Tech. Rep. 53, Istitute of Statistics Mimeographs, North Carolia State Uiversity, North Carolia State Uiversity. Grimshaw, S D., Colligs, B. J., Larse, W. A., & Hurt, C. R. (00). Elicitig Factor Importace i a Desiged Experimet. Techometrics, 43(), 33-46. Jeiso, D. (005). The elicitatio of probabilities: A review of the statistical literature. Departmet of Probability ad Statistics, Uiversity of Sheffield. 84
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