# On Estimation of Three-Component Mixture of Distributions via Bayesian and Classical Approaches

Authors: Muhammad Tahir; Ibrahim M. Almanjahie; Muhammad Abid; Ishfaq Ahmad
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this study, we model a heterogeneous population assuming the three-component mixture of the Pareto distributions assuming type I censored data. In particular, we study some statistical properties (such as various entropies, different inequality indices, and order statistics) of the three-component mixture distribution. The ML estimation and the Bayesian estimation of the mixture parameters have been performed in this study. For the ML estimation, we used the Newton Raphson method. To derive the posterior distributions, different noninformative priors are assumed to derive the Bayes estimators. Furthermore, we also discussed the Bayesian predictive intervals. We presented a detailed simulation study to compare the ML estimates and Bayes estimates. Moreover, we evaluated the performance of different estimates assuming various sample sizes, mixing weights and test termination times (a fixed point of time after which all other tests are dismissed). The real-life data application is also a part of this study.


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## Body

## 1. Introduction

In the last decade, finite mixture models have emerged as flexible models due to their applications in applied sciences, engineering, and physical sciences. As explained by Mendenhall and Hader [1], for real-life purposes, an engineer split the failures of a structure into more than one kind of causes. For example, to know the proportion of failures due to a definite reason and to recover the engineering system, Acheson and McElwee [2] separated electronic tube flops into three different faults such as mechanical faults, gaseous faults, and normal decline of the cathode.Moreover, the mixture models can also be used in a situation when the data are presented in the form of the overall mixture models. The overall mixture models are also called the direct application of the mixture models, and their applications can be seen in medicine, botany, zoology, agriculture, economics, life testing, reliability, and survival analysis. The various aspects of mixture models were discussed by Li and Sedransk [3]. The interested readers can refer the work of Harris [4], Kanji [5], and Jones and McLachlan [6] on the application of mixture models for real-life problems.The mixture models have been extensively used for heterogeneous nature of the process in comparison to the simple models. Most of the researchers have comprehensively applied mixture distributions in various real-life situations and estimated parameters using the Bayesian and classical methods. For a detailed appraisal of classical techniques for estimation and applications of mixture distributions, we refer to studies by Sultan et al. [7], Abu-Zinadah [8], and Kamaruzzaman et al. [9] among others. On the other hand, the estimation of parameters in Bayesian framework for a mixture of two distributions has been considered by many researchers [10–21]. Contrary to the two-component mixture modeling, some authors have discussed situations where data are assumed to follow a three-component mixture of suitable probability distributions [22–28].Censoring is a significant characteristic of the real data application. Due to time and cost problem, it is very difficult to continue the lifetime testing experiment till observing the last failure. Although there are many censoring schemes, the type I right censoring is commonly used in life testing experiments. In this scheme, we consider a fixed censoring time, and the values larger than the specifict (life test termination time) are observed as censored observations. Romeu [29] and Kalbfleisch and Prentice [30] explained various censoring schemes.To motivate the readers about the mixture modeling, consider a sample of sand which is based on the mixture of some minerals. With the application of mixture modeling, estimates of the proportions of various minerals in the sand can be obtained. Similarly, the grain size distributions for the different minerals can also be estimated. It is worth mentioning that mixture models can be classified into type I (if component densities of the various components belong to the same family) and type II (if the component densities belong to different families) mixtures.It has been noticed that from the recent literature that the Pareto distribution can be applied efficiently in various situations rather than other distributions to model data. The significance of Pareto distribution in forming various real phenomena is patent from the following revealed research and references mentioned therein, Abdel-All et al. [31], Ismail [32], Sankaran and Nair [33], and Nadarajah and Kotz [34]. Inspired by the wide real-life applications of mixture distributions, the main objective of this study is to develop a new three-component mixture of Pareto distributions (TCMPD) for lifetime data modeling under type I mixture. Furthermore, we also compare the maximum likelihood (ML) estimates, ML variances (MLVs), Bayes estimates, and their posterior variances (PVs) assuming type I right censored data.


## 2. The Population and the Three-Component Mixture Distribution

The finitek-components mixture density function can be defined for random variable Y as fy=∑d=1kwdfdy, where fdy is the dth component density function, wd, d=1,2,…,h is the dth mixing proportion, and ∑d=1kwd=1. A finite TCMPD is defined as(1)fy;λ1,λ2,λ3,w1,w2=w1f1y;λ1+w2f2y;λ2+1−w1−w2f3y;λ3w1,w2≥0,w1+w2≤1,where fdy;λd=λde−λd+1lny, 1<y<∞,λd>0, d=1,2,3.The cdfFy of a TCMPD is(2)Fy;λ1,λ2,λ3,w1,w2=w1F1y;λ1+w2F2y;λ2+1−w1−w2F3y;λ3,where Fdy;λd=1−e−λdlny.


## 3. Properties of the TCMPD

The statistical properties such as moments, mean, variance, and mode of the TCMPD are derived in this section.rth moment about zero: the rth moment about zero of a TCMPD is derived as(3)EYr=w1λ1λ1−r−1+w2λ2λ2−r−1+1−w1−w2λ3λ3−r−1.kth order negative moment: the kth order negative moment is derived as(4)EY−k=w1λ1λ1+k−1+w2λ2λ2+k−1+1−w1−w2λ3λ3+k−1.Factorial moments: the factorial moments can be determined as(5)EYY−1Y−2,…,Y−δ+1=∑ω=0δ−1ξω−1ωEYδ−ω,Whereξω′s is the real number. The EYδ−ω can be obtained ash(6)EYδ−ω=w1λ1λ1−δ+ω−1+w2λ2λ2−δ+ω−1+1−w1−w2λ3λ3−δ+ω−1.Mean: the mean of a TCMPD is evaluated ash(7)EY=w1λ1λ1−1−1+w2λ2λ2−1−1+1−w1−w2λ3λ3−1−1.Variance: the variance of a TCMPD is derived as(8)VarY=∑d=13wdλdλd−2−1−∑d=13wdλdλd−1−12.Median: the median (y) is determined by evaluating the following nonlinear equation for y.(9)w1e−λ1lny+w2e−λ2lny+1−w1−w2e−λ3lny=0.5.Mode: the mode (y) is obtained by solving the following nonlinear equation for y.(10)w1λ1λ1+1e−λ1+2lny+w2λ2λ2+1e−λ2+2lny+1−w1−w2λ3λ3+1e−λ3+2lny=0.Using the above expressions, mean, median, variance, and coefficient of skewness (SK) are calculated for different parameters’ values and are given in Table1.Table 1
Mean, median, variance, and coefficient of skewness of TCMPD.
λ1,λ2,λ3,w1,w2MeanMedianVarianceSK5, 6, 7, 0.1, 0.21.18171.11120.05034.12485, 6, 7, 0.2, 0.31.19331.11700.05934.39075, 6, 7, 0.3, 0.41.20501.12320.06804.49055, 6, 7, 0.4, 0.51.21671.12990.07644.50807, 6, 5, 0.2, 0.11.22831.13530.08794.71928, 7, 6, 0.3, 0.21.17621.10800.04663.82859, 8, 7, 0.4, 0.31.14291.08970.02823.327910, 9, 8, 0.5, 0.41.11981.07670.01862.98376, 5, 4, 0.5, 0.31.24171.14090.10836.52387, 6, 5, 0.5, 0.31.19331.11700.05934.39078, 7, 6, 0.5, 0.31.16141.10000.03773.65359, 8, 7, 0.5, 0.31.13871.08740.02623.2733It is observed from the entries in Table1 and Figure 1 that TCMPD is positively skewed as we have SK > 0 for all the entries arranged in Table 1. We noticed that the variance of the TCMPD was a declining function of the mixture distribution’s parameters.Figure 1
The graph of the BE and MLE of parameterλ1=3.


