Estimation of a Mixture of Two Weibull Distributions under Generalized Order Statistics

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1 IOSR Joural of Mathematics (IOSR-JM) e-issn: , p-issn: X Volume 12, Issue 2 Ver V (Mar - Apr 2016), PP wwwiosrjouralsorg Estimatio of a Mixture of Two Weibull Distributios uder Geeralized Order Statistics Neamat S Qutb 1, Samia A Adham 2,Nojoud K Dadei 3 1,2,3 (Departmet Of Statistics, Faculty Of Sciece/ Kig Abdulaziz Uiversity, Saudi Arabia) Abstract:This paper deals with the estimatio of the parameters, reliability ad hazard rate fuctios of the mixture of two Weibull distributios (MTWD), with a commo shape parameter, based o the geeralized order statistics (GOS) The maximum likelihood ad Bayes methods of estimatio are used for this purpose with stadard errors ad credible itervals The Markov Chai Mote Carlo (MCMC) method is used for obtaiig Bayes estimates uder the squared error loss fuctio Our results are specialized to progressive Type II cesorig ad Type II cesorig Comparisos are made betwee Bayesia ad maximum likelihood estimates ad betwee the two cesorig types, progressive Type II cesorig ad Type II cesorig A real data set is used for illustratio purpose Keywords: Mixture of two Weibull distributios, geeralized order statistics (GOS ), Maximum likelihood estimatio, Bayesia estimatio, Markov Chai Mote Carlo I Itroductio The fiite mixture of distributios have multiple uses i a various scietific fields such as physics, biology, medicie ad idustrial egieerig, amog others I fact, i life testig, each failure occurs may ot has oe type of failure, it could be categorized to more tha oe type Moreover, the failure time populatios could be heterogeeous sice it could be cosistig of weak compoets correspodig to short lives ad strog compoets correspodig to log lives Mixtures of Weibull distributios are widely used to model lifetime data That is because of the flexibility of the Weibull distributio i modelig both icreasig ad decreasig failure rates This flexibility maily depeds o the shape parameter; which led us, whe estimatig the distributio parameters, to let the value of this parameter to be kow ad cotrolled by the researcher carryig out the applicatio Where, it could be selected accordig to the type of failure rate that fits the data For the icreasig failure rate data, the shape parameter ca be set to a value greater tha oe While, for the decreasig failure rate data, the shape parameter ca be set to a valueless tha oe; but for the costat failure rate data, the value of the parameter will be equal to oe This study is cocered with studyig the fiite mixture of two Weibull distributios, deoted by MTWD, as a lifetime model with two ukow scale parameters ad a commo kow shape parameter Some of the most importat refereces that discussed differet types of mixtures of distributios are the moographs by [1], [2], [3] ad [4] [5]cosidered Bayesia estimatio of the mixig parameter, mea ad reliability fuctio of a mixture of two expoetial lifetime distributios based o right cesored samples[6] studied ad compared classical ad Bayesia estimates of the parameters of a fiite mixture of two Gomperetz lifetime models based o simulated data sets with Type I ad Type II cesorig [7]obtaied Bayesia predictive desity of order statistics based o fiite mixture models Based o Type I cesored samples from a fiite mixture of two trucated Type I geeralized logistic compoets,[8] computed Bayes estimates of parameters, reliability ad hazard rate fuctios[9]cosidered estimatio for the parameters of mixed expoetial distributio based o record statistics[10]cosidered Bayes