# The Mixture of the Marshall–Olkin Extended Weibull Distribution under Type-II Censoring and Different Loss Functions

Authors: Refah Alotaibi; Mervat Khalifa; Lamya A. Baharith; Sanku Dey; H. Rezk
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

To study the heterogeneous nature of lifetimes of certain mechanical or engineering processes, a mixture model of some suitable lifetime distributions may be more appropriate and appealing as compared to simple models. This paper considers a mixture of the Marshall–Olkin extended Weibull distribution for efficient modeling of failure, survival, and COVID-19 data under classical and Bayesian perspectives based on type-II censored data. We derive several properties of the new distribution such as moments, incomplete moments, mean deviation, average lifetime, mean residual lifetime, Rényi entropy, Shannon entropy, and order statistics of the proposed distribution. Maximum likelihood and Bayes procedure are used to derive both point and interval estimates of the parameters involved in the model. Bayes estimators of the unknown parameters of the model are obtained under symmetric (squared error) and asymmetric (linear exponential (LINEX)) loss functions using gamma priors for both the shape and the scale parameters. Furthermore, approximate confidence intervals and Bayes credible intervals (CIs) are also obtained. Monte Carlo simulation study is carried out to assess the performance of the maximum likelihood estimators and Bayes estimators with respect to their estimated risk. The flexibility and importance of the proposed distribution are illustrated by means of four real datasets.


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## Body

## 1. Introduction

In the history of statistics, the use of finite mixture models is very old. They were particularly used to model population heterogeneity, generalize distributional assumptions, clustering and classification, etc. The concept of the finite mixture distribution was pioneered by Newcomb [1] for modeling outliers. Eight years later, Pearson [2] considered a mixture of two univariate Gaussian distributions to estimate the parameters of the model using the method of moments (MOM) to analyze a dataset containing ratios of the forehead to body lengths for 1,000 crabs. Since then, several authors have studied finite mixture models under different scenarios. Mendenhall and Hader [3] considered exponentially distributed failure time distributions based on censored lifetime data to estimate the model parameters using the maximum likelihood method. In their study, they divided the failure population into two subpopulations, each representing a different cause or type of failure. Radhakrishna et al. [4] considered both moment and maximum likelihood estimators of the unknown parameters of a two-component mixture of generalized gamma distribution. Ahmed et al. [5] obtained approximate Bayes estimators for parameters of the mixture of two Weibull distributions under type-II censoring. Al-Hussaini et al. [6] applied both maximum likelihood and Bayes estimation methods on a two-component mixture of the Gompertz distribution based on type-I and type-II censoring. Jaheen [7] adopted both maximum likelihood and Bayesian approach to discuss the problem of estimating the parameters using the finite mixture of two exponential distributions based on record statistics. Shawky and Bakoban [8] adopted both maximum likelihood and Bayesian approach to estimate the parameters of the model, reliability, and failure rate functions of two-component finite mixtures of exponentiated gamma distribution. Abu-Zinadah [9] used maximum likelihood and Bayes estimation methods to estimate the parameters, reliability, and hazard functions of a mixture of exponentiated Pareto and exponential distribution under complete and type-II censoring schemes. Prakash [10] adopted the Bayes method to estimate the parameters from the mixture of two Weibull distributions based on informative and noninformative priors. Zhang et al. [11] proposed a mixture Weibull proportional hazard model to predict the failure of a mechanical system with multiple failure modes. They estimated the mixed model parameters by combining historical lifetime and monitoring data of all failure modes. ALgfary [12] introduced the finite mixture of two exponentiated Kumaraswamy (MEKum) distributions and obtained the maximum likelihood estimates for the vector of the parameters of the MEKum distribution. Adham and ALgfary [13] adopted the Bayesian approach to estimate the vector of parameters of the finite mixture of two-component exponentiated Kumaraswamy distribution. They also obtained Bayesian predictive density functions of future observations from the MEKum distribution. Ateya and Al Khald [14] studied the finite mixture of truncated generalized Cauchy distribution based on type-I, type-II, and progressively type-II censored samples. Aslam et al. [15] studied the three-component mixture of exponential, Rayleigh, Pareto, and Burr type-XII distributions in relation to reliability analysis. Tahir et al. [16] studied the properties of the three-component mixture of Rayleigh distributions based on doubly censored lifetime data. Kalantan and Alrewely [17] studied the two-component Laplace mixture model and obtained the estimates of the parameters using maximum likelihood and method of moments. Recently, Tahir et al. [18] also studied the three-component mixture of exponential distributions from the Bayesian perspective based on type-II doubly censoring sampling scheme. Kharazmi et al. [19] studied the 2-component mixture of Topp–Leone distribution and obtained classical and Bayes estimators based on the complete sample and the references cited therein.Wide applicability of mixture modeling motivates us to develop a two-component mixture of Marshall–Olkin extended Weibull distribution for efficient modeling of breaking stress of carbon fibers, survival times in days of 72 guinea pigs infected datasets, survival times in weeks of 33 patients suffering from acute myelogenous leukemia, and COVID-19 data belonging to Canada of 36 days. In this article, the primary objective of the paper is twofold. First, we obtain maximum likelihood estimators and corresponding approximate confidence intervals (CIs) of the unknown parameters of the 2-component mixture of the Marshall–Olkin extended Weibull (MOEW) distribution for type-II censored data. Next, we consider the Bayes estimation method. The Bayes estimators have been derived and evaluated under the assumption of two loss functions using independent gamma priors. Symmetric1001−τ% two-sided Bayes credible intervals are also obtained, and they are compared with classical CIs. To the best of our knowledge, the 2-component mixture of the MOEW distribution is not discussed before using the aforementioned methods of estimation. Through this paper, we purport to provide some guidelines on selecting the best estimator that may be of significant interest to applied statisticians/practitioners/engineers.The organization of this paper is as follows. The description of the model along with basic properties is reported in Section2. We use the maximum likelihood estimation method based on type-II censoring as a part of frequentist methodology for parameter estimation in Section 3. We have also taken into account approximate CIs in the same section. In Section 4, we have derived the Bayes estimators of the unknown parameters of the model under squared error loss (SEL) and linear exponential (LINEX) loss functions using gamma priors for both scale and shape parameters. We have also obtained two-sided Bayes probability intervals in the same section. The simulation study is carried out in Section 5. For illustrative purposes, four real datasets are analyzed in Section 6. Finally, concluding remarks are presented in Section 7.