## 4. Entropies

The entropy is a quantity of unspecified extent of evidence in a function. Here, we derive the expressions of the most commonly used entropy measures such as Shannon’s entropy,β−entropy, and Re´nyi entropy, in this section. As said by Song [35], Shannon’s entropy has a same behavior as a measure of kurtosis in equating the forms of different functions and computing substance of tails.Shannon’s entropy: Shannon’s entropy for a random variableY, which follows TCMPD, is(11)ISy=−∫1∞fy;λ1,λ2,λ3,w1,w2logfy;λ1,λ2,λ3,w1,w2dy,ISy=−∫1∞w1λ1e−λ1+1lny+w2λ2e−λ2+1lny+1−w1−w2λ3e−λ3+1lnylogw1λ1e−λ1+1lny+w2λ2e−λ2+1lny+1−w1−w2λ3e−λ3+1lnydy.Re´nyi entropy: the Rényi entropy (Rényi, [36]) is explained as(12)IRy=11−ξlog∫1∞fξy;λ1,λ2,λ3,w1,w2dy,Whereξ>0, and ξ≠1. The Rényi entropy of a 3-CMPD is(13)IRy=11−ξlog∫1∞w1λ1e−λ1+1lny+w2λ2e−λ2+1lny+1−w1−w2λ3e−λ3+1lnyξdy.β−Entropy: the β−entropy (Ullah, [37]) is written as(14)Iβy=1φ−11−∫1∞fφy;λ1,λ2,λ3,w1,w2dy,Whereφ>0, and φ≠1. The β−entropy of a TCMPD is(15)Iβy=1φ−11−∫1∞w1λ1e−λ1+1lny+w2λ2e−λ2+1lny+1−w1−w2λ3e−λ3+1lnyφdy,Which can be evaluated by the numerical integration.


## 5. Inequality Measures

The most common income inequality indices are Bonferroni curve, Gini index, and Lorenz curve.The Gini index: this index (Gini, [38]) for a TCMPD is(16)G=1EY∫1∞Fy;λ1,λ2,λ3,w1,w21−Fy;λ1,λ2,λ3,w1,w2dy,G=∫1∞w1e−λ1lny+w2e−λ2lny+1−w1−w2e−λ3lnydy−∫1∞w1e−λ1lny+w2e−λ2lny+1−w1−w2e−λ3lny2dyw1λ1λ1−1−1+w2λ2λ2−1−1+1−w1−w2λ3λ3−1−1,G=w1λ1−1−1+w2λ2−1−1+1−w1−w2λ3−1−1w1λ1λ1−1−1+w2λ2λ2−1−1+1−w1−w2λ3λ3−1−1−w122λ1−1−1+w222λ2−1−1+1−w1−w222λ3−1−1+2w1w2λ1+λ2−1−1+2w21−w1−w2λ2+λ3−1−1+2w11−w1−w2λ1+λ3−1−1w1λ1λ1−1−1+w2λ2λ2−1−1+1−w1−w2λ3λ3−1−1.The Lorenz curve: this curve (Lorenz, [39]) for a TCMPD is(17)Lp=1EY∫1yyfy;λ1,λ2,λ3,w1,w2dy,Lp=1−w1λ1λ1−1−1e−λ1−1lny+w2λ2λ2−1−1e−λ2−1lny+1−w1−w2λ3λ3−1−1e−λ3−1lnyw1λ1λ1−1−1+w2λ2λ2−1−1+1−w1−w2λ3λ3−1−1.Bonferroni curve: the Bonferroni [40] curve for a TCMPD is(18)BCp=LpFy=1−w1λ1λ1−1−1e−λ1−1lny+w2λ2λ2−1−1e−λ2−1lny+1−w1−w2λ3λ3−1−1e−λ3−1lny/w1λ1λ1−1−1+w2λ2λ2−1−1+1−w1−w2λ3λ3−1−1/1−w1e−λ1lny−w2e−λ2lny−1−w1−w2e−λ3lny.


## 6. Order Statistics

In this section, we derivegyk:n;λ1,λ2,λ3,w1,w2 which is pdf of kth order statistic yk:n, assuming a sample of size n from the TCMPD. The rth raw moment along with mean and variance of 1st and nth order statistics are obtained in this section.Probability density function ofkth order statistic: the pdf of kth order statistic is(19)gyk:n;λ1,λ2,λ3,w1,w2=n!k−1!n−k!fy;λ1,λ2,λ3,w1,w2Fy;λ1,λ2,λ3,w1,w2k−11−Fy;λ1,λ2,λ3,w1,w2n−k,where fy;λ1,λ2,λ3,w1,w2=w1f1y;λ1+w2f2y;λ2+1−w1−w2f3y;λ3,(20)w1,w2≥0,w1+w2≤1,Fy;λ1,λ2,λ3,w1,w2=w1F1y;λ1+w2F2y;λ2+1−w1−w2F3y;λ3,Fy;λ1,λ2,λ3,w1,w2k−1=1−w1e−λ1lny−w2e−λ2lny−1−w1−w2e−λ3lnyk−1,Fy;λ1,λ2,λ3,w1,w2k−1=∑a=0k−1∑b=0a∑c=0b−1ak−1aabbce−λ1a−blnye−λ2b−clnye−λ3clnyw1a−bw2b−c1−w1−w2c,1−Fy;λ1,λ2,λ3,w1,w2n−k=w1e−λ1lny+w2e−λ2lny+1−w1−w2e−λ3lnyn−k,1−Fy;λ1,λ2,λ3,w1,w2n−k=∑u=0n−k∑v=0un−kuuve−λ1n−k−ulnye−λ2u−vlnye−λ3vlnyw1n−k−uw2u−v1−w1−w2v.After little simplification, the pdf (19) of kth order statistic can be written as(21)gyk:n;λ1,λ2,λ3,w1,w2=n!k−1!n−k!∑a=0k−1∑b=0a∑c=0b−1ak−1aabbce−λ1a−blnyke−λ2b−clnyke−λ3clnykw1a−bw2b−c1−w1−w2c∑u=0n−k∑v=0un−kuuve−λ1n−k−ulnyke−λ2u−vlnyke−λ3vlnykw1n−k−uw2u−v1−w1−w2vw1λ1e−λ1+1lnyk+w2λ2e−λ2+1lnyk+1−w1−w2λ3e−λ3+1lnyk,gyk:n;λ1,λ2,λ3,w1,w2=n!k−1!n−k!∑a=0k−1∑b=0a∑c=0b−1ak−1aabbce−A01lnykw1B01−1w2C01−11−w1−w2D01−1∑u=0n−k∑v=0un−kuuve−A02lnykw1B02−1w2C02−11−w1−w2D02−1w1λ1e−λ1+1lnyk+w2λ2e−λ2+1lnyk+1−w1−w2λ3e−λ3+1lnyk,where A01=λ1a−b+λ2b−c+λ3c, B01=a−b+1, C01=b−c+1, D01=c+1,(22)A02=λ1n−k−u+λ2u−v+λ3v,B02=n−k−u+1,C02=u−v+1,D02=v+1.Probability density function of 1st order statistic: substituting k=1 in (21) and simplifying it, the pdf of 1st order statistic is(23)gy1:n;λ1,λ2,λ3,w1,w2=n∑w=11λw∑u=0n−1∑v=0un−1uuve−A1w+1lny1w1B1w−1w2C1w−11−w1−w2D1w−1,where A11=λ1n−u+λ2u−v+λ3v, B11=n−u+1, C11=u−v+1, D11=v+1,(24)A12=λ1n−1−u+λ2u−v+1+λ3v,B12=n−u,C12=u−v+2,D12=v+1,A13=λ1n−1−u+λ2u−v+λ3v+1,B13=n−u,C13=u−v+1,D13=v+2.Probability density function ofnth order statistic: substituting k=n in (21) and after little algebraic simplification, the pdf of nth order statistic is(25)gyn:n;λ1,λ2,λ3,w1,w2=n∑w=11λw∑a=0n−1∑b=0a∑c=0b−1ak−1aabbce−A2w+1lnynw1B2w−1w2C2w−11−w1−w2D2w−1,where A21=λ1a−b+1+λ2b−c+λ3c, B21=a−b+2, C21=b−c+1, D21=c+1,(26)A22=λ1a−b+λ2b−c+1+λ3c,B22=a−b+1,C22=b−c+2,D22=c+1,A23=λ1a−b+λ2b−c+λ3c+1,B23=a−b+1,C23=b−c+1,D23=c+2.rth moments, mean, and variance of 1st order statistic: the rth moment about the origin, mean, and variance of 1st order statistic are(27)EY1r=n∑w=13λw∑u=0n−1∑v=0un−1uuvw1B1w−1w2C1w−11−w1−w2D1w−1A1w−r−1,EY1=n∑w=13λw∑u=0n−1∑v=0un−1uuvw1B1w−1w2C1w−11−w1−w2D1w−1A1w−1−1,VarY1=n∑w=13λw∑u=0n−1∑v=0un−1uuvw1B1w−1w2C1w−11−w1−w2D1w−1A1w−2−1−n∑w=13λw∑u=0n−1∑v=0un−1uuvw1B1w−1w2C1w−11−w1−w2D1w−1A1w−1−12.rth moments, mean, and variance of nth order statistic: the rth moment about the origin, mean, and variance of nth order statistic are(28)EYnr=n∑w=13λw∑a=0n−1∑b=0a∑c=0b−1ak−1aabbcw1B2w−1w2C2w−11−w1−w2D2w−1A2w−r−1,EYn=n∑w=13λw∑a=0n−1∑b=0a∑c=0b−1ak−1aabbcw1B2w−1w2C2w−11−w1−w2D2w−1A2w−1−1,VarYn=n∑w=13λw∑a=0n−1∑b=0a∑c=0b−1ak−1aabbcw1B2w−1w2C2w−11−w1−w2D2w−1A2w−2−1−n∑w=13λw∑a=0n−1∑b=0a∑c=0b−1ak−1aabbcw1B2w−1w2C2w−11−w1−w2D2w−1A2w−1−12.