iferece uder a fiite mixture of two compoud Gompertz compoets model [11]cosidered estimatio forthe parameters of mixture of two compoet expoetiated gamma distributio [12]applied the order statisticso a mixture model of expoetiated Rayleigh ad expoetiated expoetial distributios [13]estimated the parameters of a two-parameter weighted Lidley distributio based o hybrid cesorig[14]itroduced the fiite mixture of two expoetiatedkumaraswamy distributios Mixtures of Weibull distributios are widely used to model lifetime data ad they have bee cosidered extesively by may authors, [15] estimated the parameters of a mixture Weibull distributio usig MLE ad Bayes estimatio uder Type I cesorig [16]studied the classificatio of the agig properties of geeralized mixtures of two or three Weibull distributios i terms of the mixig weights, scale parameters ad a commo shape parameter [17]estimated the parameters from the mixture of two Weibull distributios uder the iformative ad o-iformative priors they also determied, the Bayes predictive itervals A mixture of two ad three Weibull distributios were used to aalyze the data of failure times [18] [19]proposed a mixture Weibull proportioal hazards model to predict the failure of a mechaical system with multiple failure modes DOI: / wwwiosrjouralsorg 25 Page

2 Estimatio Of A Mixture Of Two Weibull Distributios Uder Geeralized Order Statistics II Model Descriptio The cosidered MTWD, which is produced from mixig two Weibull distributios, the first with two parameters, the scale parameterα 1 > 0, ad the shape parameter, β > 0 ad the secod also has the scale parameterα 2 > 0, ad the shape parameter, β > 0That is the scale parameters of the MTWD are α 1 ad α 2 ad the commo shape parameterisβ, which is assumed as a kow parameter, its value could be selected (greater, less or equals 1) accordig to the type of the failure rate that fits the data set uder the study The probability desity fuctio(pdf) of MTWDis give as f t = pf 1 t + 1 p f 2 t, t > 0 (1) where0 p 1,ad the PDF of the j th compoet, j = 1,2, is f j t = βtβ 1 α β exp t j α j β, t > 0 (2) The cumulative distributio fuctio(cdf), reliability fuctio(rf) ad hazard fuctio (HF) of the MTWD are,respectively, give by F t = p 1 exp t β + (1 p) 1 exp t β, (3) α 1 α 2 R t = p exp t α 1 β + (1 p) exp t α 2 β (4) where, H t = g t H 1 1 t g t H 2 t, R j t = exp t α j g t = 1 p R 2 t pr 1 t,, H j t = βtβ 1, j = 1,2 α β j (5) The mea, variace, skewess ad kurtosis of the MTWDare computed for differet values of the parameters ad are give i Table (1) Table (1): Mea, variace, skewess ad kurtosis of MTWD θ = (p, β, α 1, α 2 ) Mea variace skewess kurtosis θ = (03,1,05,15) θ = (03,2,05,15) θ = (03,35,05,15) θ = (01,3,1,2) θ = (05,3,1,2) θ = (08,3,1,2) θ = (06,3,15,28) θ = (06,3,2,28) θ = (06,3,35,28) θ = (07,2,15,11) θ = (07,2,15,28) θ = (07,2,15,34) Table (1) shows that, whe the parameters (p, α 1, α 2 ) are fixed, the the relatioship betwee the shape parameter βad each of the skewess ad kurtosis is iverse However, whe (β, α 1, α 2 ) are fixed we have positive relatioship betwee the mixig proportio parameter pad each of the skewess ad kurtosis With the fixed(p, β, α 2 ) we have egative relatioship betwee scale parameter α 1 ad each of the skewess ad kurtosis With the fixed (β, p, α 1 ) we have positive relatioship betwee scale parameter α 2 ad skewess ad kurtosis DOI: / wwwiosrjouralsorg 26 Page