## 2. Model Description

The probability density function (pdf) of the Marshall–Olkin extended Weibull distribution (MOEW) for a random variableX is defined by (see Ghitany et al. [20] and Zhang and Xie [21])(1)fx=αλθxθ−1e−λxθ1−a¯e−λxθ2,x≥0,α,λ,θ>0,where a¯=1−a, and the cumulative distribution function (cdf) of the distribution is(2)Fx=1−e−λxθ1−α¯e−λxθ,x≥0.The hazard rate function of MOEW takes the form(3)hx=λθxθ−11−α¯e−λxθ,x≥0.A density function for the mixture of two components’ densities with mixing proportionspj,j=1,2 is(4)gx=∑j=12pjfjx,wherefjx=αjθjλjxθj−1e−λjxθj1−α¯jeλjxθj−2,where pj are the mixing proportions, satisfying the conditions ∑j=12pj=1 and pj≥0; all of them are unknowns, and the pdf is plotted in Figure 1. The cdf for the mixture model is(5)Gx=∑j=12pjFjx,where(6)Fjx=1−e−λjxθj1−α¯eλjxθj.Figure 1
pdf of the two-component mixture of the Marshall–Olkin extended Weibull distribution for different values of the parameters, wherec1=α1=2,θ1=3,λ1=1,α2=2,θ2=3,λ2=1,p1=0.4, c2=α1=1,θ1=5,λ1=2,α2=2,θ2=3,λ2=5,p1=0.3, and c3=α1=1,θ1=5,λ1=2,α2=4,θ2=3,λ2=5,p1=0.2.The reliability function for the mixture model is(7)Sx=∑j=12pjRjx,where(8)Rjx=1−1−e−λjxθj1−α¯eλjxθj.Due to exponentiation of each component by a positive integer, the model is so flexible that shows different shapes of hazard rate function (hrf) of the mixture which is given by(9)Hx=∑j=12pjhjx,hjx=αjθjλjxθj−1e−λjxθj1−α¯jeλjxθj1−α¯eλjxθj−1−α¯eλjxθj1−e−λjxθj.


## 3. General Properties of the MOEW Distribution

### 3.1. Moments

Therth moment of a finite mixture of the 2-component MOEW distribution is given by(10)μr′=∑j=12pjμr=∫0∞p1α1θ1λ1xθ1+r−1e−λ11−α¯1eλ1xθ1−2+∫0∞1−p1α2θ2λ2xθ2+r−1e−λ21−α¯2eλ2xθ2−2=p1α1λ1−λ1−r/θ1Γr+θ1θ1+1−p1α2λ2−λ2−r/θ2Γr+θ2θ2,where p2=1−p1.Whenr = 1, the mean is given by(11)μ1′=p1α1λ1−λ1−1/θ1Γ1+θ1θ1+1−p1α2λ2−λ2−1/θ2Γ1+θ2θ2.The moment-generating function of the mixture MOEW distribution is given by(12)μxt=∫0∞etxfjxdx=p1θ1λ1xθ1e−λ1xθ1/α1+1−p1θ2λ2xθ2e−λ1xθ2/α2x2.


### 3.2. Incomplete Moments

Therth incomplete moment of a finite mixture of the 2-component mixture distribution is given by(13)Trz=Exr=∫0zxrfxdx=∑j=12pj∫0zxrfjxdx=∑j=12pjμjr,where Tjr=∫0zxrfjxdx is the rth incomplete moment of the jth component so that the rth incomplete moment of a finite mixture of the 2-component MOEW distribution is given by(14)Trz=∑j=12pj=∫0zp1α1θ1λ1xθ1+r−1e−λ11−α¯1eλ1xθ1−2+∫0z1−p1α2θ2λ2xθ2+r−1e−λ21−α¯2eλ2xθ2−2dx.The first incomplete moment of a finite mixture of the 2-component MOEW distribution is given by(15)T1z=∑j=12pjμj1=∫0zp1α1θ1λ1xθ1e−λ11−α¯1eλ1xθ1−2∫0z1−p1α2θ2λ2xθ2e−λ21−α¯2eλ2xθ2−2dx.


### 3.3. Mean Deviations

The mean deviations of the random variableX about the mean, μ, and the median, M, are given, respectively, by(16)δ1=∫x.x−μfx,α,λ,θdx,δ2=∫x.x−Mfx,α,λ,θdx,where δ1=p1δ1j+1−p1δ1j and δ2=p1δ2j+1−p1δ2j since δ1j=∫0∞x−μfjx,αj,λj,θjdx and δ2j=∫0∞x−Mfjx,αj,λj,θjdx.(17)δ1=∫0∞x−μfx,α,λ,θdx,δ1=∫0∞x−μp1α1λ1θ1xθ1−1e−λ1xθ11−α1¯e−λ1xθ12+1−p1α2λ2θ2xθ2−1e−λ2xθ21−α2¯e−λ2xθ22dx=p1α1λ1θ1xθ1−1e−λ1xθ11−α1¯e−λ1xθ12+1−p1α2λ2θ2xθ2−1e−λ2xθ21−α2¯e−λ2xθ22,δ2=∫0∞x−Mp1α1λ1θ1xθ1−1e−λ1xθ11−α1¯e−λ1xθ12+1−p1α2λ2θ2xθ2−1e−λ2xθ21−α2¯e−λ2xθ22dx.The median follows from the nonlinear equationFx,α,λ,θ=1/2. So, these quantities reduce to(18)δ1Fx,α,λ,θ2μFx,α,λ,θ−2T1μ,δ2=μ−2T1M,where T1(z) is the first incomplete moment of X obtained from (15),(19)δ1j=2μjFjx,αj,λj,θj−2T1μj,δ2j=μj−2T1Mj,where(20)δ1=2μp11−e−λ1x0θ11−α1¯e−λ1x0θ1+1−p11−e−λ2x0θ21−α2¯e−λ2x0θ2−2T1μ,δ1=μ−2T1M.


### 3.4. Average Lifetime and Mean Residual Lifetime Functions

The average lifetime is given by(21)tm=∫0∞1−Fx,α,λ,θdx,tm=∫0∞1−p11−e−λ1xθ11−α1¯e−λ1xθ1+1−p11−e−λ2xθ21−α2¯e−λ2xθ2dx.The application of mean residual lifetime can be seen in the paper of Guess and Proschan [22]. The mean residual lifetime is given by(22)mx0,α,λ,θ=EX−x0|X≥x0,α,λ,θ=∫X:X−x0∞X−x0fx,α,k+1λ,θPrX≥x0dx=PrX≥x0−1∫0∞yfx0+y,α,λ,θdy=1−p11−e−λ1x0θ11−α1¯e−λ1x0θ1+1−p11−e−λ2x0θ21−α2¯e−λ2x0θ2−1=∫0∞yp1α1λ1θ1x0+yθ1−1e−λ1x0+yθ11−α1¯e−λ1x0+yθ12+1−p1α2λ2θ2x0+yθ2−1e−λ2x0+yθ21−α2¯e−λ2x0+yθ22dy.Then,mx0,α,λ,θ⟶EXasx0⟶0.


### 3.5. Rényi Entropy

The Rényi entropy ofX with pdf (4) is given by(23)HRSx=11−Slog∫x.fx,α,λ,θSdx,where S∈0,1∪1,∞.(24)HRSx=11−Slog∫0∞p1α1λ1θ1xθ1−1e−λ1xθ11−α1¯e−λ1xθ12+1−p1α2λ2θ2xθ2−1e−λ2xθ21−α2¯e−λ2xθ22Sdx.It is difficult to obtainHRSx in the closed form for a finite mixture of the 2-component MOEW distribution.


