On the Bayesian Estimation for two Component Mixture of Maxwell Distribution, Assuming Type I Censored Data

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1 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 O the Bayesa Estmato for two Compoet Mxture of Maxwell strbuto, Assumg Type I Cesored ata Syed Mohs Al Kazm Sustaable evelopmet Polcy Isttute Islamabad 440, Pasta. Muhammad Aslam epartmet of Statstcs Quad--Azam Uversty Islamabad 4, Pasta. Sajd Al epartmet of Statstcs Quad--Azam Uversty Islamabad 4, Pasta. Abstract Mxture models costtute a fte ad fte umber of compoets that expla dfferet datasets. However there are may stuatos where mxture models comprse a terestg setch of dfferet aspects. I ths study we explore the dea of mxture desty uder Type I cesorg scheme. We model a heterogeeous populato by meas of two compoets mxture of the Maxwell dstrbuto. The parameters of the Maxwell mxture are estmated ad compared usg the Bayes estmates uder the square error loss fucto ad precautoary loss fucto. A cesored mxture data s smulated by probablstc mxg for the computatoal purpose. Closed form expressos for the Bayes estmators ad posteror rs are derved for the cesored sample as well as for the complete sample. Some terestg comparso ad propertes of the estmates are observed ad preseted. A real lfe data applcato has also bee dscussed. Keywords: Fte mxture of Maxwell dstrbuto; Cesored samplg; Fxed termato tme; Lmtg expresso; Elctato of Hyperparameters; Squared error loss fucto; Precautoary loss fucto.. Itroducto The Maxwell dstrbuto s a probablty dstrbuto wth applcato physcs ad chemstry. The most frequet applcato s the feld of statstcal mechacs. The temperature of ay (massve) physcal system s the result of the motos of the molecules ad atoms whch mae up the system. These partcles have a rage of dfferet veloctes, ad the velocty of ay sgle partcle costatly chages due to collsos wth other partcles. However, the fracto of a large umber of partcles wth a partcular velocty rage s early costat. The Maxwell dstrbuto of veloctes specfes ths fracto, for ay velocty rage as a fucto of the temperature of the system. Tyag ad Bhattacharya (989a, b) cosdered Maxwell dstrbuto as a lfetme model for the frst tme. They obtaed Bayes estmates ad mmum varace ubased estmators of the parameter ad relablty fucto for the Maxwell dstrbuto. Chaturved ad Ra (998) geeralzed Maxwell dstrbuto ad they obtaed Classcal ad Bayesa estmators for geeralzed dstrbuto. Beer ad Roux (5) studed Emprcal Bayes estmato for Maxwell dstrbuto. These studes gve mathematcal hadlg to Maxwell dstrbuto but gore the applcato aspect of the Maxwell dstrbuto. The motvato of usg mxture model s that curret scearo, aalysts are able to descrbe estmates, predct ad fer about the complex system of terest usg more powerful ad complex computatoal methods. But mxture model comprses a terestg setch of all these aspects. Mxture models costtute a fte ad fte umber of compoets that expla dfferet datasets. The Bayesa approach to aalyze mxture models has developed great terest betwee aalysts. Posteror dstrbuto s the worbech of Bayesa statstcas. It s obtaed whe pror formato s combed wth lelhood. Therefore pror formato s ecessary for Bayesa approach. 97

2 Cetre for Promotg Ideas, USA The pror formato s purely subjectve assessmet of a expert before ay data have bee observed. Also Berger (985) argue that whe formato s ot compact form the Bayesa aalyss usg o-formatve prors or sgle most sutable cosderato. So ths study we cosder two o-formatve prors ad two formatve prors for comparso purpose. A fte mxture of some sutable probablty dstrbutos are recommeded to study a populato that s supposed to comprse a umber of subpopulato mxed a uow proporto. A populato of lfetmes of certa electrcal elemets may be dvded to a umber of subpopulatos depedg upo the possble case. Mxture model have bee used physcal chemcal, socal scece, bologcal ad other felds. For example Sha (998) cosdered the Bayesa couterpart of the maxmum lelhood estmates of the Medehall ad Hader (958) mxture. Saleem ad Aslam (8) use the Bayesa Aalyss for the two compoets Mxture of the Raylegh dstrbuto assumg the uform ad Jeffery prors. A type I mxture s stated as the mxture of probablty desty fucto from the same famly, whle a mxture of desty fuctos from several famles s called a type II mxture. Now f we tal about the practcal stuatos, a mxture populato may have the ow compoet destes ad we eed to fer oly about the mxg weghts. O the other had, may real lfe applcatos, there are ow fuctoal forms of compoet destes wth uow parameters but mxg weghts are ow ad vce versa. I ths paper, type I mxture models wth uow parameters of the ow umber of compoet destes belogg to the same parametrc famly ad wth uow mxg weghts are cosdered. Cesorg s a uavodable feature of the lfetme applcatos ad s a form of mssg data problem. A accout of cesorg ca be see Leems (955), etz et al (97), Kle (9), Kalbflesch et al () ad Smth () that are valuable cotrbuto to survval aalyss techque for cesored ad trucated data. Jag (99) deals wth maxmum lelhood estmates usg cesored data for mxed Webull dstrbuto whle Wag et al (958) cosders the estmators for survval fucto whe cesorg tmes are ow. Cesorg s dvded to three types,.e., left, rght, ad terval cesorg s sad to be employed f lfetme of a object s greater tha a depedet radom umber. I type I (type II) rght cesorg, the lfe-test termato tme (the umber of dead objects) s pre-specfed. I ordary type I rght cesorg; the lfe-test termato tme s the same for all the objects. The lfe-tme of a object s called terval cesored f t s ow to fall a ow tme-terval. I our study, a ordary type I rght cesorg s cosdered wth a fxed-test termato tme. Now to aswers the questos that why we use verted gamma ad verted Ch square prors for mxture aalyss ad why we use cesorg o data why ot o parameter, As we ow that the Maxwell model s sewed so we should have a pror whch reflect expert owledge a better form so there should be a sewed pror for ths model. Sce Iverted Gamma s a atural cojugate pror for the Maxwell model therefore we use t. The we ch square pror whch s aother form of the verted gamma dstrbuto order to chec that may be t perform better tha verted Gamma pror. If we tal about cesorg the cesorg s a data property ad we caot apply o parameters. I ths paper, radom observatos tae from ths populato are supposed to be characterzed by oe of the two dstct uow members of a Maxwell dstrbuto. So the two compoet mxture of the Maxwell dstrbuto s recommeded to model ths populato. Rght cesorg s cosdered ad the observatos greater tha the fxed cut off cesor value, T are tae as cesored oes. The Iverse Trasform method of smulato, ad the computatos volved are coducted usg the pacages Mtab, Mathematca, SAS ad Excel. We may brea ths study to followg sectos. The Maxwell mxture model s defed Secto ad ts lelhood s developed Secto. Secto 4,5 ad 6, evaluate the Bayes estmators ad ther posteror rs uder square error loss fucto ad precautoary loss fucto usg uform, Jeffreys, Iverted Gamma ad Iverted Ch- Squared pror. Elctato of hyperparameters method s dscussed Secto 7. Lmtg expressos are derved Secto 8. I Secto 9, the smulato study s performed. Secto 0 presets the real lfe data whch are used for the evaluato of Bayes estmates. Some cocludg remars are gve last Secto.. The Mxture Model A fte mxture desty fucto wth the two compoet destes of specfed parametrc form wth uow mxg weghts ( p, p) s defed as follows g( x) = pf ( x ) + (- p)f ( x), 0 p () 98