3 Estimatio Of A Mixture Of Two Weibull Distributios Uder Geeralized Order Statistics III Maximum Likelihood Estimatio Let T 1;,m,k, T 2;,m,k,, T ;,m,k, k > 0, m = m 1,, m R, m 1,, m R, are GOS draw from the MTWD The likelihood fuctio (LF) is give i [20], for < t 1,, t <, by L θ t = k γ i where, t = t 1,, t, θεθis parameter space, ad R t m i i f t i R t k 1 f t, (6) γ i = k + i + M i > 0, M i = Substitutig (1) ad (4), i (6), the the likelihood fuctio takes the form of L θ t pr 1 t + (1 p)r 2 t k 1 pr 1 t i + p 2 R 2 t i m i ν=1 (1 p)f 1 t i + p 2 f 2 t i Takig the logarithm of (7), to obtai l θ = l L θ t k 1 l pr 1 t + (1 p)r 2 t + m i l pr 1 t i + (1 p)r 2 t i + l pf 1 t i + (1 p)f 2 t i Differetiatig (8) with respect to the parameters p ad α j, j = 1,2 ad equatig to zero gives the followig likelihood equatios l p = k 1 θ t + l α 1 = p k 1 ψ 1 t + l α 2 = (1 p) k 1 ψ 2 t + Where, for j = 1, 2 ξ 1 t i m i θ t i ξ 2 t i ψ 1 t i + + θ t i ψ 2 t i + = 0 m i ψ 1 t i m ν m i ψ 2 t i = 0 = 0 (7) (8) (9) f 1 t i f 2 t i θ t i = pf 1 t i + (1 p)f 2 t, R 1 t i R 2 t i θ t i = i pr 1 t i + (1 p)r 2 t i f j t i ψ j t i = pf 1 t i + (1 p)f 2 t, ψ j t R j t i βt β β 1 i α j i = i pr 1 t i + (1 p)r 2 t i ξ j t i = β + βt β β 1 α i α j j The solutio of the three oliear likelihood equatios i (9) usig umerical method, yields the maximum likelihood (ML) estimates p, α 1 ad α 2 The ML estimates of the R(t) ad the H(t) are give, respectively, by (4) ad (5) after replacig θ = (p, α 1, α 2 ) by their correspodig ML estimates IV Bayesia Estimatio This sectio deals with Bayesia usig cojugate ad o-iformative priors 41 Bayesia estimatio usig cojugate prior Let p, α 1 ad α 2 are idepedet radom variables such that p follows Beta(b 1, b 2 ) ad for j = 1, 2, α j to follow a iverted gamma prior distributio with PDFs, respectively, give by 1 π p = β(b 1, b 2 ) p b1 1 (1 p) b 2 1, 0 p 1, b 1, b 2 > 0 DOI: / wwwiosrjouralsorg 27 Page

4 Estimatio Of A Mixture Of Two Weibull Distributios Uder Geeralized Order Statistics π α j = γ j τ j τ α j 1 Γτ j e γ j α j, α j > 0, j wherej = 1, 2 α j > 0, (b j, τ j, γ j ) > 0 A joit prior desity fuctio of θ = (p, α 1, α 2 )is the give by π θ = π p π α 1 π α 2 π θ p b 1 1 (1 p) b j=1 α j τ j 1 e γ 2 j j=1α j It follows, from (10) ad (7), that the joit posterior desity fuctio is give by π θ t = A 1 1 p b p b 2 1 where, 2 j =1 α j τ j 1 e A 1 1 = 2 γ j j =1α j (10) pr 1 t + (1 p)r 2 t k 1 pf 1 t i + (1 p)f 2 t i θ L θ t π(θ)dθ Uder the squared error loss fuctio (SE), the Bayes estimator of a fuctio, say φ φ(p, α 1, α 2 ),is give by φ Bs = E φ t = θ, (11) φπ θ t dθ, (12) where the itegral is take over the three dimesioal space To compute the itegral we propose to cosider MCMC methods The coditioal posterior distributio of the parameters p, α 1 ad α 2 usig cojugate prior ca be computed ad writte, respectively, by π p α 1, α 2, t p b p b 2 1 pr 1 t + (1 p)r 2 t k 1 pf 1 t i + (1 p)f 2 t i π α i p, α j, t α i θ 1 1 e γ 1 α i pr 1 t + (1 p)r 2 t k 1 pf 1 t i + (1 p)f 2 t i where, i j iadj = 1,2 42 Bayesia estimatio usig o-iformative prior Assumig that all of the parameters cosistig θ are positive ad idepedet, ad that we are igorig the prior iformatio about θ so that we set improper o-iformative prior to α j, j = 1, 2, ad p as follows π p 1, π α 1 1 α 1, π α 2 1 α 2 Hece, the joit prior desity fuctio ofθ = (p, α 1, α 2 ) is the give by π θ = π 1 (p)π 2 (α 1 )π 3 (α 2 ) π θ 2 j =1 (α j ) 1 (13) Forj = 1,2,α j > 0, it follows, from (13) ad (7), that the joit posterior desity fuctio is give by where, π θ t = A j =1 α j 1 pr 1 t + (1 p)r 2 t k 1 pf 1 t i + (1 p)f 2 t i,, (14) A 2 1 = θ L θ t π(θ)dθ DOI: / wwwiosrjouralsorg 28 Page