### 3.6. Shannon Entropy

The Shannon entropy ofX is given by(25)Hsx=Ex−logfx,α,λ,θ.Thus, from (4), we can get the log-likelihood function as(26)logfjx,αj,λj,θj=logαjλj+logθjxθj−1−λjxθj−2log1−α¯j⋅exp−λjxθj.Thus, the above equation can be rewritten as(27)Hsx=−logPα1λ1+2Elog1−α¯1exp−λ1xθ1−Exlogθ1xθ1−1+λ1Exθ1+qα2λ2+2Elog1−α¯2⋅exp−λ2xθ2−Exlogθ2xθ2−1+λ2Exθ2.


### 3.7. Distribution of Order Statistics

The pdf of therth order statistic for a random sample x1,x2,x3,…,xn from the MOEW distribution with pdf and cdf given in equations (1) and (2), respectively, is given by(28)fr,nx=n!n−r!r−1!fXFXr−11−FXn−r,where(29)1−FXn−r=∑i=0n−r−1in−riFXi.Therth order statistics for a finite mixture of the 2-component MOEW distribution can be obtained by substituting equations (4) and (6) into (28); thus, we have(30)fr,nx=n!n−r!r−1!∑i=0n−r−1in−rip1α1λ1θ1xθ1−1e−λ1xθ11−e−λ1xθ1r+i−11−α1¯e−λ1xθ1r+i+1+1−p1α2λ2θ2xθ2−1e−λ2xθ21−e−λ2xθ2r+i−11−α2¯e−λ2xθ2r+i+1.


### 3.8. Maximum Likelihood Estimation Based on Type-II Censoring

Here, we discuss the maximum likelihood estimates of the unknown parameters of the 2-component mixture of the MOEW distributions. In a life testing experiment,n items from the above mixture model are employed to test and get terminated when a preassigned number of items, say r (<n), have failed. The samples obtained from such an experiment are called failure-censored samples or type-II censored samples. In the failure-censored case, data comprise the lifetime of the r items that have failed (say x1<x2<⋯<xr), and the remaining (n − r) items have survived beyond xr with the assumption that the lifetime distribution of the items is independent and identically distributed MOEW distribution; the likelihood function for the type-II censoring scheme can be written as(31)Lθ1,θ2,λ1,λ2,α1,α2,p1|x¯∝∏i=1rfjxiRjxrn−r=p1rθ1rλ1rα1r∏i=1rxiθ1−1e−λ1xiθ11−1−α1e−λ1xiθ1−21−1−e−λ1xrθ1.1−1−α1e−λ1xrθ1n−r+1−p1rθ2rλ2rα2r∏i=1rxiθ2−1e−λ2xiθ21−1−α2e−λ2xiθ2−21−1−e−λ2xrθ21−1−α2e−λ2xrθ2n−r.The corresponding log-likelihood function can be written as(32)ℓ=logL=rlogp1+rlogθ1+rlogλ1+rlogα1+θ1−1∑i=1rlogxi−λ1∑i=1rxiθ1−2∑i=1rlog1−1−α1e−λ1xiθ1+n−rlog1−1−e−λ1xrθ11−1−α1e−λ1xrθ1+rlog1−p1+rlogθ2+rlogλ2+rlogα2+θ2−1∑i=1rlogxi−λ2∑i=1rxiθ2−2∑i=1rlog1−1−α2e−λ2xiθ2+n−rlog1−1−e−λ2xrθ21−1−α2e−λ2xrθ2.The resulting normal equations are(33)∂ℓ∂α1=rα1−2∑i=1re−λ1xiθ11−1−α1e−λ1xiθ1+n−re−λ1xrθ11−1−α1e−λ1xrθ1−21−1−e−λ1xrθ1/1−1−α1e−λ1xrθ1,(34)∂ℓ∂α2=rα2−2∑i=1re−λ2xiθ21−1−α2e−λ2xiθ2+n−re−λ2xrθ21−1−α2e−λ2xrθ2−21−1−e−λ2xrθ2/1−1−α2e−λ2xrθ2,(35)∂ℓ∂θ1=rθ1+∑i=1rlogxi−λ1∑i=1rxiθ1logxi+2∑i=1r1−α1λ1logxixiθ1e−λ1xiθ11−1−α1e−λ1xiθ1−n−rα1λ1logxrxrθ1e−λ1xiθ11−1−α1e−λ1xrθ1/1−1−α1e−λ1xrθ121−1−e−λ1xrθ1/1−1−α1e−λ1xrθ1,(36)∂ℓ∂θ2=rθ2+∑i=1rlogxi−λ2∑i=1rxiθ2logxi+2∑i=1r1−α2λ2logxixiθ2e−λ2xiθ21−1−α2e−λ2xiθ2−n−rα2λ2logxrxrθ2e−λ2xiθ21−1−α2e−λ2xrθ2/1−1−α1e−λ1xrθ121−1−e−λ2xrθ2/1−1−α2e−λ2xrθ2,(37)∂ℓ∂λ1=rλ1−∑i=1rxiθ1−2∑i=1r1−α1xiθ1e−2λ1xiθ11−1−α1e−λ1xiθ1+n−rxrθ1e−λ1xiθ11−1−α1e−λ1xrθ1+1−α11−e−λ1xrθ1/1−1−α1e−λ1xrθ121−1−e−λ1xrθ1/1−1−α1e−λ1xrθ1,(38)∂ℓ∂λ2=rλ2−∑i=1rxiθ2−2∑i=1r1−α2xiθ2e−2λ2xi21−1−α2e−λ2xiθ2+n−rxrθ2e−λ2xiθ21−1−α2e−λ2xrθ2+1−α21−e−λ2xrθ2/1−1−α2e−λ2xrθ221−1−e−λ2xrθ2/1−1−α2e−λ2xrθ2,(39)∂ℓ∂p1=rp1−r1−p1.The MLEs ofΦ=α1,α2,θ1,θ2,λ1,λ2andp1 can be obtained by solving equations (32)–(39) simultaneously. Since explicit solutions cannot be obtained from the above equations, thus, we propose to use a suitable numerical technique to solve these seven nonlinear equations; however, one may use Newton–Raphson to solve these equations. This can be routinely done using R packages.


### 3.9. Approximate Confidence Intervals

In this section, under the normality property of MLEs of the parametersΦ=α1,α2,θ1,θ2,λ1,λ2,p1, we obtain the asymptotic confidence interval. The asymptotic distribution of the MLE Φ^ is Φ^−Φ⟶N0,I−1Φ, see Lawless [23], where I−1Φ, the inverse of the observed information matrix of the unknown parameters Φ=α1,α2,θ1,θ2,λ1,λ2,p1, is I−1Φ=∂2ℓ/∂Φ2α1,α2,θ1,θ2,λ1,λ2,p1=α^1,α^2,θ^1,θ^2,λ^1,λ^2,p1^−1.The above approach is used to derive approximate1001−τ% confidence intervals of the parameters Φ of the forms Φ^∓zτ/2varΦ^, where zτ/2 is the upper τ/2th percentile of the standard normal distribution.