3 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 The followg Maxwell dstrbuto s assumed for both compoets of the mxture. f x x 4 x e,,, 0, 0 x so the mxture model taes the followg form x 4 x 4 g x p x e q x e ; q p,0 p Krsha ad Mal (9) use the followg form of dstrbuto fucto: x Gx ( ), x, a e u du, s the complete gamma fucto. u a where x 0 So the correspodg dstrbuto fucto s gve by F( x) pg ( x) ( p) G ( x) x x F( x) p, ( p),. Lelhood Fucto Suppose uts from the above mxture model are used to lfe testg expermet. Let the test be coducted ad t s observe that out of test s termated as soo as the rth falure occurs ad the remag r uts are stll worg. As Medehall ad Hader (958) elghte that may real lfe stuatos oly the faled objects ca easly be detfed as member of ether subpopulato or subpopulato. So, depedg upo the cause of falure t may be observed that r ad r are detfed as members of the frst ad secod subpopulato respectvely. It s apparet that r r r ad remag r objects provde o formato about the subpopulato to whch they belog. We defe, x j as the falure tme of the jth subpopulato, where j,,,...;,; 0 xj, x j. So the lelhood fucto for the gve codto s: r r r L(,, p x) pf x j qf x j F t j j () x = x, x..., x, x, x,..., x s the observed falure tmes for the o-cesored observatos where r r r 4 x r x 4 r L(,, p x ) p x e q x e F t j j, p t q t Ft ( ),,, where t t t r r r r r r r 4 4 m r m x j j 0 m0 m j j L(,, p ) ( ) p q x x A A m m exp exp C C, () where r t, r, A xj,, 0.5 ad C,, j 99

4 Cetre for Promotg Ideas, USA 4. Bayesa Estmato usg Uformatve Prors Sce Bayesa estmato ca be appled eve whe o pror formato s avalable, so we ca say that uformatve pror s a pror whch cota o formato about parameter. Amog the techques that have bee proposed for determg uformatve prors, Jeffreys (96) suggests the most wdely use method. Box ad Tao (97), defe a uformatve pror as pror whch provdes lttle formato relatve to the expermet. Berardo ad Smth (994) use a smlar defto; they say that uformatve prors have mmal effect relatve to the data, o the fal ferece. They regard the uformatve pror as a mathematcal tool; t s o a uquely uformatve pror. Berardo (979b) argue that a uformatve pror should be regarded as referece pror,.e. a pror whch s coveet to use as a stadard aalyzg statstcal data. Gesser (984) also proposed some techques for uformatve prors. The most commo examples of uformatve prors are Uform ad Jeffreys. Both prors are used oly whe o formal pror formato s avalable. 4.. The Uform (Uformatve) Pror Bayes (76), Laplace (8) ad Gesser (984) suggest that oe may tae uform dstrbuto for the uow parameter the absece of suffcet reaso for assgg uequal probabltes to the values the parameter space had created a lot of dscomforts for the users of Bayes theorem for feretal purposes. Uform prors are partcularly easy to specfy the case of a parameter wth bouded support. The smplest stuato to cosder s whe s fte. Let ~ Uform (0, ), ~ Uform (0, ) ad p ~ U(0,).Assumg depedece, we have mproper jot pror that s proporto to a costat whch s corporated wth the above lelhood (4) ad we have jot posteror ad margal dstrbutos. The jot posteror dstrbuto usg uform pror as follows r r r r r r r 4 4 m r m x j j 0 m0 m j j p(,, p ) ( ) p q x x A A m m exp exp C C, 0,0,0 p (4) 4. Bayes Estmators usg Uform Pror Bayes estmator s a estmator or decso rule that maxmzes the posteror expected value of utlty fucto or mmzes the posteror expected value of the loss fucto. The loss fucto s the real valued fucto that clearly provdes a loss for decso a gve parameter. The square error loss fucto (SELF), L a a was proposed by Legedre (805) ad Gauss (80) to develop least square theory. Later t was used estmato problem whe ubased estmators of were evaluated terms of the rs fucto R, g whch become othg but the varace of the estmators. Norstrom (996) troduced a alteratve asymmetrc precautoary loss fucto (PLF), ad also preseted a geeral class of precautoary loss fuctos ( d) as a specal case whch s defed as L4 L(, d), Bayes estmator usg ths loss fucto s d * d E( x) ad E xl(, d) E( x) E( x) s posteror rs. The respectve margal dstrbuto yeld the followg Bayes estmators of, ad of p uder the squared error loss fucto. c c r c c r m m ( ) B, dd 0 m0 m A A E( x ) (5) r c c c c r m m ( ) B, dd 0 m0 m A A r r c c m m ( ) B, dd c c 0 m 0 m A A E( x ) (6) r r c c m m ( ) B, c c dd 0 m0 m A A