5 Estimatio Of A Mixture Of Two Weibull Distributios Uder Geeralized Order Statistics Uder the squared error loss fuctio (SE), the Bayes estimator of a fuctio, say φ φ(p, α 1, α 2 ), is give i (12) where the itegral is take over the three dimesioal space To compute the itegral we propose to cosider MCMC methods The coditioal posterior distributio of the parameters p, α 1 ad α 2 usig cojugate prior ca be computed ad writte, respectively, by π p α 1, α 2, t pr 1 t + (1 p)r 2 t k 1 where, i j iadj = 1,2 pf 1 t i + (1 p)f 2 t i π α i p, α j, t α i 1 pr 1 t + 1 p R 2 t k 1 pf 1 t i + (1 p)f 2 t i, V Numerical Computatios I this sectio, the computatios regardig the comparisos are performed ad the MTWD with a commo parameter β = 15 is cosidered A compariso betwee the ML ad Bayesia estimates, uder the squared error loss fuctio (SE), is made by usig Mote Carlo simulatio study (based o progressive Type II cesorig ad Type II cesorig) which is cosidered i subsectio 51 ad a real data set is cosidered i subsectio Mot Carlo simulatio (1) Based o progressive Type II cesored samples The estimatio of the parameters p, α 1, α 2 usig ML ad Bayesia techiques o cases whe parameter β is kow have bee obtaied, based of progressive Type II cesored sample which is special case of GOS that ca be derived from it, by choosig m i = R i ; i = 1,2,, m 1 γ i = R r + 1 (15) Therefore, the ML ad Bayes estimates of the ukow parameters ca be obtaiedby substitutig (15) i (6),ad the the likelihood reduces to the followig L θ t m pr 1 t i + (1 p)r 2 t i R i pf 1 t i + (1 p)f 2 t i We compare the performace of the ML ad Bayes estimates with their stadard deviatio (SD), bias, mea squared errors(mse) ad the width of 95 % asymptotic cofidece itervals (CI) ad the width of 95% highest posterior desity (HPD) itervals The simulatio is performed for differet choices of the sample sizes ad cesored sample sizesm The procedure is repeated for 40,000 times We computed the estimates, their SD, bias, MSE, width of asymptotic CI ad the HPD width for the ML method ad Bayesia method (based o the MCMC techique) The results are reported i Tables (2) ad (3) The HPD cofidece widths are smaller tha the widths of asymptotic CI It seems plausible to say that the performaces of the ML estimates ad Bayesia estimates are very comparable i most results The SD is computed by take the square root of the variace; however the bias is computed by take the absolute value of the differece betwee true value ofθ ad its estimateθ ad MSE is computed by add the variace to the bias squared It is clear from Tables (2) ad (3) that the computed estimates of the parameters p, α 1 ad α 2 are better whe the sample size icreases Sice, the MSE ad bias are small Oe ca see that the estimates of the complete samples are better tha the cesored oes I additio comparig the cesored samples for the cosidered sample sizes, the ML ad Bayesia estimates have smaller bias, MSE ad iterval width whe the cesorig is large That is the estimates are better tha the correspodig estimates whe the cesorig is large I geeral, the computed bias ad MSE of Bayesia estimates are smaller tha there correspodig for ML estimatesthe estimates of the reliability ad hazard rate fuctios are foud after substitutig the parameters estimates i their fuctios ad fixed time DOI: / wwwiosrjouralsorg 29 Page