## 3.1. Moments

Therth moment of a finite mixture of the 2-component MOEW distribution is given by(10)μr′=∑j=12pjμr=∫0∞p1α1θ1λ1xθ1+r−1e−λ11−α¯1eλ1xθ1−2+∫0∞1−p1α2θ2λ2xθ2+r−1e−λ21−α¯2eλ2xθ2−2=p1α1λ1−λ1−r/θ1Γr+θ1θ1+1−p1α2λ2−λ2−r/θ2Γr+θ2θ2,where p2=1−p1.Whenr = 1, the mean is given by(11)μ1′=p1α1λ1−λ1−1/θ1Γ1+θ1θ1+1−p1α2λ2−λ2−1/θ2Γ1+θ2θ2.The moment-generating function of the mixture MOEW distribution is given by(12)μxt=∫0∞etxfjxdx=p1θ1λ1xθ1e−λ1xθ1/α1+1−p1θ2λ2xθ2e−λ1xθ2/α2x2.


## 3.2. Incomplete Moments

Therth incomplete moment of a finite mixture of the 2-component mixture distribution is given by(13)Trz=Exr=∫0zxrfxdx=∑j=12pj∫0zxrfjxdx=∑j=12pjμjr,where Tjr=∫0zxrfjxdx is the rth incomplete moment of the jth component so that the rth incomplete moment of a finite mixture of the 2-component MOEW distribution is given by(14)Trz=∑j=12pj=∫0zp1α1θ1λ1xθ1+r−1e−λ11−α¯1eλ1xθ1−2+∫0z1−p1α2θ2λ2xθ2+r−1e−λ21−α¯2eλ2xθ2−2dx.The first incomplete moment of a finite mixture of the 2-component MOEW distribution is given by(15)T1z=∑j=12pjμj1=∫0zp1α1θ1λ1xθ1e−λ11−α¯1eλ1xθ1−2∫0z1−p1α2θ2λ2xθ2e−λ21−α¯2eλ2xθ2−2dx.


## 3.3. Mean Deviations

The mean deviations of the random variableX about the mean, μ, and the median, M, are given, respectively, by(16)δ1=∫x.x−μfx,α,λ,θdx,δ2=∫x.x−Mfx,α,λ,θdx,where δ1=p1δ1j+1−p1δ1j and δ2=p1δ2j+1−p1δ2j since δ1j=∫0∞x−μfjx,αj,λj,θjdx and δ2j=∫0∞x−Mfjx,αj,λj,θjdx.(17)δ1=∫0∞x−μfx,α,λ,θdx,δ1=∫0∞x−μp1α1λ1θ1xθ1−1e−λ1xθ11−α1¯e−λ1xθ12+1−p1α2λ2θ2xθ2−1e−λ2xθ21−α2¯e−λ2xθ22dx=p1α1λ1θ1xθ1−1e−λ1xθ11−α1¯e−λ1xθ12+1−p1α2λ2θ2xθ2−1e−λ2xθ21−α2¯e−λ2xθ22,δ2=∫0∞x−Mp1α1λ1θ1xθ1−1e−λ1xθ11−α1¯e−λ1xθ12+1−p1α2λ2θ2xθ2−1e−λ2xθ21−α2¯e−λ2xθ22dx.The median follows from the nonlinear equationFx,α,λ,θ=1/2. So, these quantities reduce to(18)δ1Fx,α,λ,θ2μFx,α,λ,θ−2T1μ,δ2=μ−2T1M,where T1(z) is the first incomplete moment of X obtained from (15),(19)δ1j=2μjFjx,αj,λj,θj−2T1μj,δ2j=μj−2T1Mj,where(20)δ1=2μp11−e−λ1x0θ11−α1¯e−λ1x0θ1+1−p11−e−λ2x0θ21−α2¯e−λ2x0θ2−2T1μ,δ1=μ−2T1M.


## 3.4. Average Lifetime and Mean Residual Lifetime Functions

The average lifetime is given by(21)tm=∫0∞1−Fx,α,λ,θdx,tm=∫0∞1−p11−e−λ1xθ11−α1¯e−λ1xθ1+1−p11−e−λ2xθ21−α2¯e−λ2xθ2dx.The application of mean residual lifetime can be seen in the paper of Guess and Proschan [22]. The mean residual lifetime is given by(22)mx0,α,λ,θ=EX−x0|X≥x0,α,λ,θ=∫X:X−x0∞X−x0fx,α,k+1λ,θPrX≥x0dx=PrX≥x0−1∫0∞yfx0+y,α,λ,θdy=1−p11−e−λ1x0θ11−α1¯e−λ1x0θ1+1−p11−e−λ2x0θ21−α2¯e−λ2x0θ2−1=∫0∞yp1α1λ1θ1x0+yθ1−1e−λ1x0+yθ11−α1¯e−λ1x0+yθ12+1−p1α2λ2θ2x0+yθ2−1e−λ2x0+yθ21−α2¯e−λ2x0+yθ22dy.Then,mx0,α,λ,θ⟶EXasx0⟶0.


## 3.5. Rényi Entropy

The Rényi entropy ofX with pdf (4) is given by(23)HRSx=11−Slog∫x.fx,α,λ,θSdx,where S∈0,1∪1,∞.(24)HRSx=11−Slog∫0∞p1α1λ1θ1xθ1−1e−λ1xθ11−α1¯e−λ1xθ12+1−p1α2λ2θ2xθ2−1e−λ2xθ21−α2¯e−λ2xθ22Sdx.It is difficult to obtainHRSx in the closed form for a finite mixture of the 2-component MOEW distribution.


## 3.6. Shannon Entropy

The Shannon entropy ofX is given by(25)Hsx=Ex−logfx,α,λ,θ.Thus, from (4), we can get the log-likelihood function as(26)logfjx,αj,λj,θj=logαjλj+logθjxθj−1−λjxθj−2log1−α¯j⋅exp−λjxθj.Thus, the above equation can be rewritten as(27)Hsx=−logPα1λ1+2Elog1−α¯1exp−λ1xθ1−Exlogθ1xθ1−1+λ1Exθ1+qα2λ2+2Elog1−α¯2⋅exp−λ2xθ2−Exlogθ2xθ2−1+λ2Exθ2.


