Bayesian Modeling of 3-Component Mixture of Exponentiated Inverted Weibull Distribution under Noninformative Prior

Department of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan Department of Mathematics, Air University, Islamabad, Pakistan Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia Statistical Research and Studies Support Unit, King Khalid University, Abha 62529, Saudi Arabia Department of Mathematics and Statistics, Faculty of Basic and Applied Sciences, International Islamic University, Islamabad, Pakistan


Introduction
Mixture modeling exists in many situations, particularly whenever we have more than one subpopulation. Mixture densities have beautiful properties to solve the complex problems in an easier manner. Recently, [1] explored 3component mixture modeling of exponentiated Weibull distribution under the Bayesian approach. Exponentiated inverted Weibull distribution (EIWD) has wide application in reliability theory. e authors in [2] used maximum likelihood and Bayes methods to derive parameters of EIWD under the type II censoring scheme. Later on, parameters of EIWD under type II censoring are discussed in [3]. e author in [4] derived Bayes and classical estimators of EIWD using noninformative prior. e authors in [5] proposed 2parameter model of EIWD. Bayesian analysis of shape parameter of EIWD under different loss functions (LFs) is discussed in [6]. Bayes and E-Bayes estimators of EIWD using conjugate prior under different loss functions are estimated in [7]. e authors in [8] studied three-parameter weighted EIWD. e number of components in a mixture distribution is due to heterogeneity of the parent population. It is often restricted to be finite, although in some cases the component may be infinite. As compared to simple modeling, it provides more attractive description of different statistical frameworks. Mixture modeling has wide applications in survival analysis. Recently, the authors in [9] studied 3-component mixture model of Pareto distribution by using type I right censoring scheme. Bayesian estimation and properties of 3component mixture of Rayleigh distribution are discussed in [10]. e authors in [11] explored 3-component mixture of exponential distribution under different loss functions. Moreover, the authors in [12] performed Bayesian estimation for finite mixture of exponential, Rayleigh, and Burr type XII distribution.
Motivated by the abovementioned studies of 3-component mixture and EIWD, we investigate the Bayesian modeling of 3-component mixture of EIWD in this study.
e main focus of this paper is to highlight efficient Bayes estimators (BEs) of component and proportional parameter(s). For this reason, two symmetric and two asymmetric LFs are used with noninformative priors, uniform prior (UP), and Jeffreys prior (JP), to obtain such results. e estimators are derived by applying the type I right censoring scheme. e rest of the paper is designed as follows. In the next section, the 3-component mixture model of EIWD is designed. In Section 3, we illustrate the proposed BEs of different parameters under several LFs. e limiting expressions and simulation study are discussed in Sections 4 and 5. Real data analysis is presented in Section 6. In Section 7, conclusions are provided.

The 3-Component Mixture of the EIWD
For the shape parameter of a random variable X, the pdf (probability density function) with the cdf (cumulative distribution function) of EIWD can be illustrated as (1) Here, for EIWD, θ i is defined as shape parameter.
With w 1 and w 2 mixing proportion, a finite 3-component mixture model can be written as For mixing proportion parameters and different component values, a 3-component mixture model of the EIWD is shown in Figure 1.
For 3-component mixture, the cdf is written as

Posterior Distribution Using the Noninformative Priors
e prior information plays an important role to differentiate between classical and Bayesian inference. e probability distribution which characterizes uncertainty of the parameter, prior to the existing information which is studied, is classified as prior distribution. A prior distribution is differentiated as noninformative prior, if it is smooth comparative towards likelihood function, whereas an informative prior (IP) is defined as a prior that has a contact towards the posterior distribution and is not the subject by the likelihood function. In this section, posterior distributions with the likelihood are obtained assuming the noninformative priors (NIPs) (UP and JP).

Likelihood Function.
Assume that from the 3-component mixture modeling of EIWD n units are consumed in a life assessment process with fixed t (test termination time). Suppose that the selected trial reveals that n units from r failed till fixed t and the n-r where the rest of the units are still in running phase. It is noted that, due to the failures, from r failures, r 1 failures classified are related to subpopulation I, r 2 failures belong to subpopulation II, and r 3 failures are related to subpopulation III. Now, the total uncensored sample points are classified as r � r 1 , r 2 , and r 3 , whereas the rest of the sample points n-r are considered as censored. Here, we have defined the time to failure, of the ith unit relating to lth subpopulation as x oi , 0 < X oi ≤ t, where l � 1, . . ., 3 and i � 1, . . ., r l , and t is defined as time of test termination. e likelihood of a 3-component mixture is stated as where and x � (x 11 , . . . , x 1r 1 , x 21 , . . . , x 2r 2 , x 31 , . . . , x 3r 3 ) for the uncensored observations are the failure times and Φ � (θ 1 , θ 2 , θ 3 , w 1 , w 2 ).

BEs and PRs under LFs.
e real-valued function which illustrates a loss for estimator over the exact value of parameter is defined as loss function (LF). e current section discussed BEs and posterior risks (PRs) over four different LFs, that is, squared error loss function (SELF), quadratic loss function (QLF), precautionary loss function (PLF), and DeGroot loss function (DLF).