6 Estimatio Of A Mixture Of Two Weibull Distributios Uder Geeralized Order Statistics Table (2): Classical ad Bayesia estimates of the parameters, RF ad HF at t 0 = 05 with their SD, bias ad MSE for progressive Type II cesored samples from the MTWD with parametersp = 05, α 1 = 01 ad α 2 = 07 m θ ML (SD) Bayesia CP (SD) NP (SD) p (01448) (00316) (00009) ( ) (00022) ( ) α (00284) 0028 (00015) (00019) ( ) (00009) ( ) α (01254) (00458) (00009) 0001 ( ) (00006) ( ) R(t 0 ) H(t 0 ) p (01114) (00265) 04964(00016) ( ) (00014) ( ) α (00323) 0008 (00011) (00022) ( ) (00010) ( ) α (02383) (01458) (00015) ( ) (00010) ( ) R(t 0 ) H(t 0 ) p (00971) (00097) (00007) 0001 ( ) (00011) ( ) α (00151) 0034 (000138) (00026) ( ) (00010) ( ) α (01086) (003471) (00011) ( ) (00008) ( ) R(t 0 ) H(t 0 ) p (00752) (00129) (00010) (13e-06) (00008) (21e-06) α (00413) (00049) (00016) (156e-05) (00010) (75e-06) α (13268) (117326) (00026) (17e-05) (00018) (12e-05) R(t 0 ) H(t 0 ) p (00994) (001232) (00014) ( ) (00009) ( ) α (00188) (000066) (00013) ( ) (00017) ( ) α (00689) (006576) (00012) ( ) (00011) ( ) R(t 0 ) H(t 0 ) p (00344) (01083) (00012) (63e-06) (00026) (25e-05) α (00368) (00037) (00011) (11e-05) (00009) (37e-06) α (05472) (72533) (00019) (701e-06) (00021) (45e-06) R(t 0 ) H(t 0 ) (CP ad NP deote Bayesia estimates whe cojugate ad o-iformative priors are cosidered, respectively) DOI: / wwwiosrjouralsorg 30 Page

7 Estimatio Of A Mixture Of Two Weibull Distributios Uder Geeralized Order Statistics Table (3):Average 95% cofidece/hpd itervals ad their width uder differet progressive Type II cesorig for p = 05, α 1 = 01 ad α 2 = 07 m θ CI Bayes HPDC HPDN p (01128,06806) (04985,05017) (04975,05051) [00075] [05678] [00031] α 1 (00162,01278) (00928,00996) (00979,01015) [00035] [01115] [00068] α 2 (02804,07721) (06967,07007) (06976,07002) [00026] [04917] [00039] 25 p (03784, 07959) (04994,05022) (04955,05005) [00049] [04175] [00028] α 1 (00624, 01608) (00996,01065) (00981,01022) [00041] [00984] [00068] α 2 (03759, 17711) (06953,07014) (06982,07020) [00038] [13952] [00060] p (03287,07096) (04975,05003) (04974,05014) [00039] [03809] [00028] α 1 (00364,00956) [00591] (01001,01087) [00085] (00997,010330) [00035] α 2 (03357,076152) (06996,07035) (06968,07001) [00032] [04257] [00039] 25 p (02673,05623) (04985,05023) (04994,05024) [00030] [02950] [00038] α 1 (00758,02379) (00941,01002) (00954,00993) [00038] [00060] α 2 (12574,64583) (06917,07004) (06940,07005) [00064] [52009] [00087] p (02558,06455) (04972,05020) (04966,05004) [00037] [03897] [00048] α 1 (00453,01193) (00997,01043) (00990,01052) [00062] [00740] [00046] α 2 (03179,05881) (06996,07041) (06948,06991) [00042] [02702] [00045] 25 p (01050,02401) (04953,05002) (05001,05087) [00085] [01351] [00049] α 1 (00763,02209) (00951,00992) (00963,01001) [00037] [01445] α 2 (22644,44096) [21451] [00041] (06946,07010) [00063] (06951,07024) [00072] (HPDC ad HPDN deote the HPD itervals whe cojugate ad o-iformative priors are cosidered respectively) (2) Based o Type II cesored samples The estimatio of the parameters p, α 1, α 2 usig the ML ad Bayesia techiques o cases whe the parameter β is kow have bee obtaied, based o Type II cesored sample which is special case of GOS, that ca be derived from it by choosig m i = 0 ; i = 1,2,, r 1 (16) k = r + 1 By substitutig (16) i (6), the likelihood equatios reduced to the followig L θ t R(t r ) r f(t i ) It is clear from Tables (4) ad (5) that the computed estimates of the parameters p, α 1 ad α 2 are better whe the sample size icreases Sice, the MSE ad bias are small Oe ca see that the estimates of the complete samples are better tha the cesored oes I additio, comparig the cesored samples for the cosidered sample sizes, the ML ad Bayesia estimates have smaller bias, MSE ad iterval width whe the cesorig is large That is the estimates are better tha the correspodig estimates whe the cesorig is small I geeral, the computed bias ad MSE of Bayesia estimates are smaller tha there correspodigcomputedbias ad MSE for the ML estimates r DOI: / wwwiosrjouralsorg 31 Page