## 3.7. Distribution of Order Statistics

The pdf of therth order statistic for a random sample x1,x2,x3,…,xn from the MOEW distribution with pdf and cdf given in equations (1) and (2), respectively, is given by(28)fr,nx=n!n−r!r−1!fXFXr−11−FXn−r,where(29)1−FXn−r=∑i=0n−r−1in−riFXi.Therth order statistics for a finite mixture of the 2-component MOEW distribution can be obtained by substituting equations (4) and (6) into (28); thus, we have(30)fr,nx=n!n−r!r−1!∑i=0n−r−1in−rip1α1λ1θ1xθ1−1e−λ1xθ11−e−λ1xθ1r+i−11−α1¯e−λ1xθ1r+i+1+1−p1α2λ2θ2xθ2−1e−λ2xθ21−e−λ2xθ2r+i−11−α2¯e−λ2xθ2r+i+1.


## 3.8. Maximum Likelihood Estimation Based on Type-II Censoring

Here, we discuss the maximum likelihood estimates of the unknown parameters of the 2-component mixture of the MOEW distributions. In a life testing experiment,n items from the above mixture model are employed to test and get terminated when a preassigned number of items, say r (<n), have failed. The samples obtained from such an experiment are called failure-censored samples or type-II censored samples. In the failure-censored case, data comprise the lifetime of the r items that have failed (say x1<x2<⋯<xr), and the remaining (n − r) items have survived beyond xr with the assumption that the lifetime distribution of the items is independent and identically distributed MOEW distribution; the likelihood function for the type-II censoring scheme can be written as(31)Lθ1,θ2,λ1,λ2,α1,α2,p1|x¯∝∏i=1rfjxiRjxrn−r=p1rθ1rλ1rα1r∏i=1rxiθ1−1e−λ1xiθ11−1−α1e−λ1xiθ1−21−1−e−λ1xrθ1.1−1−α1e−λ1xrθ1n−r+1−p1rθ2rλ2rα2r∏i=1rxiθ2−1e−λ2xiθ21−1−α2e−λ2xiθ2−21−1−e−λ2xrθ21−1−α2e−λ2xrθ2n−r.The corresponding log-likelihood function can be written as(32)ℓ=logL=rlogp1+rlogθ1+rlogλ1+rlogα1+θ1−1∑i=1rlogxi−λ1∑i=1rxiθ1−2∑i=1rlog1−1−α1e−λ1xiθ1+n−rlog1−1−e−λ1xrθ11−1−α1e−λ1xrθ1+rlog1−p1+rlogθ2+rlogλ2+rlogα2+θ2−1∑i=1rlogxi−λ2∑i=1rxiθ2−2∑i=1rlog1−1−α2e−λ2xiθ2+n−rlog1−1−e−λ2xrθ21−1−α2e−λ2xrθ2.The resulting normal equations are(33)∂ℓ∂α1=rα1−2∑i=1re−λ1xiθ11−1−α1e−λ1xiθ1+n−re−λ1xrθ11−1−α1e−λ1xrθ1−21−1−e−λ1xrθ1/1−1−α1e−λ1xrθ1,(34)∂ℓ∂α2=rα2−2∑i=1re−λ2xiθ21−1−α2e−λ2xiθ2+n−re−λ2xrθ21−1−α2e−λ2xrθ2−21−1−e−λ2xrθ2/1−1−α2e−λ2xrθ2,(35)∂ℓ∂θ1=rθ1+∑i=1rlogxi−λ1∑i=1rxiθ1logxi+2∑i=1r1−α1λ1logxixiθ1e−λ1xiθ11−1−α1e−λ1xiθ1−n−rα1λ1logxrxrθ1e−λ1xiθ11−1−α1e−λ1xrθ1/1−1−α1e−λ1xrθ121−1−e−λ1xrθ1/1−1−α1e−λ1xrθ1,(36)∂ℓ∂θ2=rθ2+∑i=1rlogxi−λ2∑i=1rxiθ2logxi+2∑i=1r1−α2λ2logxixiθ2e−λ2xiθ21−1−α2e−λ2xiθ2−n−rα2λ2logxrxrθ2e−λ2xiθ21−1−α2e−λ2xrθ2/1−1−α1e−λ1xrθ121−1−e−λ2xrθ2/1−1−α2e−λ2xrθ2,(37)∂ℓ∂λ1=rλ1−∑i=1rxiθ1−2∑i=1r1−α1xiθ1e−2λ1xiθ11−1−α1e−λ1xiθ1+n−rxrθ1e−λ1xiθ11−1−α1e−λ1xrθ1+1−α11−e−λ1xrθ1/1−1−α1e−λ1xrθ121−1−e−λ1xrθ1/1−1−α1e−λ1xrθ1,(38)∂ℓ∂λ2=rλ2−∑i=1rxiθ2−2∑i=1r1−α2xiθ2e−2λ2xi21−1−α2e−λ2xiθ2+n−rxrθ2e−λ2xiθ21−1−α2e−λ2xrθ2+1−α21−e−λ2xrθ2/1−1−α2e−λ2xrθ221−1−e−λ2xrθ2/1−1−α2e−λ2xrθ2,(39)∂ℓ∂p1=rp1−r1−p1.The MLEs ofΦ=α1,α2,θ1,θ2,λ1,λ2andp1 can be obtained by solving equations (32)–(39) simultaneously. Since explicit solutions cannot be obtained from the above equations, thus, we propose to use a suitable numerical technique to solve these seven nonlinear equations; however, one may use Newton–Raphson to solve these equations. This can be routinely done using R packages.


## 3.9. Approximate Confidence Intervals

In this section, under the normality property of MLEs of the parametersΦ=α1,α2,θ1,θ2,λ1,λ2,p1, we obtain the asymptotic confidence interval. The asymptotic distribution of the MLE Φ^ is Φ^−Φ⟶N0,I−1Φ, see Lawless [23], where I−1Φ, the inverse of the observed information matrix of the unknown parameters Φ=α1,α2,θ1,θ2,λ1,λ2,p1, is I−1Φ=∂2ℓ/∂Φ2α1,α2,θ1,θ2,λ1,λ2,p1=α^1,α^2,θ^1,θ^2,λ^1,λ^2,p1^−1.The above approach is used to derive approximate1001−τ% confidence intervals of the parameters Φ of the forms Φ^∓zτ/2varΦ^, where zτ/2 is the upper τ/2th percentile of the standard normal distribution.


## 4. Bayesian Estimation Using the Gamma Prior Distribution

In this section, the Bayes estimates of the model parameters are obtained under the assumption that the random variablesΦ have independent gamma prior distributions (see Dey et al. [24–26]) with hyperparameters akandbk,k=1,2,3,4,5,6,7,and are given by(40)fΦ;a,b=bkakΓakΦak−1e−bkΦ,Φ>0,where Φ=αj,θj,λj,pj. By multiplying (31) with (40), the joint posterior density for the vector of parameters Φ given the data becomes(41)πΦ|x¯∝Lx|ΦfΦ;ak,bk.Thus,(42)πΦ|x¯∝n−r.p1rθ1rλ1rα1r∏i=1rxiθ1−1e−λ1xiθ11−1−α1e−λ1xiθ1−21−1−e−λ1xrθ11−1−α1e−λ1xrθ1+n−r1−p1rθ2rλ2rα2r∏i=1rxiθ2−1e−λ2xiθ21−1−α2e−λ2xiθ2−21−1−e−λ2xrθ21−1−α2e−λ2xrθ2bkakΓakΦak−1e−bkΦ.Marginal distributions ofΦ can be obtained by integrating with respect to the nuisance parameters. Next, we consider the loss function that will be used to derive the estimators from the marginal posterior distributions.