## 7. Parametric Estimation

Here, we discuss the parameter estimation methods. In particular, we use the ML and Bayesian methods of estimation to estimate unknown parameters under type I censored data. Before discussing the parameter estimation, we construct the likelihood function.Assumen units, with fixed t as test termination time, from the TCMPD are used in a real-life experiment. Let y1,y2,…,yξ be the ordered values that can only be observed. The n−ξ remaining largest values are censored from the study, i.e., their exact failure time cannot be recorded due to time constraint. So, y11,…,y1ξ1, y21,…,y2ξ2, and y31,…,y3ξ3 are failed observations relating to subpopulations I, II, and III, respectively. The observation yξ is assumed to be censored from each component, whereas the numbers ξ1, ξ2, and ξ3 of failed values are obtained from the subpopulations. The n1−ξ1, n2−ξ2, and n3−ξ3 observations are assumed to be censored values from subpopulations, whereas r1+r2+r3=r. So, the likelihood function using type I censored sample, y=y1=y11,…,y1ξ1, y2=y21,…,y2ξ2, and y3=y31,…,y3ξ3, for a TCMPD is Lλ1,λ2,λ3,w1,w2|y∝∏k=1ξ1w1f1y1k∏k=1ξ2w2f2y2k∏k=1ξ31−w1−w2f3y3k1−Ftn−ξ. After substitution and simplification, the likelihood function of TCMPD becomes(29)Lλ1,λ2,λ3,w1,w2|y∝∑m1=0n−ξ∑m2=0m1n−ξm1m1m2e−λ1n−ξ−m1lnt+∑k=1ξ1lny1ke−λ2m1−m2lnt+∑k=1ξ2lny2ke−λ3m2lnt+∑k=1ξ3lny3k×λ1ξ1λ2ξ2λ3ξ3w1n−ξ−m1+ξ1w2m1−m2+ξ21−w1−w2m2+ξ3.


### 7.1. ML Estimators and Variances

The ML estimators of TCMPD for parameterΦ=λ1,λ2,λ3,w1,w2 are derived from the solution of nonlinear equations (30)–(34). The equations have been derived by partially differentiating the natural logarithm of the likelihood function as(30)∂lnLΦ|y∂λ1=ξ1λ1−∑k=1r1lny1k−n−ξtw1e−λ1lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt=0,(31)∂lnLΦ|y∂λ2=ξ2λ2−∑k=1r2lny2k−n−ξtw2e−λ2lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt=0,(32)∂lnLΦ|y∂λ3=ξ3λ3−∑k=1r3lny3k−n−ξt1−w1−w2e−λ3lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt=0,(33)∂lnLΦ|y∂w1=ξ1w1−ξ31−w1−w2+n−ξe−λ1lnt−e−λ3lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt=0,(34)∂lnLΦ|y∂w2=ξ2w2−ξ31−w1−w2+n−ξe−λ2lnt−e−λ3lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt=0.It is very tough to find out an explicit solution from the nonlinear equations (30)–(34); therefore, to obtain ML estimators, any mathematical or statistical software such as Mathematica (Wolfram, [41]) can be used to solve them by an iterative procedure.LetΦ=λ1,λ2,λ3,w1,w2, and by using multivariate central limit theorem, that is, Φ^∼ΝΦ,I−1Φ, one can get asymptotic variances, where the variance is given on diagonal of the inverted Fisher information matrix which is expectation of the negative Hessian as(35)IΦ=−E∂2lnLΦ|y∂λ12∂2lnLΦ|y∂λ1∂λ2∂2lnLΦ|y∂λ1∂λ3∂2lnLΦ|y∂λ1∂w1∂2lnLΦ|y∂λ1∂w2∂2lnLΦ|y∂λ2∂λ1∂2lnLΦ|y∂λ22∂2lnLΦ|y∂λ2∂λ3∂2lnLΦ|y∂λ2∂w1∂2lnLΦ|y∂λ2∂w2∂2lnLΦ|y∂λ3∂λ1∂2lnLΦ|y∂λ3∂λ2∂2lnLΦ|y∂λ32∂2lnLΦ|y∂λ3∂w1∂2lnLΦ|y∂λ3∂w2∂2lnLΦ|y∂w1∂λ1∂2lnLΦ|y∂w1∂λ2∂2lnLΦ|y∂w1∂λ3∂2lnLΦ|y∂w12∂2lnLΦ|y∂w1∂w2∂2lnLΦ|y∂w2∂λ1∂2lnLΦ|y∂w2∂λ2∂2lnLΦ|y∂w2∂λ3∂2lnLΦ|y∂w2pλ1∂2lnLΦ|y∂w22,where(36)∂2lnLΦ|y∂λ12=−ξ1λ12+w2e−λ2lnt+1−w1−w2e−λ3lntn−ξt2w1e−λ1lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt2,∂2lnLΦ|y∂λ22=−ξ2λ22+w1e−λ1lnt+1−w1−w2e−λ3lntn−ξt2w2e−λ2lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt2,∂2lnLΦ|y∂λ32=−ξ3λ32+w1e−λ1lnt+w2e−λ2lntn−ξt21−w1−w2e−λ3lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt2,∂2lnLΦ|y∂w12=−ξ1w12−ξ31−w1−w22−n−ξe−λ1lnt−e−λ3lnt2w1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt2,∂2lnLΦ|y∂w22=−ξ2w22−ξ31−w1−w22−n−ξe−λ2lnt−e−λ3lnt2w1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt2.Next, we discuss the Bayesian estimation for the estimation of unknown parameters.