8 Estimatio Of A Mixture Of Two Weibull Distributios Uder Geeralized Order Statistics Table (4):Classical ad Bayesia estimates of the parameters, RF ad HF at t 0 = 05 with their SD, MSE ad bias for Type II cesored samples from the MTWD with parametersp = 05, α 1 = 01 ad α 2 = 07 m θ ML (SD) Bayes CP (SD) NP (SD) p (01270) (003285) (00008) ( ) (00018) ( ) α (00257) (000113) (00013) ( ) (00015) ( ) α (01332) (002715) (00021) ( ) (00022) ( ) R(t 0 ) H(t 0 ) p (01747) (003459) (00007) ( ) (00009) ( ) α (00472) (000247) (00023) ( ) (00013) ( ) α (02101) (004479) (00010) ( ) (00019) ( ) R(t 0 ) H(t 0 ) p (01590) (002528) (00017) ( ) (00018) ( ) α (00432) (000260) (00018) ( ) (00014) ( ) α (01575) (002752) (00013) ( ) (00014) ( ) R(t 0 ) H(t 0 ) p (01032) (001183) (00032) ( ) (00012) ( ) α (00236) (000061) (00009) ( ) (00013) ( ) α (02076) (005319) (00012) ( ) (00018) ( ) R(t 0 ) H(t 0 ) p (00828) (000069) (00013) ( ) (00010) ( ) α (00177) ( ) (00015) ( ) (00010) ( ) α (00945) (001411) (00013) ( ) (00018) ( ) R(t 0 ) H(t 0 ) p (00715) (006796) (00013) ( ) (00012) ( ) α (00225) (000098) (00014) ( ) (00015) ( ) α (00843) (001711) (00011) ( ) (00013) ( ) R(t 0 ) H(t 0 ) DOI: / wwwiosrjouralsorg 32 Page

9 Estimatio Of A Mixture Of Two Weibull Distributios Uder Geeralized Order Statistics Table (5): Average 95% cofidece/hpd itervals with their width uder differet Type II cesorig for fixed p = 05, α 1 = 01 ad α 2 = 07 m θ CI Bayes HPDC HPDN p (01216,06198) (04985,05016) [00030] (04953,05014) [00060] [04981] α 1 (00280,01288) (00977,01023) [00046] (00985,01038) [00052] [01007] α 2 (03419,08641) (06998,07064) [00066] (06995,07064) [00069] [05221] 15 p (00938,07787) (04969,05000) [00031] (04967,05007) [00039] [06848] α 1 (00229,02083) (00919,01005) [00085] (00979,01029) [00050] [01853] α 2 (02626,10865) (06991,07030) [00038] (06928,06997) [00068] [08239] p (02398,07630) (04999,05064) [00065] (04984,05044) [00059] [05231] α 1 (00560,01982) (00947,01013) [00065] (00997,01047) [00050] [01421] α 2 (03888,09069) (06956,07007) [00051] (06991,07039) [00048] [05181] 30 p (02958,06354) (04996,05101) [00104] (04994,05036) [00042] [03395] α 1 (00541,01318) (00983,01019) [00035] (00999,01049) [00050] [00777] α 2 (04590,11420) (06977,07022) [00044] (06945,07012) [00067] [06830] p (03712,06437) (04945,05000) [00054] (04969,05013) [00043] [02725] α 1 (00644,01229) (00927,00987) [00060] (00978,01015) [00037] [00585] α 2 (04724,07835) (06979,07026) [00046] (06974,07035) [00060] [03111] 50 p (01317,03669) (04998,05038) [00040] (04996,05041) [00045] [02352] α 1 (00412,01154) (00945,00999) [00054] (00974,01030) [00055] [00741] α 2 (04613,07387) [02773] (06972,07024) [00051] (06994,07040) [00046] 52 Real data aalysis I this subsectio, a real data set has bee aalyzed This data set represets the time betwee failures of secodary reactor pumps This data set has bee origially discussed by [21]ad [22] The chace of the failure of the secodary reactor pump is of the icreasig ature i early stage of the experimet ad after that it decreases It has bee checked by [23] that flexible Weibull distributio is well fitted model to this data set The times betwee failures of 23 secodary reactor pumps are as follows: 2160, 0150, 4082, 0746, 0358, 0199, 0402, 0101, 0605, 0954, 1359, 0273, 0491, 3465, 0070, 6560, 1060, 0062, 4992, 0614, 5320, 0347, ad 1921 Table (6): Classical ad Bayesia estimates of the parameters, RF ad HF at t 0 = 05 with their SD for the real data set uder progressive Type II cesorig m θ ML (SD) Bayes CP (SD) NP (SD) p (01593) (00030) (00012) α (01427) (00018) (00010) Complete α (09763) (00018) (00033) R(t 0 ) H(t 0 ) p (01329) (00012) (00022) α (01348) (00013) (00020) 18 α (12420) (00018) (00015) R(t 0 ) H(t 0 ) p (01367) (00014) (00018) α (01541) (00010) (00009) 15 α (10288) (00014) (00013) R(t 0 ) H(t 0 ) DOI: / wwwiosrjouralsorg 33 Page