### 4.1. Bayes Estimators of the Vector of ParametersΦ

In this section, we derive the Bayes estimators of the model parameters under symmetric as well asymmetric loss functions. A much known symmetric loss function is the squared error loss (SEL) function which is defined as(43)LΦ^−Φ=Φ−Φ^2.The popularity of this loss function is due to its relationship to least squares theory; it also makes the calculations simpler. Under the SELF in (43), the Bayes estimates of any function Φ=αj,θj,λj,pjcan be derived as(44)Φ^SEL=EΦ|t¯=A=πr2∫ΦΦ−Φ^2πΦ|x¯dΦ.All the above integrals have no closed form; so, we employ the numerical method to estimate the parameters. A useful asymmetric loss function, known as LINEX loss function, was introduced by Varian [27] and widely used by several authors (see Zellner [28] and Pandey and Rai [29]). We noticed that the LINEX loss function does not perform well for the estimation of the scale parameter in the whole parametric space, but performs well for a certain specified value of Φ. Basu and Ebrahimi [30] also suggested that the LINEX loss function is proper for the location parameter, and it appears not to be suitable for the estimation of the scale parameter. The linear exponential (LINEX) is an asymmetric loss function defined as(45)LΦ^−Φ=eνΦ−Φ^−νΦ−Φ^−1..Under the LINEX loss function, the Bayes estimators of any functionΦ=αj,θj,λj,p can be written as(46)Φ^LINEX=EΦ|t¯=∫Φ.eνΦ−Φ^−νΦ−Φ^−1πΦ|x¯dΦ.All the above integrals have no closed form. So, they are solved by the analytical method.


### 4.2. Credible Intervals

In this section, a symmetric1001−τ% two-sided Bayes probability interval estimate of Φ, denoted by LΦ,UΦ, is obtained by satisfying the following expression:(47)pLt¯<Φ<Ut¯=∫Lt¯Ut¯πθ,β,λ|t¯dΦ=1−τ.Since it is difficult to find the intervalLΦ and UΦ analytically, thus, we apply suitable numerical techniques to solve this nonlinear equation.


## 4.1. Bayes Estimators of the Vector of ParametersΦ

In this section, we derive the Bayes estimators of the model parameters under symmetric as well asymmetric loss functions. A much known symmetric loss function is the squared error loss (SEL) function which is defined as(43)LΦ^−Φ=Φ−Φ^2.The popularity of this loss function is due to its relationship to least squares theory; it also makes the calculations simpler. Under the SELF in (43), the Bayes estimates of any function Φ=αj,θj,λj,pjcan be derived as(44)Φ^SEL=EΦ|t¯=A=πr2∫ΦΦ−Φ^2πΦ|x¯dΦ.All the above integrals have no closed form; so, we employ the numerical method to estimate the parameters. A useful asymmetric loss function, known as LINEX loss function, was introduced by Varian [27] and widely used by several authors (see Zellner [28] and Pandey and Rai [29]). We noticed that the LINEX loss function does not perform well for the estimation of the scale parameter in the whole parametric space, but performs well for a certain specified value of Φ. Basu and Ebrahimi [30] also suggested that the LINEX loss function is proper for the location parameter, and it appears not to be suitable for the estimation of the scale parameter. The linear exponential (LINEX) is an asymmetric loss function defined as(45)LΦ^−Φ=eνΦ−Φ^−νΦ−Φ^−1..Under the LINEX loss function, the Bayes estimators of any functionΦ=αj,θj,λj,p can be written as(46)Φ^LINEX=EΦ|t¯=∫Φ.eνΦ−Φ^−νΦ−Φ^−1πΦ|x¯dΦ.All the above integrals have no closed form. So, they are solved by the analytical method.


## 4.2. Credible Intervals

In this section, a symmetric1001−τ% two-sided Bayes probability interval estimate of Φ, denoted by LΦ,UΦ, is obtained by satisfying the following expression:(47)pLt¯<Φ<Ut¯=∫Lt¯Ut¯πθ,β,λ|t¯dΦ=1−τ.Since it is difficult to find the intervalLΦ and UΦ analytically, thus, we apply suitable numerical techniques to solve this nonlinear equation.