5 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 c c r c c r m m ( ) B, dd 0 m0 m A A E( p x ) (7) r c c c c r m m ( ) B, dd 0 m0 m A A where,. r m, r m, r m ad r m. Smlarly Bayes 4 estmators usg precautoary loss fucto ca also be derved accordgly. 4.. Posteror Rss assumg Uform Pror The posteror rss of, ad p usg the uform pror are gve as ( ) ( ) ( p ) r r c c m m ( ) B, d c c d 0 m 0 m A A x r r c c E m m ( ) B, c c dd m A A r c c m m ( ) B, c c dd m A A r c c E x (9) m m ( ) B, c c dd m A A r c c m m ( ) B4, d c c d m A A E p x (0) r c c m m ( ) B, c c dd m A A E p x are gve s equatos (5),(6) ad (7) respectvely. 0 m0 r x (8) 0 m0 x r 0 m 0 r 0 m0 x r 0 m0 E x ad where E x, Note that here all tegrals are evaluated umercally usg Mathematca 6.0. The posteror rss uder precautoary loss fucto ca also be derved smlar maer. 5. The Jeffreys (Uformatve) Pror Jeffreys pror s aother form of uformatve pror whch s also called referece pror, t s based o Fsher formato matrx habtually lead to a famly of mproper prors. Uder some regularty codtos specally case of oe parameter t does ot reveal lac of owledge, Jeffreys pror llustrate the sort of pror owledge whch would mae the data as posteror domat as possble. The posteror dstrbuto based o Jeffreys pror may the be used as a bechmar or a referece for the class of posteror dstrbuto whch may be obtaed from other prors. For the Maxwell model, gve Secto, let the Jeffrey prors g( ) I( ), f( x ) I( ) E where, are g( ) 0 g( ) 0 ad g ( p ), 0< p <. By assumg depedece we obta a jot pror g(,, p) whch s corporated wth the lelhood (4) to yeld the jot posteror ad margal dstrbutos. So the Jot posteror dstrbuto s gve as r r r r r r r 4 4 m r m x j j 0 m0 m j j P(,, p ) ( ) p q x x A A m m exp exp C C, 0,0,0 p () 5. Bayes Estmators usg the Jeffrey Pror The respectve margal dstrbuto yeld the followg Bayes estmators of, ad of p uder the squared error loss fucto. 0

6 Cetre for Promotg Ideas, USA c c r c c r m m ( ) B, dd 0 m0 m A A E( x ) () r c c c c r m m ( ) B, dd 0 m0 m A A r r c c m m ( ) B, dd c c 0 m 0 m A A E( x ) r r c c m m ( ) B, c c dd 0 m0 m A A () r r c c m m ( ) B, d d c c 0 m 0 m A A E( p x ) r r c c m m ( ) B, c c dd 0 m0 m A A (4) 5.. Posteror Rs assumg Jeffrey Pror Uder SELF the posteror rss of, ad p usg Jeffreys pror are provded as ( ) ( ) ( p x) r r c c m m ( ) B, c c dd 0 m 0 m A A x r r c c E m m ( ) B, c c dd m A A r c c m m ( ) B, c c dd m A A r c c E x (6) m m ( ) B, c c dd m m A A r c c m m ( ) B, dd c c m A A E p x (7) r c c m m ( ) B, dd c c m A A E p x are gve s equatos (),() ad (4) respectvely. The 0 m0 r x (5) 0 m0 x r r m0 r 0 m0 E x ad where E x, posteror rs uder precautoary loss fucto usg Jeffrey pror ca also be derved as SELF. 6. Iformatve Prors A formatve pror expresses specfc, defte formato about a varable. The terms "pror" ad "posteror" are geerally relatve to a specfc datum or observato. I case of a formatve pror, the use of pror formato s equvalet to addg a umber of observatos to a gve sample sze, ad therefore leads to a reducto of the varace or posteror rs of the Bayes estmates. 6.. The Cojugate Pror I probablty theory ad statstcs, the Cojugate pror (Iverted Gamma dstrbuto) s a two-parameter famly of cotuous probablty dstrbuto o the postve real le, whch s the dstrbuto of the recprocal of a varable dstrbuted accordg to the gamma dstrbuto. Let Iverted Gamma ( a, b ), Iverted Gamma ( a, b ) ad P U(0,),assumg depedece, we have a jot pror a b a b b b a a ( a) ( a) g(,, p) e. e whch s corporated wth the lelhood gve (4). 0

7 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 p(,, p x) ( ) p q exp x b / r r r r m r m r j 0 m0 m a j r a r exp / j m m x j b,, 0,0 p (8) The margal dstrbutos wth respect to parameters q, q ad p are elaborated as p( x) p( x) p( p x) A b r r c a m m ( ), e d c a c a 0 m 0 m A b 0 r r c a c a m m ( ), c a c a dd 0 m0 m A b A b A b r r c a m m ( ), e d c a c a 0 m 0 m A b 0 r r c a c a m m ( ), c a c a dd 0 m0 m A b A b r c a c a r r m r m m m ( ) p q dd c a c a 0 m 0 m A b A b r r c a c a m m ( ), c a c a dd 0 m0 m A b A b 6... Bayes Estmators usg the Cojugate Pror, (9), (0). () Bayes estmator s a estmator or decso rule that maxmzes the posteror expected value of utlty fucto or mmzes the posteror expected value of the loss fucto. The respectve margal dstrbuto yeld the followg Bayes estmators of q, q ad p uder the squared error loss fucto. E x E x E p x r c a c a r m m ( ), dd c a c a 0 m 0 m A b A b r r c a c a m m ( ), c a c a dd 0 m0 m A b A b r c a c a r m m ( ), dd c a c a 0 m 0 m A b A b r r c a c a m m ( ), c a c a dd 0 m0 m A b A b r c a c a r m m ( ), d d c a c a 0 m 0 m A b A b r r c a c a m m ( ), c a c a dd 0 m0 m A b A b Where, A, ad defed above, whle a, =,..,4 are defed as r m, r m, r m ad r m Posteror Rs usg the Cojugate Pror, (), (). (4) Whe presetg a Statstcal estmate, t s ecessary to dcate the accuracy of the estmates. The Bayesa measure of the accuracy of a estmate s the posteror rs of the estmate. The expressos of the Bayes posteror rss uder square error loss fucto are 0