10 Estimatio Of A Mixture Of Two Weibull Distributios Uder Geeralized Order Statistics Table (7):Average 95% cofidece/hpd itervals with their width for the real data set uder progressive Type II cesorig m θ CI Bayes HPDC HPDN p (02400, 08645) (05475, 05575) (05515, 05557) [06244] [00099] [00041] Complete α 1 (01246, 06841) (04000, 04062) (04029, 04067) [05595] [00062] [00038] α 2 (11119, 49390) (30227, 30282) (30245, 30348) [38270] [00054] [00102] p (01780, 06990) [05210] (04379, 04421) [00042] (04382, 04451) [00069] 18 α 1 (01222, 06507) [05285] (03863, 03913) [00050] (03803, 03870) [00067] α 2 (14699, 63385) [48685] (38987, 39055) [00067] (39028, 39081) [00052] p (00840, 06201) (03518, 03568) (03525, 03602) [05360] [00050] [00076] 15 α 1 (00553, 06595) (03554, 03593) (03559, 03591) [06042] [00038] [00031] α 2 (13053, 53383) (33174, 33223) (33188, 33237) [40330] [00049] [00048] Table (8):Classical ad Bayesia estimates of the parameters, RF ad HRF at t 0 = 05 with their SD forthereal data set uder Type II cesorig m θ ML(SD) Bayes CP(SD) NP(SD) p (01593) (00015) (00016) α (01427) (00007) (00019) Complete α (09763) (00012) (00015) R(t 0 ) H(t 0 ) p (01530) (00008) (00009) α (01636) (00011) (00016) 20 α (22741) (00009) (00016) R(t 0 ) H(t 0 ) Table (9):Average 95% cofidece/hpd itervals with their width for real data set uder Type II cesorig m Par CI Bayes (MCMC) HPDC HPDN p (02400, 08645) (05525, 05575) (05534, 05596) [06244] [00049] [00061] Complete α 1 (01246, 06841) (04028, 04060) (04030, 04092) [05595] [00032] [00062] α 2 (11119, 49390) (30225, 30266) (30209, 30265) [38270] [00040] [00055] p (03296, 09295) (06286, 06318) (06277, 06312) [05999] [00031] [00035] 20 α 1 (01460, 07874) (04614, 04657) (04654, 04706) [06413] α 2 (00551, 89696) [89145] [00043] (45090, 45128) [00037] [00051] (45119, 45178) [00059] VI Coclusio I this paper, some achievemets have bee doe for the MTWD model, with a commo shape parameter, based o GOS The GOS approach is used for obtaiig the statistical ifereces of MTWD All the results that have bee obtaied for GOS of MTWD ca be specialized to ay special cases of GOS I this paper, the progressive Type II cesored sample ad Type II cesored sample are used as special cases of GOS The ML estimates are obtaiedusigthelm (No-Liear Miimizatio) routie of R 303 to compute the oliear ML equatio Bayesia estimates based o SE loss fuctio are obtaied by usig MCMC techique uder two cases, first with the cojugate prior distributios ad the other case with the o-iformative prior distributios The simulatio have bee studied uder differet sizes ad differet cesorig schemes i DOI: / wwwiosrjouralsorg 34 Page

11 Estimatio Of A Mixture Of Two Weibull Distributios Uder Geeralized Order Statistics progressive Type II cesored samples ad Type II cesored samples I may situatios, Bayesia estimates are more efficiet tha ML estimates I geeral, Bayesia estimatio is better tha ML because the MSE of Bayesia is less tha MSE of ML Bayesia estimatio with cojugate prior ad o-iformative prior have the same quality because the SD, bias, MSE ad width are similar But i some cases usig cojugate prior is better tha o-iformative prior because the MSE i cojugate prior is less tha MSE i o-iformative prior ad this appears i parameter p computatios For each whe m is large, the estimates have smaller MSE tha their correspodig, whe m is small Usually for each m whe is small, the