## 5. Simulation Study and Comparisons

Here, we have carried out Monte Carlo simulation study to assess the performance of the maximum likelihood estimators and Bayes estimators with respect to their estimated risk. Here, for the simulation study, we have considered the parameter values asa1=1.9,5,a2=2,4,θ1=1.3,4,θ2=1.6,3,λ1=1,2.5,λ2=1.3,3.5,andp1=0.5,0.6 and different values of the mixing proportion. We set sample sizes n = 20, 40, and 80.Probabilistic mixing is used here to generate the mixture data. For each observation, a random numberu is generated from the uniform (0, 1) distribution. If u < p1, the observation is taken randomly from F1 (the MOEW distribution with parameters a1,θ1, and λ1; otherwise, from F2 (the MOEW distribution with parameters a2,θ2, and λ2). The choice of the censoring failure is made in such a way that the censoring rate of the resultant sample is approximately 10%. To implement censored samplings, the observations x11,…,x1r and x21,…,x2r, of failed items come from first and second subpopulations, respectively. The rest of the observations, which are greater than x1r and x2r, have been assumed to be censored from each component.The simulated datasets have been obtained using the following steps:Step 1: generate a uniform random numberu corresponding to each observationStep 2: ifu<p1, take the observation x11,…,x1n1 from the first subpopulation; otherwise, from the second subpopulation x21,…,x2n2Step 3: determine the test termination points on the right, that is,xrStep 4: the observations which are greater thanxr have been considered to be censored from each component (type-II censoring)To avoid an extreme sample, we simulate 5000 datasets each of sizen. The abbreviations used in the tables are estimate, estimated risk, and length of CIs based on the maximum likelihood method, Bayes estimates based on the squared loss function, and Bayes estimates based on the LINEX loss function. The Bayes estimates, estimated risk, and length of the confidence interval are computed using R package. These results are reported in Table 1. We assume that the prior distributions follow gamma distribution with hyperparameters akandbk,k=1,2,3,4,5,6,7, a1,b1=1.3,1.8,a2,b2=2.1,1.7,a3,b3=2.3,2.5,a4,b4=1.6,2.2,a5,b5=0.7,2.3,a6,b6=2.7,0.8,a7,b7=1.9,2.4.Table 1
The estimate, estimated risk, and length of the interval, Bayes method based on the squared loss function, and Bayes method based on the LINEX function when the prior distribution is gamma at different sample sizes.
n204080MLEsParameterEstimateEstimated riskLengthEstimateEstimated riskLengthEstimateEstimated riskLengthα12.22669620.455222931.78444111.92550290.261506881.02508811.71969750.173381000.6796115.44807500.746434512.92596954.87380220.45190731.77144444.66223340.294815901.1556571α22.60506910.46961281.84084852.36229730.329970541.29346082.19652630.21342740.83661984.95933110.742548332.91073604.61061230.43595771.70892284.41589060.280535771.0996800θ12.18420480.786394153.24107301.83864150.34367881.34719631.70592490.189377500.74234614.82917331.518234465.95136974.16354120.73807152.89318733.87622120.406415291.5931187θ22.22417090.89045263.58512221.94108340.40023031.56887401.81207310.233500510.91530524.59221801.323007625.18609464.11842710.73962912.89929293.81276300.392107271.5370323λ11.781231320.512795472.05997571.73702240.3279791.28568681.74245580.222299350.87139744.16721001.277105135.00616013.75289990.75125982.94488423.31002110.442603201.7349727λ21.63697910.567171752.23610351.48591410.30432381.19292771.39663390.187699520.73576865.04071801.562414616.12455274.34498700.87544583.43168453.80527240.518940372.0342089p10.49971870.079014530.3097310.49930520.05588370.21906010.9942090.039519370.15491310.47972180.079008470.30970750.50031560.05588110.21904990.50006050.039520830.1549188Bayes estimates based on the squared loss functionα12.17291810.0045522930.97845511.086459050.002276110.489227550.5432295250.0011380550.2446137754.9809010.058294611.93425852.49045050.02914730.96712911.2452250.0145740.483565α22.0566290.46961281.12973551.02831450.23480640.564867750.5141570.1174030.2824344.52831120.073265482.49382602.26415560.036632741.2469131.1320780.0183160.623457θ11.70439840.0926884152.38641560.85219920.04634420751.19320780.42610.0231720.5966044.12619440.06928463262.53681352.06309720.034642311.26840651.0315490.0173210.634203θ22.13682090.08452692.18512221.068410450.04226361.09256110.5342050.0211320.5462814.31231800.0323007621.98609462.15615990.016150360.99304731.078080.0080750.496524λ11.29268750.0512795472.05997570.646343750.025638971.029987850.3231720.0128190.5149943.96831900.497105133.00616010.98415951.98415951.503080050.992080.0992080.75154λ21.43983210.067171752.23610350.719916050.719916051.11805150.3599580.3599580.5590264.284371201.562414610.92455272.14218560.0140218560.462276351.0710930.0070110.231138p10.49919578570.079014530.63097310.249597850.034959780.315486550.1247990.01247990.1577430.63899780.079008470.03097920.31949890.021949890.01548960.1597490.0197490.007745Bayes estimates based on the LINEX functionα12.2729180.1045521.0784551.1864590.1022760.5892280.643230.1011380.3446145.0809010.1582952.0342592.5904510.1291471.0671291.3452250.1145740.583565α22.1566290.5696131.2297361.1283150.3348060.6648680.6141570.2174030.3824344.6283110.1732652.5938262.3641560.1366331.3469131.2320780.1183160.723457θ11.8043980.1926882.4864160.9521990.1463441.2932080.52610.1231720.6966044.2261940.1692852.6368142.1630970.1346421.3684071.1315490.1173210.734203θ22.2368210.1845272.2851221.168410.1422641.1925610.6342050.1211320.6462814.4123180.1323012.0860952.256160.116151.0930471.178080.1080750.596524λ11.3926880.151282.1599760.7463440.1256391.1299880.4231720.1128190.6149944.0683190.5971053.106161.084162.084161.603081.092080.1992080.85154λ21.5398320.1671722.3361040.8199160.8199161.2180520.4599580.4599580.6590264.3843711.6624151.0245532.2421860.1140220.5622761.1710930.1070110.331138p10.5991960.1790150.7309730.3495980.134960.4154870.2247990.112480.2577430.7389980.1790080.1309790.4194990.121950.115490.2597490.1197490.107745From Table1, we observe that, as sample size increases, estimated values of the parameters converge to the true values, and Bayes posterior risk tends to decrease. We also observe that, as sample size increases, the length of the classical confidence interval and Bayes credible interval decreases. It is to be noted that the Bayes estimates perform better than maximum likelihood estimates. In comparison of loss functions, the squared loss function provides better results than the LINEX loss function.