### 7.2. The Joint Prior and Posterior Distributions

Now, we discuss the Bayesian estimation of the unknown parameters. This method allows us to obtain an updated form of the knowledge which is calculated by combining the current and the prior knowledge. In particular, we use uniform and Jeffreys as noninformative priors, which are used when little or no formal prior knowledge on the parameters of concern is available. Box and Taio [42] stated a noninformative prior which gives slight information relative to the testing experiment. Bernardo and Smith [43] defined a noninformative prior has the least influence relative to the data. Jeffreys prior suggested by Jeffreys [44] is obtained by evaluating the Fisher information.


#### 7.2.1. The Joint Prior and Posterior Distributions Assuming the UP

We take the improper UP for component parameters, i.e.,λd∼U0,∞, d=1,2,3. Also, we take the UP for the unknown proportion parameters, i.e., ws∼U0,1, s=1,2. Under the assumption of independence of parameters λ1,λ2,λ3,w1, and w2, the joint prior distribution is(37)ψ1λ1,λ2,λ3,w1,w2∝1.By combining the likelihood function and the joint UP prior, i.e.,ψ1λ1,λ2,λ3,w1,w2, we obtain the joint posterior distribution of parameters λ1,λ2,λ3,w1, and w2 given data y as(38)ξ1λ1,λ2,λ3,w1,w2|y=Lλ1,λ2,λ3,w1,w2|yψ1λ1,λ2,λ3,w1,w2|y∫w2∫w1∫λ3∫λ2∫λ1Lλ1,λ2,λ3,w1,w2|yψ1λ1,λ2,λ3,w1,w2|ydλ1dλ2dλ3dw1dw2,ξ1λ1,λ2,λ3,w1,w2|y=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2e−B11λ1e−B21λ2e−B31λ3w1A01−1w2B01−11−w1−w2C01−1E1λ11−A11λ21−A21λ31−A31,where A11=ξ1+1, A21=ξ2+1, A31=ξ3+1,(39)B11=n−ξ−m1lnt+∑k=1ξ1lny1k,B21=m1−m2lnt+∑k=1ξ2lny2k,B31=m2lnt+∑k=1ξ3lny3k,A01=n−ξ−m1+ξ1+1,B01=m1−m2+ξ2+1,C01=m2+ξ3+1,E1=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA11ΓA21ΓA31B11−A11B21−A21B31−A31BA01,B01,C01.


#### 7.2.2. The Joint Prior and Posterior Distributions Assuming the JP

The JP for component parametersλd, d=1,2,3, is pλd∝Iλd, where Iλd is the Fisher information and is Iλd=−E∂2lnLλd;yd/∂λd2. The prior distributions of w1 and w2 are assumed as the UP, i.e., ws∼U0,1, s=1,2. So, the joint prior distribution is(40)ψ2λ1,λ2,λ3,w1,w2|y∝1λ1λ2λ3.Using the JP, the joint posterior distribution ofλ1,λ2,λ3,w1, and w2 that give data y is(41)ξ2λ1,λ2,λ3,w1,w2|y=Lλ1,λ2,λ3,w1,w2|yψ2λ1,λ2,λ3,w1,w2|y∫w2∫w1∫λ3∫λ2∫λ1Lλ1,λ2,λ3,w1,w2|yψ2λ1,λ2,λ3,w1,w2|ydλ1dλ2dλ3dw1dw2,ξ2λ1,λ2,λ3,w1,w2|y=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2e−B12λ1e−B22λ2e−B32λ3w1A02−1w2B02−11−w1−w2C02−1E2λ11−A12λ21−A22λ31−A32,where A12=ξ1, A22=ξ2, A32=ξ3,(42)B12=n−ξ−m1lnt+∑k=1ξ1lny1k,B22=m1−m2lnt+∑k=1ξ2lny2k,B32=m2lnt+∑k=1ξ3lny3k,A02=n−ξ−m1+ξ1+1,B02=m1−m2+ξ2+1,C02=m2+ξ3+1,E2=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA12ΓA22ΓA32B12−A12B22−A22B32−A32BA02,B02,C02.


### 7.3. Bayes Estimators and Posterior Variances

The Bayes estimators of the component and mixing proportion, i.e.,λ1,λ2,λ3,w1, and w2 using the UP are obtained as(43)λ^α|y=1E1∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓAα1+1ΓAβ1ΓAγ1BA01,C01BB01,A01+C01Bα1Aα1+1Bβ1Aβ1Bγ1Aγ1,where α,β, and γ are defined as (i) α=1,β=2,γ=3 (ii) α=2,β=1,γ=3, and (iii) α=3,β=1,γ=2.(44)w^δ|y=1E1∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA11ΓA21ΓA31Bϒ01,C01BΛ01+1,ϒ01+C01B11A11B21A21B31A31,where δ,ϒ, and Λ are defined as (i) δ=1,ϒ=B,Λ=A and (ii) δ=2,ϒ=A,Λ=B.To measure the accuracy and efficiency of the Bayes estimators, we normally calculate the PVs of the parameters. For the PVs ofλ1,λ2,λ3,w1, and w2, using the UP are derived as(45)Varλ^α|y=1E1∑m1=0n−ξ∑m2=0m1n−ξm1m1m2m1m2ΓAα1+2ΓAβ1ΓAγ1BA01,C01BB01,A01+C01Bα1Aα1+2Bβ1Aβ1Bγ1Aγ1−1E1∑m1=0n−r∑m2=0m1n−ξm1m1m2ΓAα1+1ΓAβ1ΓAγ1BA01,C01BB01,A01+C01Bα1Aα1+1Bβ1Aβ1Bγ1Aγ12,Varw^δ|y=1E1∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA11ΓA21ΓA31Bϒ01,C01BΛ01+2,ϒ01+C01B11A11B21A21B31A31−1E1∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA11ΓA21ΓA31Bϒ01,C01BΛ01+1,ϒ01+C01B11A11B21A21B31A312.The Bayes estimators ofλ1,λ2,λ3,w1, and w2 using the JP are obtained as(46)λ^α|y=1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓAα2+1ΓAβ2ΓAγ2BA02,C02BB02,A02+C02Bα2Aα2+1Bβ2Aβ2Bγ2Aγ2,w^δ|y=1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA12ΓA22ΓA32Bϒ02,C02BΛ02+1,ϒ02+C02B12A12B22A22B32A32.Also, the PVs of the parametersλ1,λ2,λ3,p1, and p2using the JP are derived as(47)Varλ^α|y=1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓAα2+2ΓAβ2ΓAγ2BA02,C02BB02,A02+C02Bα2Aα2+2Bβ2Aβ2Bγ2Aγ2−1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓAα2+1ΓAβ2ΓAγ2BA02,C02BB02,A02+C02Bα2Aα2+1Bβ2Aβ2Bγ2Aγ22,Varw^δ|y=1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA12ΓA22ΓA32Bϒ02,C02BΛ02+2,ϒ02+C02B12A12B22A22B32A32−1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA12ΓA22ΓA32Bϒ02,C02BΛ02+1,ϒ02+C02B12A12B22A22B32A322.


### 7.4. The Bayesian Prediction

One of the main objectives of statistical modeling is the prediction of the future values. The Bayesian methodology allows us to obtain this in a natural way. In particular, the posterior predictive distribution (PPD) comprises the knowledge about future valueX=Yn+1 given data y. Al-Hussaini et al. [45], Bolstad [46], and Bansal [47] have discussed the usefulness of the prediction and predictive distribution comprehensively in the Bayesian framework.