estimates have smaller MSE tha their correspodig, whe is large Fially, the estimatio based o the progressive Type II cesorig is better tha the correspodig estimatio based o cesorigtype II because the MSE i the progressive Type II cesorig is less tha the MSE i the cesorig Type II i the same value of ad m Refereces [1] BSEveritt ad DJ Had,Fiite Mixture Distributios(Uiversity Press, Cambridge,1981) [2] DM Titterigto, AFMSmith ad UE Makov,Statistical Aalysis of Fiite Mixture Distributios (Wiley, New York, 1985) [3] SA Adham, O fiite mixture of two-compoet Gompertz lifetime model, Master s Thesis, Kig Abdulaziz Uiversity, Jeddah, Saudi Arabia,1996 [4] GJMcLachla ad KE Basford,Mixture Models( Ifereces ad Applicatios to Clusterig Marcel Dekker, New York,1988) [5] AS Papadapoulos, ad WJPadgett, (1986) O Bayes estimatio for mixtures of two expoetial-life-distributios from rightcesored samples IEEE Tras Relia, Vol 35(1),1986, [6] EKAL-Hussaii, GR AL-DayiaadSAAdham, O fiite mixture of two-compoet Gompertz lifetime model, Joural of statistical comutatio ad simulatio, Vol 67, 2000, 1-20 [7] EKAL-Hussaii, Bayesia predictive desity of order statistics based o fiite mixture models J Statist Pla If, Vol 113,2003, [8] EKAL-Hussaiiad SF Ateya,Bayes estimatios uder a mixture of trucated Type I geeralized logistic compoets model J Statist Th Appl, Vol 4 (2),2005, [9] ZF Jahee, O record statistics from a mixture of two expoetial distributios J Statist Comput Simul, Vol 75,2005, 1-11 [10] RA AL-Jarallahad EK AL-Hussaii, Bayes iferece uder a ifiite mixture of two compoud Gompertz compoets model J Statist Comput Simul, Vol 77(11), 2007, [11] AI Shawky ad RA Bakoba, O ifiite mixture of two-compoet expoetiated gamma distributio, Joural of Applied Scieces Research, Vol 5(10), 2009, [12] RA Bakoba, A Study o Mixture of Expoetiated Rayleigh ad Expoetiated Expoetial Distributios Based o Order Statistics Joural of Mathematics ad System Sciece Vol 2 (2012),2012, [13] B Al-Zahraiad M Gidwa,Parameter estimatio of a two-parameter Lidley distributio uder hybrid cesorig It J Syst Assur Eg Maag, DOI /s ,2013 [14] AA ALgfary,O fiite mixture of expoetiatedkumaraswamy distributios, Master s Thesis, Kig Abdulaziz Uiversity, Jeddah- Kigdom of Saudi Arabia, 2015 [15] K Che, AS Papadopoulos ad P Tamer,Obayes estimatio for mixtures of two Weibull distributios uder Type I cesorig Microelectroics Reliability,Vol 29(4), 1988, [16] M Fraco, JM Vivo ad N Balakrisha,Reliability properties of geeralized mixtures of Weibull distributios with a commo shape parameter Joural of Statistical Plaig ad Iferece, Vol 141, 2011, [17] G Prakash, Bayes estimatio for a mixture of the Weibull distributios, Iteratioal Joural of Mathematics ad Scietific Computig Vol 2(1), 2012, [18] AMRazali, ad AA Al-Wakeel, Mixture Weibull distributios for fittig failure times data,applied Mathematics ad Computatio, Vol 219,2013, [19] Q Zhag, CHua, ad Xu, A mixture Weibull proportioal hazard model for mechaical system failure predictio utilisig lifetime ad moitorig data,mechaical Systems ad Sigal Processig, Vol 43,2014, [20] U Kamps, A cocept of geeralized order statistics Teuber Stuttgart, 1995 [21] M SSuprawhardaa, SPrayoto ad Sagadji, Total time o test plot aalysis for mechaical compoets of the rsg-gas reactor Atom Idoes, Vol 25 (2),1999 [22] S KSigh, U Sigh ad V K Sharma,Bayesia estimatio ad predictio for flexible Weibull model uder Type II cesorig 2013 Statistics, scheme Joural of Probability ad [23] M Bebbigto, CD Lai ad R Zitikis, A flexible Weibull Extesio,Reliability Egieerig ad SystemSafety, Vol 92 (6), 2007, DOI: / wwwiosrjouralsorg 35 Page