8 Cetre for Promotg Ideas, USA ( x) 04 r c a c a r m m ( ), dd c a c a 0 m 0 m A b A b r r c a c a m m ( ), c a c a dd 0 m0 m A b A b r c a c a (5) E x r ( ), dd c a c a 0 m 0 m A b A b E r r c a c a ( ), c a c a dd 0 m0 m A b A b ( x) x r c a c a r m m ( ), dd c a c b 0 m 0 m A b A b E p r r c a c a m m ( ), c a c a dd 0 m0 m A b A b ( p x) x where E ( q x ), E ad E p x (6) (7) are specfed equatos (), () ad (4) respectvely. Here we use x umercal tegrato order to evaluate tegrals gve varaces equatos. Smlarly we ca estmate the Bayes estmators ad Bayes posteror rss uder precautoary loss fucto usg Cojugate pror. 6.. The Iverted Ch-square Pror The verted ch-square dstrbuto s the dstrbuto of a radom varable whose multplcatve (recprocal) has a ch-square dstrbuto. It s also ofte defed as the dstrbuto of a radom varable whose recprocal dvded by ts degrees of freedom s a ch-square dstrbuto. Let Iverted Ch squre( a, b ), q : Iverted Ch- squre( a, b ) ad P U(0,) assumg depedece, we have a jot prors, a a b b b b a a a a a a g,, p e. e So the jot posteror dstrbuto gets the followg form,whch s corporated wth the lelhood gve (4). A A r r r m r m m m c c 0 m0 m p(,, p x) ( ) p q e e. (8) where,,,. Now the margal dstrbutos of the correspodg parameters q, q ad p are metoed as p( x) p( x) c c A r r c m m ( ), e c d 0 m0 m A 0 r c c r m m ( ), c dd 0 m0 m A A c c A r r c m m ( ), e c d 0 m0 m A 0 r c c r m m ( ), c dd 0 m0 m A A, (9), (0)

9 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 p( p x) c c r c c c c r r m r m m m ( ) p q dd 0 m0 m A A r c c r m m ( ), dd 0 m0 m A A 6... Bayes Estmators usg the Iverted Ch-square Pror. () The uder the squared error loss fucto, respectve margal dstrbuto yeld the followg Bayes estmators of, ad of p E x E x E p x Where, A, ad c c c c c c r c c r m m ( ), dd c 0 m 0 m A A r r c c m m ( ), c dd 0 m0 m A A r c c r m m ( ), dd c 0 m 0 m A A r r c c m m ( ), c dd 0 m0 m A A r c c c c r m m ( ), d d 0 m 0 m A A r r c c m m ( ), dd 0 m0 m A A a,,..,4 are defed above Posteror Rs usg the Iverted Ch-square Pror, (), (). (4) The expressos for the varaces of the Bayes estmators uder square error loss fucto are r r c c m m ( ), dd c c 0 m 0 m A A ( x) r E x r c c m m ( ), c c dd 0 m0 m A A (5) r r c c m m ( ), dd c c 0 m 0 m A A ( x) r r c c E x m m ( ), c c dd 0 m0 m A A (6) r r c c m m ( ) 4, dd c c 0 m 0 m A A ( p x) E p x r r c c m m ( ), c c dd 0 m0 m A A (7) where E x, E x ad x E p are metoed equatos (), () ad (4) respectvely. Here we use umercal tegrato order to evaluate tegrals gve varace expressos. The Bayes estmator ad posteror rs ca also be derved uder precautoary loss fucto usg Ch-Squared prors. 05

10 Cetre for Promotg Ideas, USA 7. Elctato of Hyperparameters Accordg to Garthwate et al. (4) elctato s the process of formulatg a perso s owledge ad belefs about oe or more ucerta quattes to a (jot) probablty dstrbuto for those quattes. I the cotext of Bayesa statstcal aalyss, t arses most usually as a method for specfyg the pror dstrbuto for oe or more uow parameters of a statstcal model. Aslam () purposed four methods for elctato, three methods for two treatmets ad oe method for geeral treatmets. Parameters volve the Bayes estmates ad varaces by usg both verted gamma pror ad verted ch-square pror are elcted accordg to method of elctato va pror predctve approach whch s also oe of them, where pror predctve dstrbutos usg verted gamma pror ad verted ch-square pror are derved by usg followg formula,,,,. p y p p p y p d d dp 0 Accordg to the expert probabltes we cosder four tervals for the elctato, ad the set of hyperparameters wth mmum values are chose to be the elcted values of the hyperparameters. The resultat pror predctve dstrbutos for the mxture of Maxwell model are as follows 7. Elctato usg Cojugate pror The pror predctve dstrbuto equato usg Cojugate pror s a a a a b 4 b 4 p y = y, y,, y 0 a a a a ( b y ) ( b y ) Sce we have to elct four parameters therefore we cosdered four tervals. The set of hyperparameters wth mmum values are cosdered to be the elcted values of the hyperparameters. Usg the pror predctve dstrbuto gve (8), the experts probabltes are assumed to be 0., 0., 0. ad 0. whch are assocated wth the tervals 0 y 0 ad 0. y 0 0. y 0 ad 0. y 40 respectvely as a a y, y, dy 0., (9) 0 a a b 4 4 b a 0. a a ( ) a b y ( b y ) a a y, y, dy 0.. (40) 0 a a b 4 4 b a 0. a a ( ) a b y ( b y ) a a y, y, dy 0. 0 a a b 4 4 b a 0. a a ( ) a b y ( b y ) a a y, y, dy a a b 4 4 b a 0. a a ( ) a b y ( b y ) (8) (4) (4) For elctg the hyperparameters a, a, b ad b, the equatos (9) to (4) are smultaeously solved through the computer program developed SAS pacage usg the PROC SYSLIN commad ad the values of the hyperparameters a, a, b ad b are foud to be 0.486, , ad respectvely. 06

11 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 7. Elctato usg Iverted Ch-Square pror The pror predctve dstrbuto equato usg verted ch-squared pror s a a a a b 4 b 4 p y = y, y,, 0 a y a a a a b a b ( y ) ( y ) (4) Usg smlar crtera as defed for verted gamma pror, the values of the hyperparameters a, a, b ad b are.7054, 0.48, 0.66 ad respectvely. 8. Lmtg expressos SupposeT, all observatos whch are slot our aalyss become ucesored, ad cosequetly r teds to, r teds to the uow ad r to the uow. Accordgly, the sum of formato eclosed the sample become creasg, as a result varaces of the estmates become dmsh. The expressos for the complete sample Bayes estmates ad ther varaces are smplfed as Table : The Lmtg Expresso for the Bayes Estmators usg Uform ad Jeffreys Prors Parameters Bayes estmates (Uform) Bayes estmates(jeffreys) lm x T lm x T x j j ( ) 4 x j j ( ) 4 lm x T lm x T x j j ( ) x j j ( ) p lm p x lm p x T T Table : The Lmtg Expresso for the Bayes Posteror Rss usg Uform ad Jeffreys Prors Parameters Uform Pror Jeffreys prors lm ( ) T lm ( ) T 8( x j ) j x ( 4) ( 6) 8( x j ) j x ( 4) ( 6) ( )( ) ( ) ( ) lm ( ) T lm ( ) T 8( x j ) j x ( ) ( 4) 8( x j ) j x ( ) ( 4) ( )( ) ( ) ( ) p lm ( p x ) lm ( p x ) T T 07