#### 7.4.1. The Posterior Predictive Distribution

For the future valueX=Yn+1, the PPD using the UP and the JP is(48)fx|y=∫w2∫w1∫λ3∫λ2∫λ1fx|λ1,λ2,λ3,w1,w2ξvλ1,λ2,λ3,w1,w2|ydλ1dλ2dλ3dw1dw2,(49)fx|y=1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1v+1ΓA2vΓA3vBA0v+1,C0vBB0v,A0v+C0v+1B1v+lnxA1v+1B2vA2vB3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2v+1ΓA3vBA0v,C0vBB0v+1,A0v+C0v+1B1vA1vB2v+lnxA2v+1B3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2vΓA3v+1BA0v,C0v+1BB0v,A0v+C0v+1B1vA1vB2vA2vB3v+lnxA3v+1.In equation, we considerv=1 for UP and v=2 for JP, respectively.


#### 7.4.2. Bayesian Predictive Intervals

To obtain the Bayesian predictive intervals (BPIs), letL and U are the two lower and upper endpoints of the BPI, which are obtained from (49). A 1001−α% BPI (L, U) using UP and JP can be obtained by simplifying the given expression:(50)∫1L1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1v+1ΓA2vΓA3vBA0v+1,C0vBB0v,A0v+C0v+1B1v+lnxA1v+1B2vA2vB3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2v+1ΓA3vBA0v,C0vBB0v+1,A0v+C0v+1B1vA1vB2v+lnxA2v+1B3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2vΓA3v+1BA0v,C0v+1BB0v,A0v+C0v+1B1vA1vB2vA2vB3v+lnxA3v+1dx=α2,∫U∞1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1v+1ΓA2vΓA3vBA0v+1,C0vBB0v,A0v+C0v+1B1v+lnxA1v+1B2vA2vB3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2v+1ΓA3vBA0v,C0vBB0v+1,A0v+C0v+1B1vA1vB2v+lnxA2v+1B3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2vΓA3v+1BA0v,C0v+1BB0v,A0v+C0v+1B1vA1vB2vA2vB3v+lnxA3v+1dx=α2.


## 7.1. ML Estimators and Variances

The ML estimators of TCMPD for parameterΦ=λ1,λ2,λ3,w1,w2 are derived from the solution of nonlinear equations (30)–(34). The equations have been derived by partially differentiating the natural logarithm of the likelihood function as(30)∂lnLΦ|y∂λ1=ξ1λ1−∑k=1r1lny1k−n−ξtw1e−λ1lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt=0,(31)∂lnLΦ|y∂λ2=ξ2λ2−∑k=1r2lny2k−n−ξtw2e−λ2lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt=0,(32)∂lnLΦ|y∂λ3=ξ3λ3−∑k=1r3lny3k−n−ξt1−w1−w2e−λ3lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt=0,(33)∂lnLΦ|y∂w1=ξ1w1−ξ31−w1−w2+n−ξe−λ1lnt−e−λ3lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt=0,(34)∂lnLΦ|y∂w2=ξ2w2−ξ31−w1−w2+n−ξe−λ2lnt−e−λ3lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt=0.It is very tough to find out an explicit solution from the nonlinear equations (30)–(34); therefore, to obtain ML estimators, any mathematical or statistical software such as Mathematica (Wolfram, [41]) can be used to solve them by an iterative procedure.LetΦ=λ1,λ2,λ3,w1,w2, and by using multivariate central limit theorem, that is, Φ^∼ΝΦ,I−1Φ, one can get asymptotic variances, where the variance is given on diagonal of the inverted Fisher information matrix which is expectation of the negative Hessian as(35)IΦ=−E∂2lnLΦ|y∂λ12∂2lnLΦ|y∂λ1∂λ2∂2lnLΦ|y∂λ1∂λ3∂2lnLΦ|y∂λ1∂w1∂2lnLΦ|y∂λ1∂w2∂2lnLΦ|y∂λ2∂λ1∂2lnLΦ|y∂λ22∂2lnLΦ|y∂λ2∂λ3∂2lnLΦ|y∂λ2∂w1∂2lnLΦ|y∂λ2∂w2∂2lnLΦ|y∂λ3∂λ1∂2lnLΦ|y∂λ3∂λ2∂2lnLΦ|y∂λ32∂2lnLΦ|y∂λ3∂w1∂2lnLΦ|y∂λ3∂w2∂2lnLΦ|y∂w1∂λ1∂2lnLΦ|y∂w1∂λ2∂2lnLΦ|y∂w1∂λ3∂2lnLΦ|y∂w12∂2lnLΦ|y∂w1∂w2∂2lnLΦ|y∂w2∂λ1∂2lnLΦ|y∂w2∂λ2∂2lnLΦ|y∂w2∂λ3∂2lnLΦ|y∂w2pλ1∂2lnLΦ|y∂w22,where(36)∂2lnLΦ|y∂λ12=−ξ1λ12+w2e−λ2lnt+1−w1−w2e−λ3lntn−ξt2w1e−λ1lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt2,∂2lnLΦ|y∂λ22=−ξ2λ22+w1e−λ1lnt+1−w1−w2e−λ3lntn−ξt2w2e−λ2lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt2,∂2lnLΦ|y∂λ32=−ξ3λ32+w1e−λ1lnt+w2e−λ2lntn−ξt21−w1−w2e−λ3lntw1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt2,∂2lnLΦ|y∂w12=−ξ1w12−ξ31−w1−w22−n−ξe−λ1lnt−e−λ3lnt2w1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt2,∂2lnLΦ|y∂w22=−ξ2w22−ξ31−w1−w22−n−ξe−λ2lnt−e−λ3lnt2w1e−λ1lnt+w2e−λ2lnt+1−w1−w2e−λ3lnt2.Next, we discuss the Bayesian estimation for the estimation of unknown parameters.


## 7.2. The Joint Prior and Posterior Distributions

Now, we discuss the Bayesian estimation of the unknown parameters. This method allows us to obtain an updated form of the knowledge which is calculated by combining the current and the prior knowledge. In particular, we use uniform and Jeffreys as noninformative priors, which are used when little or no formal prior knowledge on the parameters of concern is available. Box and Taio [42] stated a noninformative prior which gives slight information relative to the testing experiment. Bernardo and Smith [43] defined a noninformative prior has the least influence relative to the data. Jeffreys prior suggested by Jeffreys [44] is obtained by evaluating the Fisher information.


### 7.2.1. The Joint Prior and Posterior Distributions Assuming the UP

We take the improper UP for component parameters, i.e.,λd∼U0,∞, d=1,2,3. Also, we take the UP for the unknown proportion parameters, i.e., ws∼U0,1, s=1,2. Under the assumption of independence of parameters λ1,λ2,λ3,w1, and w2, the joint prior distribution is(37)ψ1λ1,λ2,λ3,w1,w2∝1.By combining the likelihood function and the joint UP prior, i.e.,ψ1λ1,λ2,λ3,w1,w2, we obtain the joint posterior distribution of parameters λ1,λ2,λ3,w1, and w2 given data y as(38)ξ1λ1,λ2,λ3,w1,w2|y=Lλ1,λ2,λ3,w1,w2|yψ1λ1,λ2,λ3,w1,w2|y∫w2∫w1∫λ3∫λ2∫λ1Lλ1,λ2,λ3,w1,w2|yψ1λ1,λ2,λ3,w1,w2|ydλ1dλ2dλ3dw1dw2,ξ1λ1,λ2,λ3,w1,w2|y=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2e−B11λ1e−B21λ2e−B31λ3w1A01−1w2B01−11−w1−w2C01−1E1λ11−A11λ21−A21λ31−A31,where A11=ξ1+1, A21=ξ2+1, A31=ξ3+1,(39)B11=n−ξ−m1lnt+∑k=1ξ1lny1k,B21=m1−m2lnt+∑k=1ξ2lny2k,B31=m2lnt+∑k=1ξ3lny3k,A01=n−ξ−m1+ξ1+1,B01=m1−m2+ξ2+1,C01=m2+ξ3+1,E1=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA11ΓA21ΓA31B11−A11B21−A21B31−A31BA01,B01,C01.