BEs and PRs among SELF. For
We obtain BEs and PRs assuming NIPs for the component and proportion parameters θ 1 , θ 2 , θ 3 , w 1 , and w 2 under SELF as

Mathematical Problems in Engineering
where v � 1 for the UP and v � 2 for the JP.

BEs and PRs Assuming the UP and the JP under QLF.
QLF is symmetric LF and mostly used in the least square theory. It needs more care to tackle due to its variance properties and being symmetric, where L(β, d) � α(β − d) 2 is the QLF. We can, respectively, define the BE and the PR under QLF as By applying this concept using GP and ILP, the derivation of BEs and PRs is where v � 1 and v � 2 for the UP and JP.

BEs and PRs for the UP and the JP among PLF.
PLF is an asymmetric LF, which firstly introduced by [20]. e general form of PLFs which is also a special case can be Mathematical Problems in Engineering e derived results for BEs and PRs under the said priors and LF are observed as where v � 1 for the UP and v � 2 for the JP, respectively.

BEs and PRs Assuming the UP and the JP under DLF.
Another asymmetric LF introduced by [21] is DLF, written as L(β, d) � ((β − d)/d) 2 . e BEs and the PRs among DLF are obtained as e derived BEs and PRs using the assumed priors are Mathematical Problems in Engineering where v � 1 for UP and for JP v � 2.

Limiting Expressions
In regard to the uncensored sampling scheme, limiting expressions have wide applications. When test termination time t ⟶ ∞, it can be observed that uncensored points r approach n (sample size) and r l belongs to n l , where l � 1, . . ., 3. e points that are censored have turned out to be uncensored then and information given in the sample has also raised here. As a result, the effectiveness of the BEs is also raised due to the consideration of all the points in sample. us, limiting expressions for the NIPs can easily be obtained. e limiting forms under UP and JP for the BEs and PRs are reported in Table 1. In Table 2, results of simulation study are presented for n � 50, 100, 200, 500 and(θ 1 , θ 2 , θ 3 , w 1 , w 2 ) � (4, 3, 2, 0.5, 0.3).

Simulation Study
From Table 2, it is examined that BEs assuming all stated NIPs and LFs are larger for the small n as compared to the highest n for t. It is noted that, for t, the variation of the BEs from supposed components is near to zero with the rise in n. However, the PRs assuming the mentioned priors and LFs decrease with the rise in n. In regard to the estimation of   erefore, results indicate that JP is the most suited prior over the UP for this study. On the ground of simulation formulation under the studied LFs, the DLF is considered the best for evaluating the component part of the parameters and SELF is observed to be efficient for proportion parameters.

Practical Application
EIWD has extensive uses in the area of tensile strength of carbon fiber. erefore, in this view of analysis, we have used the data information of 100 sample points of tensile strength of carbon fiber. Earlier, this dataset is also studied by [22], which later has been considered by [5]. is dataset is based on the tensile strength of 100 observations of carbon fiber and is as follows: 3. To demonstrate the proposed methodology, we have braked the given uncensored dataset in the 3-component parts according to right type I censoring scheme with rate of failure r � 91. It is unidentified which parametric part fails till a failure happens at or before t � 9. Having run the program in Mathematica software, the total tests are formulated 100 times: n � 100, x 2k � 4.14, Almost, here 9% censored observations are used. BEs and the PRs assuming the NIPs are reported in Table 3.
From Table 3, it is noted that simulated results are compatible to real data study.
ere seem to be a few exceptions which are the reasons of the small set of data information. Both simulation and real data application results among the JP are the most accurate compared to the results under UP. However, DLF in contrast to other three LFs (SELF, QLF, and PLF) illustrates enhanced findings for the proportional part of parameters.

Conclusion
In this paper, the Bayesian formulation of the 3-component mixture model of EIWD under right type I censoring technique is studied. e comprehensive simulation process is built to evaluate and demonstrate several significant features about the BEs of the 3-component mixture model of EIWD assuming the NIPs under the different symmetric and asymmetric LFs (SELF, PLF, DLF, and QLF). Overestimation and underestimation of mixture proportion are inversely related to the sample size and are directly proportional to censoring rate. A small sample size and a large censoring rate cause the higher level of overestimation. But this effect can be reduced by using a large sample size. Posterior densities are derived and notified that they are in closed forms. e second aim of this paper was the selection of appropriate LF and prior for the inference of mixture parameters at different n and t. To judge the performance, we derived different posterior summaries, like BEs and their respective PRs by assuming different n and t. e limiting terms for the BEs and PRs of the shape parameters which are unknown here are also obtained among the said LFs (SELF, QLF, DLF, and PLF) assuming the NIPs (JP and UP). e compatible results are observed for simulation and real dataset analysis, to evaluate the performance of BEs. e contact of several n and t is estimated for BEs. e outcomes revealed that, for the component parameters, the order of best BEs is as follows: DLF < PLF < SELF < QLF, and on behalf of the proportion part of parameters, this categorizing is observed as SELF < PLF < DLF < QLF. us, we conclude that the efficient and most preferable prior is JP over the UP due to the reason of minimum risk. DLF performed better Data Availability e quantitative data related to carbon tensile strength used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.