12 Cetre for Promotg Ideas, USA Table : Bayes Estmators usg formatve prors ad ML estmator as T Parameters Bayes estmates (IG) * Bayes estmates(ic) E x E x xj b j ( ) ( a ) xj b j ( ) ( a ) xj b j ( ) E x ( a ) xj b j ( ) E x ( a ) p E p x E p x f we use these estmators, our posteror rs wll be small because we are usg complete formato of data as compared to cesored oe. Table 4: Lmtg Expresso for the Posteror rss of formatve prors as T Parameters Cojugate Pror Ch Square pror 8( xj b) j x a a ( ) ( ) ( 4) ( xj b) j x a a ( ) ( ) ( 4) 8( xj b) ( xj b) j j ( x ) ( x ) ( a ) ( a 4) a a ( p )( ) ( p x ( )( ) ) ( p x ) ( ) ( ) ( ) ( 4) ( ) ( ) * Where IG & IC represets Iverted Gamma dstrbuto ad Iverted Ch-square dstrbuto respectvely. 9. Smulato Study A thorough smulato study was coceded order to vestgate the performace of the Bayes estmators, mpact of sample sze ad cesorg rate the ft of model. Sample szes,,,,5 were geerated accordg to the crtera suggested by Krsha ad Mal (9) (for smple Maxwell dstrbuto) from the two compoet mxture of Maxwell dstrbuto wth parameter, ad p such that (, ) (0.5,0.8),(0.8,.5) p Probablstc mxg was used to geerate the mxture data. For each observato a radom ad (0.5, 0.4) umber was geerated from the uform o (0,) dstrbuto. If p, the observato was tae radomly from f ( x ) (the Maxwell dstrbuto wth parameter ) ad f p, the observato was tae radomly from f ( x ) (the Maxwell dstrbuto wth parameter ). Rght cesorg s carred out usg a fxed cesorg tme t. All observatos whch are greater tha T are stated as cesored oes. fferet fxed cesorg tmes t are chose to evaluate the mpact of cesorg rate o the estmates. The choce of the cesorg tme s made such a way that the cesorg rate the resultg sample to be approxmately 0% or 0%. For each of the dfferet combato of parameters, sample sze ad cesorg rate, dfferet sze of samples were geerated usg route Excel. I each case oly falures are detfed to be a member of ether Subpopulato or Subpopulato of the mxture. For each of the 0 samples, the Bayes estmates were computed usg a route Mathematca ad the results are preseted Table 5- gve appedx.the smulato study (appedx) provdes us some terestg propertes of the Bayes estmates. The propertes of the estmates are hghlghted term of sample szes, sze of mxg proporto parameters, sze of the compoet destes parameters, dfferet loss fuctos ad cesorg rates. It s observed that due to cesorg, the posteror rss of all three mxture parameters are reduced wth a crease sample sze. 08

13 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 Oe ca easly observe that the parameters of the compoet destes are geerally over-estmated wth a few exceptos case of the secod compoet. The extet of over-estmato s hgher case of the frst compoet desty parameter. O the other had the estmates of the mxg proporto parameter are observed to be uderestmated wth few values wth crease sample sze. Aother mportat pot cocerg about choce of loss fucto, SELF has less posteror rs tha PLF, however uderestmato some extet s preveted PLF. If we mae comparso betwee both uformatve (Uform ad Jeffreys) prors the due to less posteror rs the Jeffreys pror s more preferable tha the uform pror. Also comparso betwee formatve prors, the IC (Iverted Ch-square) provdes us less Bayes posteror rs tha IG (Iverted Gamma) pror so IC pror s more sutable for ths case. I over all comparso of formatve prors o the behalf of less posteror rs are more preferable tha oformatve prors ad especally the IC formatve pror s more preferable preset study. 0. Real Lfe Applcato The burg velocty s the velocty of a lamar flame uder stated codtos of composto, temperature, ad pressure. The burg velocty s a mportat parameter whch characterzes the hbto effcecy of halogecotag addtve employed as flame retardats. The burg velocty decreases wth creasg hbtor cocetrato. It ca be determed by aalyzg the pressure tme profles the sphercal vessel ad were checed by drect observato of flame propagato. The data related to the burg velocty of dfferet chemcal materals avalable at the webste ( com/msts.pdf.). ata partto for mxture dstrbuto s gve Appedx. Table : BEs ad PRs usg UP ad JP uder SELF for real data. Pror UP JP E( E( E( p E( E( p Cesorg tme= (9894) (5469.9) (0.44) (9477) (585.8) ( ) (9758.5) (0.54) (8585) (65758.) (954.) (9956.) (0.54) (85.) (8894.9) (6584.9) (666) (0.498) (8767.8) ( ) Cesorg tme= 70 p E( E( E( p E( E( (940) (6470.4) (0.89) (4796) (60.) (9786) (99998.) (0.498) (8844.0) (995.5) ( ) (94) (0.498) (9064.9) (9675.8) (687.4) (7048) (0.465) (6755.5) (6) E( p (0.44) 0.40 (0.5) (0.54) 0.54 (0.54) E( p 0.6 (0.89) (0.498) 0.5 (0.498) (0.464) 09

14 Cetre for Promotg Ideas, USA Table 4: BEs ad PRs usg UP ad JP uder PLF for real data. Pror UP JP E( E( E( p E( E( p Cesorg tme= (44.99) (4.54) (0.0569) (6.768) (.744) (57.49) (55.090) (0.058) (.054) (4.880) (55.9) ( ) (0.57) (5.4906) (5.96) (40.695) (90.70) (0.770) (5.5687) (54.658) Cesorg tme= 70 p E( E( E( p E( E( (7.477) (5.9790) (0.0448) (.09) (4.5868) ( ) (5.458) (0.554) (5.984) (.575) (5.9) (57.5) (0.946) (.0446) (5.9) ( ) (86.079) (0.78) (8.874) ( ) E( p (0.0569) (0.0944) (0.055) (0.57) E( p 0.7 (0.0448) (0.554) 0.56 (0.946) (0.78) Table 5: BEs ad PRs usg Cojugate ad IC Prors uder SELF for real data. Pror Cojugate Pror Iverted Ch-square Pror, E( E( E( p E( E( E( p p Cesorg tme = (854) (6.) (0.44) (99456) (4785.4) (0.44) (698) (45547.) (0.5) (458) (475.5) (0.5) (75.7) (7656.) (0.54) (7054.6) (5697.9) (0.54) (5567.) (654) (0.498) 5. (5757.8) (674.) Cesorg tme = 70 p E( E( E( p E( E( (9670) (5767.) (0.89) (7706) (549.7) (8840) (89964.) (0.498) (8804.) (89.) (7857.8) (8597) (0.498) (780.6) (6609.) (4559.) (40456) (0.467) (999.7) (46) (0.498) E( p 0.6 (0.89) (0.498) 0.5 (0.498) (0.4647) 0