### 7.2.2. The Joint Prior and Posterior Distributions Assuming the JP

The JP for component parametersλd, d=1,2,3, is pλd∝Iλd, where Iλd is the Fisher information and is Iλd=−E∂2lnLλd;yd/∂λd2. The prior distributions of w1 and w2 are assumed as the UP, i.e., ws∼U0,1, s=1,2. So, the joint prior distribution is(40)ψ2λ1,λ2,λ3,w1,w2|y∝1λ1λ2λ3.Using the JP, the joint posterior distribution ofλ1,λ2,λ3,w1, and w2 that give data y is(41)ξ2λ1,λ2,λ3,w1,w2|y=Lλ1,λ2,λ3,w1,w2|yψ2λ1,λ2,λ3,w1,w2|y∫w2∫w1∫λ3∫λ2∫λ1Lλ1,λ2,λ3,w1,w2|yψ2λ1,λ2,λ3,w1,w2|ydλ1dλ2dλ3dw1dw2,ξ2λ1,λ2,λ3,w1,w2|y=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2e−B12λ1e−B22λ2e−B32λ3w1A02−1w2B02−11−w1−w2C02−1E2λ11−A12λ21−A22λ31−A32,where A12=ξ1, A22=ξ2, A32=ξ3,(42)B12=n−ξ−m1lnt+∑k=1ξ1lny1k,B22=m1−m2lnt+∑k=1ξ2lny2k,B32=m2lnt+∑k=1ξ3lny3k,A02=n−ξ−m1+ξ1+1,B02=m1−m2+ξ2+1,C02=m2+ξ3+1,E2=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA12ΓA22ΓA32B12−A12B22−A22B32−A32BA02,B02,C02.


## 7.2.1. The Joint Prior and Posterior Distributions Assuming the UP

We take the improper UP for component parameters, i.e.,λd∼U0,∞, d=1,2,3. Also, we take the UP for the unknown proportion parameters, i.e., ws∼U0,1, s=1,2. Under the assumption of independence of parameters λ1,λ2,λ3,w1, and w2, the joint prior distribution is(37)ψ1λ1,λ2,λ3,w1,w2∝1.By combining the likelihood function and the joint UP prior, i.e.,ψ1λ1,λ2,λ3,w1,w2, we obtain the joint posterior distribution of parameters λ1,λ2,λ3,w1, and w2 given data y as(38)ξ1λ1,λ2,λ3,w1,w2|y=Lλ1,λ2,λ3,w1,w2|yψ1λ1,λ2,λ3,w1,w2|y∫w2∫w1∫λ3∫λ2∫λ1Lλ1,λ2,λ3,w1,w2|yψ1λ1,λ2,λ3,w1,w2|ydλ1dλ2dλ3dw1dw2,ξ1λ1,λ2,λ3,w1,w2|y=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2e−B11λ1e−B21λ2e−B31λ3w1A01−1w2B01−11−w1−w2C01−1E1λ11−A11λ21−A21λ31−A31,where A11=ξ1+1, A21=ξ2+1, A31=ξ3+1,(39)B11=n−ξ−m1lnt+∑k=1ξ1lny1k,B21=m1−m2lnt+∑k=1ξ2lny2k,B31=m2lnt+∑k=1ξ3lny3k,A01=n−ξ−m1+ξ1+1,B01=m1−m2+ξ2+1,C01=m2+ξ3+1,E1=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA11ΓA21ΓA31B11−A11B21−A21B31−A31BA01,B01,C01.


## 7.2.2. The Joint Prior and Posterior Distributions Assuming the JP

The JP for component parametersλd, d=1,2,3, is pλd∝Iλd, where Iλd is the Fisher information and is Iλd=−E∂2lnLλd;yd/∂λd2. The prior distributions of w1 and w2 are assumed as the UP, i.e., ws∼U0,1, s=1,2. So, the joint prior distribution is(40)ψ2λ1,λ2,λ3,w1,w2|y∝1λ1λ2λ3.Using the JP, the joint posterior distribution ofλ1,λ2,λ3,w1, and w2 that give data y is(41)ξ2λ1,λ2,λ3,w1,w2|y=Lλ1,λ2,λ3,w1,w2|yψ2λ1,λ2,λ3,w1,w2|y∫w2∫w1∫λ3∫λ2∫λ1Lλ1,λ2,λ3,w1,w2|yψ2λ1,λ2,λ3,w1,w2|ydλ1dλ2dλ3dw1dw2,ξ2λ1,λ2,λ3,w1,w2|y=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2e−B12λ1e−B22λ2e−B32λ3w1A02−1w2B02−11−w1−w2C02−1E2λ11−A12λ21−A22λ31−A32,where A12=ξ1, A22=ξ2, A32=ξ3,(42)B12=n−ξ−m1lnt+∑k=1ξ1lny1k,B22=m1−m2lnt+∑k=1ξ2lny2k,B32=m2lnt+∑k=1ξ3lny3k,A02=n−ξ−m1+ξ1+1,B02=m1−m2+ξ2+1,C02=m2+ξ3+1,E2=∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA12ΓA22ΓA32B12−A12B22−A22B32−A32BA02,B02,C02.


## 7.3. Bayes Estimators and Posterior Variances

The Bayes estimators of the component and mixing proportion, i.e.,λ1,λ2,λ3,w1, and w2 using the UP are obtained as(43)λ^α|y=1E1∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓAα1+1ΓAβ1ΓAγ1BA01,C01BB01,A01+C01Bα1Aα1+1Bβ1Aβ1Bγ1Aγ1,where α,β, and γ are defined as (i) α=1,β=2,γ=3 (ii) α=2,β=1,γ=3, and (iii) α=3,β=1,γ=2.(44)w^δ|y=1E1∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA11ΓA21ΓA31Bϒ01,C01BΛ01+1,ϒ01+C01B11A11B21A21B31A31,where δ,ϒ, and Λ are defined as (i) δ=1,ϒ=B,Λ=A and (ii) δ=2,ϒ=A,Λ=B.To measure the accuracy and efficiency of the Bayes estimators, we normally calculate the PVs of the parameters. For the PVs ofλ1,λ2,λ3,w1, and w2, using the UP are derived as(45)Varλ^α|y=1E1∑m1=0n−ξ∑m2=0m1n−ξm1m1m2m1m2ΓAα1+2ΓAβ1ΓAγ1BA01,C01BB01,A01+C01Bα1Aα1+2Bβ1Aβ1Bγ1Aγ1−1E1∑m1=0n−r∑m2=0m1n−ξm1m1m2ΓAα1+1ΓAβ1ΓAγ1BA01,C01BB01,A01+C01Bα1Aα1+1Bβ1Aβ1Bγ1Aγ12,Varw^δ|y=1E1∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA11ΓA21ΓA31Bϒ01,C01BΛ01+2,ϒ01+C01B11A11B21A21B31A31−1E1∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA11ΓA21ΓA31Bϒ01,C01BΛ01+1,ϒ01+C01B11A11B21A21B31A312.The Bayes estimators ofλ1,λ2,λ3,w1, and w2 using the JP are obtained as(46)λ^α|y=1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓAα2+1ΓAβ2ΓAγ2BA02,C02BB02,A02+C02Bα2Aα2+1Bβ2Aβ2Bγ2Aγ2,w^δ|y=1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA12ΓA22ΓA32Bϒ02,C02BΛ02+1,ϒ02+C02B12A12B22A22B32A32.Also, the PVs of the parametersλ1,λ2,λ3,p1, and p2using the JP are derived as(47)Varλ^α|y=1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓAα2+2ΓAβ2ΓAγ2BA02,C02BB02,A02+C02Bα2Aα2+2Bβ2Aβ2Bγ2Aγ2−1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓAα2+1ΓAβ2ΓAγ2BA02,C02BB02,A02+C02Bα2Aα2+1Bβ2Aβ2Bγ2Aγ22,Varw^δ|y=1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA12ΓA22ΓA32Bϒ02,C02BΛ02+2,ϒ02+C02B12A12B22A22B32A32−1E2∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA12ΓA22ΓA32Bϒ02,C02BΛ02+1,ϒ02+C02B12A12B22A22B32A322.