## 6. Real Data Analysis

In this section, we use four real-life datasets to illustrate the importance and flexibility of the MMOEW distribution. We compare the fits of the new MMOEW distribution with some other competitive models, such as Weibull (W), exponentiated Weibull (EW), exponentiated exponential (EE), Marshall–Olkin Weibull (MOW), and Marshall–Olkin extended Weibull (MOEW) distributions. The comparisons are done based on some measures of goodness of fit, namely, the maximized log-likelihood under the model−ℓ^, Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan–Quinn information criterion (HQIC), consistent Akaike information criterion (CAIC), and Kolmogorov–Smirnov (KS) statistic with its p value (PV). We observe that all the distributions in Tables 2–5 show a reasonably good fit for the given four datasets. The plots of empirical and fitted cdfs (Figures 2–5) also support the results in Tables 2–5. However, according to the cited statistics, the MMOEW model fits dataset II better than the other models.Table 2
Comparison among MMOEW, EW, MOEW, Weibull, EE, and MOW based on MLEs, the measures AIC, BIC, HQIC, CAIC, and KS, andp value for real data I.
DistributionPar.Estimates−ℓ^AICBICHQICCAICKSp valueMMOEWθ^12.9380.13174.26192.49181.64183.380.090.3730α^10.62λ^10.06θ^20.39α^20.88λ^20.71ρ^0.93EWθ^2.40141.69289.39297.20292.55293.290.060.84α^1.49λ^0.38MOEWθ^2.60142.12290.24298.06293.41294.150.120.1282α^1.81λ^0.08MOWθ^2.81142.08290.15297.97293.32294.060.050.9684α^1.33λ^0.36Weibullθ^3.00142.03288.05293.26290.16290.660.050.9604λ^0.34EEα^12.81150.48304.96310.17307.07307.570.080.5089λ^1.16Table 3
Comparison among MMOEW, EW, MOEW, Weibull, EE, and MOW based on MLEs, the measures AIC, BIC, HQIC, CAIC, and KS, andp value for real data II.
DistributionPar.Estimates−ℓ^AICBICHQICCAICKSp valueMMOEWθ^13.0790.91195.82211.75202.16203.780.070.8695α^10.51λ^10.15θ^21.02α^22.54λ^20.33ρ^0.78EWθ^1.55104.70215.40222.23218.12218.810.090.6466α^1.79λ^0.69MOEWθ^1.86111.13228.27235.10230.98231.680.090.5932α^1.72λ^0.40MOWθ^2.40107.47220.94227.77223.66224.360.090.5301α^0.46λ^0.42Weibullθ^2.11111.12226.24230.79228.05228.520.080.6951λ^0.52EEα^5.04102.46208.92213.47210.73211.190.070.8247λ^1.33Table 4
Comparison among MMOEW, EW, MOEW, Weibull, EE, and MOW based on MLEs, the measures AIC, BIC, HQIC, CAIC, and KS, andp value for real data III.
DistributionPar.Estimates−ℓ^AICBICHQICCAICKSp valueMMOEWθ^13.34108.74231.48241.95237.82236.720.150.4473α^11.04λ^10.26θ^20.53α^23.22λ^20.28ρ^0.02EWθ^0.58154.20314.41318.90315.92316.65240.0890.9546α^1.27λ^0.04MOEWθ^0.71154.09314.17318.66315.68316.420.200.1588α^0.73λ^0.07MOWθ^0.63154.20314.39318.88315.90316.640.100.8945α^1.41λ^0.04Weibullθ^0.67154.10312.20315.19313.21313.690.090.9474λ^0.03EEα^0.58153.93311.85314.85312.86313.350.150.4131λ^0.02Table 5
Comparison among MMOEW, EW, MOEW, Weibull, EE, and MOW based on MLEs, the measures AIC, BIC, HQIC, CAIC, and KS, andp value for real data IV.
DistributionPar.Estimates−ℓ^AICBICHQICCAICKSp valueMMOEWθ^14.92−61.62137.24148.32141.11142.780.090.9144α^11.26λ^10.002θ^22.68α^22.88λ^21.59ρ^0.93EWθ^5.19−72.34150.68155.43152.34153.060.090.8747α^0.88λ^0.29MOEWθ^5.57−68.33142.66147.41144.31145.030.110.8012α^0.49λ^0.0006MOWθ^6.53−75.96157.92162.67159.57160.290.080.9614α^0.27λ^0.25Weibullθ^4.86−68.32140.64142.22141.74142.220.090.8893λ^0.29Figure 2
Empirical and fitted cdfs for the breaking stress data for MMOEW, MOEW, EW, Weibull, EE, and MOW distributions.Figure 3
Empirical and fitted cdfs for survival times in days of 72 guinea pigs infected with virulent tubercle bacilli for MMOEW, MOEW, EW, Weibull, EE, and MOW distributions.Figure 4
Empirical and fitted cdfs for survival times of 33 patients suffering from acute myelogenous leukemia for MMOEW, MOEW, EW, Weibull, EE, and MOW distributions.Figure 5
Empirical and fitted cdfs for the COVID-19 data belonging to Canada of 36 days for MMOEW, MOEW, EW, Weibull, EE, and MOW distributions.Dataset I: the first dataset consists of 100 observations of breaking stress of carbon fibers. This dataset is obtained from Nichols and Padgett [31]. These data are stated as follows: 0.98, 5.56, 5.08, 0.39, 1.57, 3.19, 4.90, 2.93, 2.85, 2.77, 2.76, 1.73, 2.48, 3.68, 1.08, 3.22, 3.75, 3.22, 3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 4.42, 2.40, 3.15, 2.67, 3.31, 2.81, 2.56, 2.17, 4.91, 1.59, 1.18, 2.48, 2.03, 1.69, 2.43, 3.39, 3.56, 2.83, 3.68, 2.00, 3.51, 0.85, 1.61, 3.28, 2.95, 2.81, 3.15, 1.92, 1.84, 1.22, 2.17, 1.61, 2.12, 3.09, 2.97, 4.20, 2.35, 1.41, 1.59, 1.12, 1.69, 2.79, 1.89, 1.87, 3.39, 3.33, 2.55, 3.68, 3.19, 1.71, 1.25, 4.70, 2.88, 2.96, 2.55, 2.59, 2.97, 1.57, 2.17, 4.38, 2.03, 2.82, 2.53, 3.31, 2.38, 1.36, 0.81, 1.17, 1.84, 1.80, 2.05, and 3.65.Dataset II: the second dataset consists of survival times in days of 72 guinea pigs infected with virulent tubercle bacilli. This dataset is taken from Bjerkedal [32]. These data are illustrated as follows: 0.1, 0.33, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 07, 1.08, 1.08, 1.08, 1.09, 1.12, 1.13, 1.15, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 1.36, 1.39, 1.44, 1.46, 1.53, 1.59, 1.6, 1.63, 1.63, 1.68, 1.71, 1.72, 1.76, 1.83, 1.95, 1.96, 1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54, 2.54, 2.78, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.58, and 5.55.Dataset III: the third dataset consists of the survival times in weeks of 33 patients suffering from acute myelogenous leukemia. This dataset is taken from Mahmoudi [33]. These data are illustrated as follows: 65, 156, 100, 134, 16, 108, 121, 4, 39, 143, 56, 26, 22, 1, 1, 5, 65, 56, 65, 17, 7, 16, 22, 3, 4, 2, 3, 8, 4, 3, 30, 4, and 43.Dataset IV: the fourth dataset consists of drought mortality rate. The data represent COVID-19 data belonging to Canada of 36 days from 10 April to 15 May, 2020 (see the link https://covid19.who.int/). The data are as follows: 3.1091, 3.3825, 3.1444, 3.2135, 2.4946, 3.5146, 4.9274, 3.3769, 6.8686, 3.0914, 4.9378, 3.1091, 3.2823, 3.8594, 4.0480, 4.1685, 3.6426, 3.2110, 2.8636, 3.2218, 2.9078, 3.6346, 2.7957, 4.2781, 4.2202, 1.5157, 2.6029, 3.3592, 2.8349, 3.1348, 2.5261, 1.5806, 2.7704, 2.1901, 2.4141, and 1.9048.


## 7. Concluding Remarks

In this paper, we have introduced a two-component mixture model based on Marshall–Olkin extended Weibull distributions. Maximum likelihood and Bayes methods of estimation have been used to estimate the parameters of the mixture model. The numerical evidence shows that Bayes estimates perform better than the maximum likelihood estimates. Our simulated results follow the consistency property. The length of Bayes credible intervals is shorter than classical ones. From the simulation study, we may conclude that the Bayesian estimation has an advantage because of its small posterior risks as compared to the MLE method. If we compare the estimates with respect to loss functions, SELF performs better as compared to the LINEX loss function. Finally, for precise estimation of the unknown parameters of the Marshall–Olkin extended Weibull mixture model, Bayes method of estimation is preferable over maximum likelihood estimation, especially when the suitable prior information of the unknown parameters is available. The contents of the study may be useful in different fields where lifetime models are used for analysis of more than one causal factor of failure and where the data are type-II censored. The scope of this study may also be extended to other censoring schemes as well as for more than two-component mixture models.



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*Source: 6654101-2021-05-19.xml*