15 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 Table 6: BEs ad PRs usg cojugate ad IC prors uder PLF for real data. Pror Cojugate Pror Iverted Ch-square Pror, E( E( E( p E( E( E( p p Cesorg tme = (7.968) (.857) (0.0569) (.9086) (.58) (0.0569) (6.708) (9.997) (0.0944) (6.7) (7.9) (0.0944) (48.66) (45.4) (0.055) (46.89) (.458) (0.055) (6.788) (79.) (0.770) (5.60) Cesorg tme = (6.8488) p E( E( E( p E( E( (09.707) (9.986) (0.484) (98.996) (9.6) (5.9054) ( ) (0.554) (54.05) (44.05) (46.4) (5.0759) (0.946) (46.796) (8.86) (7.8) ( ) (0.759) (.46) (75.559) (0.4980) E( p 0.90 (0.484) (0.0554) 0.56 (0.946) (0.759) From Tables. -6, oe ca easly made comparso betwee results of uform pror ad Jeffery pror wth ther respectve posteror rs whch are gve parethess ad ca cocludes that Jeffery pror has less varace (posteror rs) as compare to the uform pror. Partcularly whe we use cesorg tme = 65 ad cesorg tme = 70 our results are more precse. If we compare both formatve prors, the IC pror has less posteror rs tha the IG. I the same way the comparso betwee uformatve ad formatve prors, the IC provdes less posteror rs so IC pror s more sutable pror. 0.. Graphcal presetato of Margal posteror estes The graphs of the margal posteror dstrbutos for the parameters usg Uform ad Jeffrey prors for real data set. M P X arg al of Poste ror Curves UP JP Fg Fg.

16 Cetre for Promotg Ideas, USA P p X 5 M arg al of p Poste ror M P X arg al of Poste ror Curves UP JP Curves IG IC Fg. M P X arg al of Poste ror p P p X Fg.4 M arg al of p Poste ror Curves IG IC Curves IG IC Fg.5 Fg.6 All graphs show smlar patter.e. postve sewed, wth mer dfferece. p. Cocluso The smulato study has dsplayed some fascatg propertes of the Bayes estmates. The posteror rs of the parameter estmates seems to be farly large cases whe the values of the parameters are large ad farly small for relatvely smaller values of parameters. O the other had ay case the posteror rs of estmates of both parameter ad are reduced as the sample sze creases. A further terestg observato about cesorg the posteror rs of the estmates of ad s that creasg or decreasg the proporto of a compoet the mxture reduces (creasg) the aalogous parameter s estmate. The cosequece of cesorg o s the form of overestmato (uderestmato) f s less tha or s greater tha. To be more precse, larger degree of cesorg tme results bgger szes of over or uderestmato. O the other had the parameter p s ether uderestmated or overestmated depedg upo the values of ad. To be more precse, p s over estmated or uderestmated wheever or.the level of ths over or uder estmato s drectly proportoal to amout of cesorg rates ad versely proportoal to the sample sze. Also the level of over or uder estmato s more tesve for larger parameter values of p. Further, the crease sample sze reduces the posteror rs of estmate of p.the crease proporto of a compoet the mxture does ot guaraty the reducto posteror rs of p. As the cut off sesor value gets fty, the complete sample estmators ad posteror rss are greatly smplfed. Also posteror rss of the complete sample estmates are expected to be reduced further as these are clear from the effect of cesorg tme. All the Bayes estmates get more precse wth the crease sample sze such that posteror rs usg Jeffreys pror s less tha the posteror rs of Uform pror. I real lfe example, the estmates ad are uder estmated but much greater tha the respectve sample mea lfetme hours what s expected cesored samples.