## 7.4. The Bayesian Prediction

One of the main objectives of statistical modeling is the prediction of the future values. The Bayesian methodology allows us to obtain this in a natural way. In particular, the posterior predictive distribution (PPD) comprises the knowledge about future valueX=Yn+1 given data y. Al-Hussaini et al. [45], Bolstad [46], and Bansal [47] have discussed the usefulness of the prediction and predictive distribution comprehensively in the Bayesian framework.


### 7.4.1. The Posterior Predictive Distribution

For the future valueX=Yn+1, the PPD using the UP and the JP is(48)fx|y=∫w2∫w1∫λ3∫λ2∫λ1fx|λ1,λ2,λ3,w1,w2ξvλ1,λ2,λ3,w1,w2|ydλ1dλ2dλ3dw1dw2,(49)fx|y=1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1v+1ΓA2vΓA3vBA0v+1,C0vBB0v,A0v+C0v+1B1v+lnxA1v+1B2vA2vB3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2v+1ΓA3vBA0v,C0vBB0v+1,A0v+C0v+1B1vA1vB2v+lnxA2v+1B3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2vΓA3v+1BA0v,C0v+1BB0v,A0v+C0v+1B1vA1vB2vA2vB3v+lnxA3v+1.In equation, we considerv=1 for UP and v=2 for JP, respectively.


### 7.4.2. Bayesian Predictive Intervals

To obtain the Bayesian predictive intervals (BPIs), letL and U are the two lower and upper endpoints of the BPI, which are obtained from (49). A 1001−α% BPI (L, U) using UP and JP can be obtained by simplifying the given expression:(50)∫1L1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1v+1ΓA2vΓA3vBA0v+1,C0vBB0v,A0v+C0v+1B1v+lnxA1v+1B2vA2vB3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2v+1ΓA3vBA0v,C0vBB0v+1,A0v+C0v+1B1vA1vB2v+lnxA2v+1B3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2vΓA3v+1BA0v,C0v+1BB0v,A0v+C0v+1B1vA1vB2vA2vB3v+lnxA3v+1dx=α2,∫U∞1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1v+1ΓA2vΓA3vBA0v+1,C0vBB0v,A0v+C0v+1B1v+lnxA1v+1B2vA2vB3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2v+1ΓA3vBA0v,C0vBB0v+1,A0v+C0v+1B1vA1vB2v+lnxA2v+1B3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2vΓA3v+1BA0v,C0v+1BB0v,A0v+C0v+1B1vA1vB2vA2vB3v+lnxA3v+1dx=α2.


## 7.4.1. The Posterior Predictive Distribution

For the future valueX=Yn+1, the PPD using the UP and the JP is(48)fx|y=∫w2∫w1∫λ3∫λ2∫λ1fx|λ1,λ2,λ3,w1,w2ξvλ1,λ2,λ3,w1,w2|ydλ1dλ2dλ3dw1dw2,(49)fx|y=1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1v+1ΓA2vΓA3vBA0v+1,C0vBB0v,A0v+C0v+1B1v+lnxA1v+1B2vA2vB3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2v+1ΓA3vBA0v,C0vBB0v+1,A0v+C0v+1B1vA1vB2v+lnxA2v+1B3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2vΓA3v+1BA0v,C0v+1BB0v,A0v+C0v+1B1vA1vB2vA2vB3v+lnxA3v+1.In equation, we considerv=1 for UP and v=2 for JP, respectively.


## 7.4.2. Bayesian Predictive Intervals

To obtain the Bayesian predictive intervals (BPIs), letL and U are the two lower and upper endpoints of the BPI, which are obtained from (49). A 1001−α% BPI (L, U) using UP and JP can be obtained by simplifying the given expression:(50)∫1L1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1v+1ΓA2vΓA3vBA0v+1,C0vBB0v,A0v+C0v+1B1v+lnxA1v+1B2vA2vB3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2v+1ΓA3vBA0v,C0vBB0v+1,A0v+C0v+1B1vA1vB2v+lnxA2v+1B3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2vΓA3v+1BA0v,C0v+1BB0v,A0v+C0v+1B1vA1vB2vA2vB3v+lnxA3v+1dx=α2,∫U∞1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1v+1ΓA2vΓA3vBA0v+1,C0vBB0v,A0v+C0v+1B1v+lnxA1v+1B2vA2vB3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2v+1ΓA3vBA0v,C0vBB0v+1,A0v+C0v+1B1vA1vB2v+lnxA2v+1B3vA3v+1xEv∑m1=0n−ξ∑m2=0m1n−ξm1m1m2ΓA1vΓA2vΓA3v+1BA0v,C0v+1BB0v,A0v+C0v+1B1vA1vB2vA2vB3v+lnxA3v+1dx=α2.


## 8. Monte Carlo Simulation

Here, we tabulate a comprehensive simulation to check the performance of different estimation methods. As an analytical comparison of the Bayes and ML estimators is not possible, a Monte Carlo simulation is performed to measure the performance of the Bayes and ML estimators under different aspects. Through the following steps, we obtained the maximum likelihood estimates (MLEs), MLVs, Bayes estimates (BEs), and PVs as(1)
A sample of sizen from mixtures may be taken as(i)
Generate a sample ofw1n values randomly from f1y;λ1(ii)
Generate a sample ofw2n values randomly from f2y;λ2(iii)
Generate a sample of1−w1−w2n values randomly from f3y;λ3(2)
Select the values that are larger than fixedt as censored values(3)
For a large number of times, say 1000, repeat steps 1 and 2 for the fixedt, n, and parametric values(4)
Evaluate the MLEs and MLVs ofλ1,λ2,λ3,p1, and p2 using nonlinear equations based on the samples obtained in step 3(5)
Calculate the BEs and PVs based on 1000 repetitionsThe above procedure is adopted for 50, 100, and 200 as sample sizesn, parametric values λ1,λ2,λ3,w1,w2=3,4,5,0.5,0.3,5,6,7,0.5,0.3, and t=1.5,2. The choice of t=1.5,2 was done to have 8–20% censored rate in the resulting sample. Moreover, to have a perceptible indication about the behavior of the MLEs, BEs, MLVs, and PVs, we depicted some graphs in Figures 1–4 and Tables 2 and 3.Figure 2
The graph of the BE and MLE of parameterλ2=6.Figure 3
The graph of the BE and MLE of parameterw1=0.5.Figure 4
The graphs of the BE and MLE of parameterw2=0.3.Table 2
MLVs and PVs withλ1=3,λ2=4,λ3=5,w1=0.5,w2=0.3, and t=1.5,2.
Variancestnλ^1λ^2λ^3w^1w^2MLV1.5502.3202144.7154208.9142100.0077940.0068611001.0709122.3940204.3122000.0047010.0039872000.5048411.2201802.1754200.0026990.0022672500.8775002.3125504.8328600.0052450.0044591000.4029511.0692002.2425100.0028060.0023712000.1829210.5407101.0852600.0014480.001219PV (UP)1.5502.5163705.52571010.381600.0080280.0069681001.1106902.5005404.5498700.0047740.0040552000.5044181.2607202.2638100.0027610.0023032500.9308622.4290005.2803100.0052550.0044621000.4180801.1091202.2762400.0028090.0023752000.1832320.5623811.1140900.0014500.001219PV (JP)1.5502.2484904.6786008.6671300.0077400.0068051001.0520802.3545304.2874300.0046810.0039812000.5031921.2148902.1494400.0026980.0022592500.8591032.2573004.7652900.0052410.0044561000.4001171.0674302.2252300.0028050.0023692000.1826050.5432101.0749600.0014480.001218Table 3
MLVs and PVs withλ1=5,λ2=6,λ3=7,w1=0.5,w2=0.3, and t=1.5,2.
Variancestnλ^1λ^2λ^3w^1w^2MLV1.5502.6721505.99421011.455200.0054590.0046721001.2214602.8987205.1975300.0029470.0024832000.5729001.4199402.6214000.0015460.0012932501.4842103.5321007.4251200.0047280.0039961000.6799301.6355403.3945500.0024620.0020752000.3233100.8069901.5574300.0012560.001057PV (UP)1.5502.8158606.27430012.698200.0054850.0046751001.2437402.9914105.4790600.0029650.0024922000.5974481.4504002.6584000.0015560.0013062501.5143503.8674508.3878000.0047310.0040001000.6966891.7003003.4726900.0024630.0020762000.3292320.8163731.5939100.0012570.001057PV (JP)1.5502.6251405.72588011.200900.0054510.0046701001.2172802.8148205.1024900.0029410.0024802000.5710801.4103802.5922800.0015420.0012892501.4625803.4618307.1829800.0047270.0039951000.6773301.6078403.3462600.0024610.0020752000.3232800.8067091.5473600.0012560.001056From Figures1–4, it is observed that parameters λ1,λ2,λ3, and w2 are overestimated, but w1 is underestimated at different values of t and n in both estimation methods, i.e., ML and Bayesian. Also, the degree of underestimation of λ1,λ2,λ3,w1, and w2 is higher for a small n at various values of t, and an opposite behavior was observed for a large t at a given n. Furthermore, the parameters λ1,λ2,λ3,w1, and w2 were observed overestimated to a larger extent when the true values of λ1,λ2, and λ3 were smaller at different values of t for a fixed n. In addition, the similar pattern has been observed at different values of n for a fixed t. The difference of the MLEs and BEs of parameters λ1,λ2,λ3,w1, and w2 from the nominal values becomes the minimum with the increase of t and n.It can be seen from Tables2 and 3 that at different values of t, the difference between the MLVs and the PVs (assuming the UP and the JP) diminishes by increasing sample size. The same remark is true for a large t at different values of n. Also, noticed that the MLVs and PVs of w1 and w2 are larger for smaller values of λ1,λ2, and λ3 at different values of t and n. Also, it is pointed out that the performance of the Bayes estimators using JP is best than Bayes estimators using UP and ML estimators based on lesser associated PVs.To the extent that the selection of an appropriate prior, Tables2 and 3 revealed that the JP outperforms as compared to the UP because the variance of JP is smaller than the UP.Table4 and Figure 5 showcase the 90% BPI using the UP and the JP. It is pointed out that the width of 90% BPI increases with a decrease in n. The same conclusion was observed with a smaller t for varying values of n. The 90% BPIs, for larger values of λ1,λ2, and λ3, were observed narrow at various values of t and n. Moreover, the BPIs using the JP were observed wider than the BPIs obtained by assuming the UP in the simulation study.Table 4
The BPI (L, U) with λ1=3,λ2=4,λ3=5,, w1=0.5,w2=0.3, and t=1.5,2.
tnUPJPLULU1.5501.01242.70821.01343.00041001.01082.42461.01122.70942001.01292.38661.01332.56102501.01302.52271.01392.69821001.01162.39751.01212.46092001.01322.34051.01342.3687Figure 5
The graph of widths (d = L − U) of the BPIs with λ1 = 5, λ2 = 6, λ3 = 7, w1=0.05,w2 and t = 1.5, 2.


## 9. Real Data Application

To illustrate the proposed methodology, the mixture lifetime data,z=z11,z12,…,z1r1,z21,z22,…,z2r2,z31,z32,…,z3r3 in thousand hours, was taken from Davis [48] on three factors, i.e., V805 Transmitter, Transmitter, and V600 Indicator Tube used in aircraft sets. For exponential distributed mixture data z, the suitable transformation y=expz gives the Pareto distributed mixture data y. So, the mention transformation permits to utilize the given mixture data z for using the suggested ML and Bayesian estimation techniques. Thus, the proposed mixture of the Pareto distributions can be a fair choice to model the abovementioned data. Moreover, it is unidentified that which factor fails until a failure arises at or before 0.6 hours. To calculate the MLEs, MLVs, BEs, and PVs, the data summary is(51)n=1340,∑k=1r1lny1k=∑k=1r1z1k=134.080,r1=866,∑k=1r2lny2k=∑k=1r2z2k=50.375,r2=337,∑k=1r3lny3k=∑k=1r3z3k=16.250,r3=83,r=1286,n−r=54.The MLEs, MLVs, BEs, and PVs assuming the UP and the JP are presented in Table5.Table 5
MLEs, MLVs, BEs, and PVs assuming the UP and the JP with real-life data.
Estimates and variancesλ^1λ^2λ^3w^1w^2MLE5.67025.96993.58540.66890.2594MLV0.05140.16490.38100.000181460.00015131BE (UP)5.66735.98133.64700.66910.2594PV (UP)0.05190.16530.39390.000181260.00015134BE (JP)5.67215.96643.55390.66870.2594PV (JP)0.05120.16450.37990.000181510.00015129From Table5, it is clear that the performance of the BEs using JP is the best as compared to the MLEs, as the variance of BEs is smaller than the counterpart. Moreover, the BEs using the UP have smaller variances for estimating the unknown parameters. Also, the JP was observed superior to the UP due to smaller associated PV for estimating the unknown parameters.


## 10. Conclusion

In this study, we proposed TCMPD to model lifetime data. Parameter estimation assuming the type I censoring have been considered using the ML and Bayesian estimation methods. For Bayesian estimation, we assumed the noninformative priors and expressions of the Bayes estimators for the mixing proportion (w1 and w2) and component parameters (λ1,λ2, and λ3) and PVs were derived. To examine the relative presentation of the Bayes and ML estimators under different scenario, a Monte Carlo simulation has been done. To illustrate a practical presentation of proposed mixture distribution, an example has also been analyzed.From simulated results and depicted graphs, it has been noticed that an increase int under a fixed n yields very efficient Bayes and ML estimators. It is also pointed that parameters λ1,λ2,λ3,w1, and w2 are overestimated (underestimated) to a small (larger) extent with relatively larger (smaller) value of n (value of t). More specifically, the amount of overestimation (underestimation) of parameters is smaller for a relatively large parameter value. As the value of n (value of t) increases (decreases), the PVs decrease (increase) for a fixed t (fixed n). To address the problem of selection of a suitable prior, one can observe that the JP has smaller PVs than the UP. The results depend on real-life mixture data that also support the Monte Carlo simulation study. Finally, it is concluded that the Bayes estimators using the JP performed better as compared to the ML estimators because of smaller variances.



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