17 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 The estmate of the mxg proporto parameter p s the same as that of the correspodg Medehall ad Hader (958) estmate. I case of formatve prors, posteror rss usg Iverted Ch-Square Pror are less tha the posteror rss of Cojugate (Iverted Gamma) Pror. So o based o smulato study, we suggest at least sample sze for ths type study. The posteror rss uder SELF are less tha the posteror rss uder PLF; however uderestmato s preveted PLF. I future ths wor ca be exteded usg mxture of trucated Maxwell dstrbuto ad tag beta pror for mxg compoet. Acowledgemet Prof. r Hare Krsha (Chaudhary Chara Sgh Uversty. Meerut (U.P.Ida), r Abd Suler (Sustaable evelopmet Polcy Isttute) ad revewers of IJAS, who deserves specal thas for ther ecouragemet ad provdg us useful commets order to presets ths research wor a beautful maer. Refereces Aslam, M. (), A applcato of pror predctve dstrbuto to elct the pror desty. Joural of Statstcal Theory ad Applcatos, Vol. (), pp Bayes, T. (76): A essay towards solvg a problem the doctre of chaces. Phlo Tras. Joural of the Royal Statstcal Socety, 5, (Reprted (958) Bometra 45, 9-5) Beer, A. ad Roux, J.J. (5), Relablty characterstcs of the Maxwell dstrbuto: a Bayes estmato study, Comm. Stat. (Theory & Meth), Vol. 4 No., pp Berardo, J.M. (979b): Referece Posteror strbuto for Bayesa Iferece (wth dscusso). Joural of the Royal Statstcal Socety, Seres B, 4, -47. Berardo, J.M. ad Smth, A.F.M. (994): Bayesa Theory. Chchester: Joh Wley &Sos. Box, G.E.P. ad Tao, G.C. (97): Bayesa Iferece Statstcs Aalyss. New Yor: Wley & Sos. Chaturved, A. ad Ra, U,(998), Classcal ad Bayesa relablty estmato of the geeralzed Maxwell falure dstrbuto, Joural of Statstcal Research Vol., pp -0. Garthwate H. P., Kadae, B. J. ad O Haga, A. (4), Elctato, Worg paper Uversty of Sheffeld, avalable at Gauss, C.F. (80). Methode des Modres Carres Morrre sur la Combatos des Observatos. Traslated by J. Bertrad (955). Mallet-Bacheler, Pars. Gesser, S. (984): O Pror strbuto for Bary Trals. The Amerca Statstca, 8(4) pp 4-5. J. Berger, Statstcal decso theory ad Bayesa aalyss, Sprger Verlag, Ic., New Yor, 985. J. Kalbflesch, R. Pretce, The Statstcal Aalyss of Falure Tme ata, Joh Wley & Sos, Ic., New Yor,. J. Kle, M. Moeschberger, Survval Aalyss Techques for Cesored ad Trucated ata, Sprger-Verlag, Ic., New Yor, 997. Jeffreys., H. (96): Theory of Probablty. Oxford, UK: Claredo Press K. etz, M. Gal, K. Krceberbg, B. Sger, Itroducto to Survval Aalyss, Sprger-Verlag, Ic., New Yor, 985. Krsha, H ad Mal, M. (9): Relablty estmato Maxwell dstrbuto wth Type-II cesored data. Iteratoal Joural of Qualty & relablty maagemet Vol.6 No., pp L. Leems, Relablty: Probablstcs Models ad Statstcal Methods, Pretce-Hall, Eglewood Clffs, New Jersey, 955. Laplace, P.. (8): Theore Aalytque des Probabltes, Pars Courar. The Secod, thrd, ad fourth edtos appeared 84, 88 ad 80 respectvely. It s reprted euvres completes du Laplace, 847. Pars: Gauther-Vllars. Legedre, A. (805): Nouvelles Methods Pour La etermato des Orbtes des Cometes Courcer, Pars. Medehall, W, Hader, R. A. (958). Estmato of parameter of mxed expoetally dstrbuto falure tme dstrbuto from cesored lfe test data. Bomera 45 (-4), 4-. Norstrom, J. G. (996), The use of precautoary loss fuctos rs aalyss, IEEE Tras. Relab. Vol. 45, N (), PP P.Smth,AalyssofFalureadSurvvalata,Chapma&Hall/CRC,New Yor,.

18 Cetre for Promotg Ideas, USA S. Jag,. Kececoglu, Maxmum lelhood estmates from cesored data for mxed-webull dstrbutos, IEEE Trasactos o Relablty 4 () (99) Saleem, M. ad Aslam, M. (9), O Bayesa Aalyss of the Ralegh Survval Tme assumg the Radom Cesor Tme. Pa. J. Statst. Vol. 5().pp. 7-8 Saleem, M. Aslam, M. ad Ecoomus, P. (). O the Bayesa aalyss of the mxture of the power dstrbuto usg the complete ad cesored sample. Joural of Appled Statstcs,Vol.7, No., pp Saara PG, Nar, MT (5). O fte mxture of Pareto dstrbutos. CSA Bullet, New elh. Sha, S.K. (998). Bayesa Estmato, New Age Iteratoal (P) lmted,publsher, New elh Tyag, R.K. ad Bhattacharya, S.K. (989a), Bayes estmato of the Maxwell s velocty dstrbuto fucto, Ststca, Vol.9. No 4, pp.56-7 Tyag, R.K. ad Bhattacharya, S.K. (989b), A ote o the MVU estmato of relablty for the Maxwell falure dstrbuto, Estadstca, Vol.4 No. 7. Appedx Followg tables represets the Bayes estmates (BEs) ad Posteror Rs (PRs) usg dfferet formatve ad oformatve prors uder dfferet loss fuctos. Table 5: BEs ad PRs usg UP ad JP uder SELF whe p=0.5. Pror UP JP E( E( E( p E( E( Cesorg tme=0.6, =0.5, = (0.0596) (0.4) (0.076) (0.04) ( ) (0.07) (0.04) (0.054) (0.05) (0.064) (0.04) (0.045) (0.00) (0.) (0.004) (0.89) (0.047) (0.095) (0.09) (0.5) Cesorg tme=, =0.8, =.5 E( E( E( p E( E( (0.5476) (0.957) ( ) (0.478) (0.55) (0.406) (0.069) ( ) (0.094) (0.695) (0.09) (0.065) (0.09) (0.00) (0.04) (0.56) (0.09) (0.4) (0.6) (0.09) E( p (0.0664) (0.045) 0.85 (0.056) 0.46 (0.09) E( p (0.048) (0.049) (0.4) (0.074) 4

19 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 Table 6: BEs ad PRs usg UP ad JP uder SELF whe p=0.40. Pror UP JP E( E( E( p E( E( Cesorg tme=0.6, =0.5, = (0.0574) ( ) (0.089) (0.07) ( ) (0.075) ( ) ( ) (0.006) (0.0667) (0.0896) (0.0058) (0.0749) (0.98) (0.049) (0.9) (0.064) (0.0857) (0.5) (0.064) Cesorg tme=, =0.8, =.5 E( E( E( p E( E( (0.06) (.5678) (0.0607) (0.) (.6955) (0.0845) (0.5455) (0.67) (0.0546) (0.4705) (0.079) (0.045) (0.79) (0.058) (0.7) (0.0548) (0.0) (0.58) (0.4) (0.88) E( p (0.06) (0.060) (0.946) (0.45) E( p (0.0468) (0.546) (0.) 0.4 (0.4) Table 7: BEs ad PRs usg UP ad JP uder PLF whe p=0.5. Pror Uform Pror (UP) Jeffreys Pror (JP), E( E( E( p E( E( Cesorg tme=0.6, =0.5, = ( ) ( ) (0.05) ( ) (0.76) ( ) (0.0748) (0.0579) (0.055) (0.784) ( ) ( ) (0.0859) (0.0708) (0.596) 0.5 (0.5957), Cesorg tme =, =0.8, = (0.0540) (0.045) (0.057) (0.080) E( E( E( p E( E( (0.5570) (0.09) ( ) (0.5966) (0.54) (0.487) (0.8585) (0.099).659 (0.05).5656 (0.045). (0.05) 0.09 (0.5560) ( ) ( ) 0.97 (0.0658) (0.055) (0.0859).6668 (0.069) (0.089).5469 (0.749) E( p (0.07) (0.589) ( ) ( ) E( p (0.5999) ( ) ( ) (0.068) 5

20 Cetre for Promotg Ideas, USA Table 8: BEs ad PRs usg UP ad JP uder PLF whe p=0.40. Pror Uform Pror (UP) Jeffreys Pror (JP), E( E( E( p E( E( E( p Cesorg tme=0.6, =0.5, = (0.095) (0.087) (0.568) ( ) ( ) (0.60) ( ) (0.0580) (0.874) (0.0568) ( ) (0.0989) ( ) (0.0669) (0.0475) ( ) (0.0) 0.8 (0.955), Cesorg tme =, =0.8, = ( ) (0.0807) (0.044) E( E( E( p E( E( (0.545) (0.0966) (0.045) ( ).0967 ( ) (0.708) (0.0804).5988 (0.6609) (0.0689) (0.0447) (0.455) (0.699) (0.0944) ( ) ( ) (0.0444) (0.77).6945 (0.6895).5888 (0.09).588 (0.5856) (0.065) (0.0447) ( ) E( p 0.49 (0.004) (0.097) (0.4) 0.44 (0.997) Table 9: BEs ad PRs usg Cojugate ad IC uder SELF whe p= Pror Cojugate Pror Iverted Ch-square Pror, E( E( E( p E( E( E( p Cesorg tme=0.6, =0.5, = (0.67) (0.0679) (0.04) (0.6) (0.065) (0.08) (0.79) (0.085) (0.084) (0.77) (0.06) (0.097) (0.586) (0.08) (0.04) (0.579) (0.98) (0.98) (0.49) 0.7 (0.4) (0.46) (0.7) 0.8 (0.) 0.88 (0.) (0.46) 0. (0.09), Cesorg tme =, =0.8, = (0.5) (0.) E( E( E( p E( E( (0.745) (0.0687) (0.09) (0.864) (0.05) (0.085) (0.068) (0.0987) (0.06).6 (0.067) (0.80) (0.99) (0.89) (0.44) 0.56 (0.04) (0.74) (0.0678) (0.075) (0.70) (0.0).5847 (0.55) (0.0).588 (0.064).55 (0.0578).56 (0.7) (0.) 0.46 (0.07) E( p (0.8) (0.74) 0.57 (0.77) 0.74 (0.4) (0.074)

21 Iteratoal Joural of Appled Scece ad Techology Vol. No. ; Jauary 0 Table 0: BEs ad PRs usg Cojugate ad IC uder SELF whe p=0.40. Pror Cojugate Pror Iverted Ch-square Pror, E( E( E( p E( E( E( p Cesorg tme=0.6, =0.5, = (0.0868) (0.0640) (0.0459) (0.6) (0.065) (0.08) (0.06) (0.04) (0.067) (0.77) (0.06) (0.97) (0.57) (0.058) (0.98) (0.579) (0.98) (0.) (0.856) (0.08) (0.) (0.46) (0.5) (0.07) (0.76) (0.7) (0.857) (0.5) (0.064) (0.7) Cesorg tme =, =0.8, =.5 E( E( E( p E( E( E( p (0.069) (0.769) (0.8) ( ) (0.64) (0.7) (0.04) (0.5608) (0.58) (0.068) (0.70) (0.) (0.087) (0.044) (0.59) (0.85) (0.06) (0.) (0.6) (0.) (0.048).87 (0.765) (0.) (0.9) (0.0965) (0.08) (0.08).745 (0.694) (0.6) (0.6) Table : BEs ad PRs usg Cojugate ad IC uder PLF whe p=0.5. Pror Cojugate Pror Iverted Ch-square Pror, E( E( E( p E( E( E( p Cesorg tme=0.6, =0.5, = (0.0999) (0.0947) ( ) ( ) ( ) (0.0756) (0.0) (0.040) (0.0946) (0.004) ( ) ( ) (0.088) (0.096) (0.040) (0.674) (0.0546) (0.048) (0.898) (0.46) (0.047) (0.7849) (0.88) (0.086) (0.4459) (0.44) ( ) (0.46) (0.8) (0.0848), Cesorg tme =, =0.8, =.5 E( E( E( p E( E( E( p (0.905) (0.5) ( ) (0.8840) ( ) (0.0899) ( ) (0.064) (0.076) ( ) (0.0694) (0.084) (0.0604) (0.089) (0.0067) ( ) (0.47) ( ) (0.57) (0.09) (0.75).657 (0.84) (0.0608) ( ) (0.849) (0.59) (0.408).54 (0.680) (0.06) (0.0646) 7

22 Cetre for Promotg Ideas, USA Table : BEs ad PRs usg Cojugate ad IC prors uder PLF whe p=0.40. Pror Cojugate Pror Iverted Ch-square Pror, E( E( E( p E( E( E( p Cesorg tme=0.6, =0.5, = (0.048) (0.0475) ( ) (0.0695) (0.58) (0.0775) (0.0908) (0.0695) (0.0489) (0.06) ( ) (0.0759) ( ) 0.40 (0.0448) (0.4) ( ) (0.004) (0.674) (0.7849) 0.87 (0.9909) Cesorg tme =, =0.8, = ( ) (0.0706) (0.0546) (0.88) (0.0449) E( E( E( p E( E( (0.074) ( ) (0.0658) (0.0448) (0.0).6590 (0.07) (0.606).5409 (0.0777).7875 (0.9485).47 (0.89) 0.48 (0.0578) (0.045) (0.75) (0.9) 0.47 (0.788) Real data set for mxture of Maxwell model as followg: Cesorg tme= (0.064) (0.008) ( ) (0.046) (0.070) ( ) ( ) (0.04).686 (0.908).4054 (0.468) r r j j j j ( ) (0.0685) (0.0847) (0.059) (0.) E( p (0.044) (0.888) 0.49 (0.79) (0.707) (0.76) p 0.5, 4, 4, r, r, x 888, x 789, 0.070, r r j j j j p 0.45, 5,, r 0, r, x 485, x 56958, , r r j j j j p 0., 8, 8, r, r, x 5568, x 49589, 0.665, r r j j j j p 0.60, 6, 4, r 8, r 5, x 6867, x 6906, 0.484, Cesorg tme= 70. r r j j j j p 0.5, 4, 4, r, r 5, x 6, x 9040, 0.0, r r j j j j p 0.45, 5,, r, r 5, x 598, x 6598, 0.849, r r j j j j p 0., 8, 8, r 5, r, x 65, x 5945, 0.670, r r j j j j p 0.60, 6, 4, r 0, r 7, x 7847, x 4609, 0.55,