Classical and Bayesian Inference on Finite Mixture of Exponentiated Kumaraswamy Gompertz and Exponentiated Kumaraswamy Fréchet Distributions under Progressive Type II Censoring with Applications

Abstract

A finite mixture of exponentiated Kumaraswamy Gompertz and exponentiated Kumaraswamy Fréchet is developed and discussed as a novel probability model. We study some useful structural properties of the proposed model. To estimate the model parameters under the classical method, we use the maximum likelihood estimation using a progressive type II censoring scheme. Under the Bayesian paradigm the estimation is carried out with gamma priors under a progressive type II censored samples with squared error loss function. To demonstrate the efficiency of the proposed model based on progressively type II censoring, a simulation study is carried out. Three actual data sets are used as an example, demonstrating that the suggested model in the new class fits better than the existing finite mixture models available in the literature.

1. Introduction

The literature on statistical distributions in the continuous domain can be supplemented in a number of ways. Adding one or more additional parameters to a baseline distribution, as proposed in [1], and it was subsequently developed by those who utilized a mixture of two normal distributions to fit a data set of crab measurements, which pioneered the use of finite mixture models in statistical research. Since then, mixture models have attracted a lot of interest, owing to the wide range of applications in which they are used. A non-exhaustive collection of references is provided below that utilized the concept of the several different mixing strategies to produce new probability models: [2,3,4]. Mixture distributions are a valuable statistical tool with more flexibility for analyzing and interpreting probabilistic called random occurrences in a potentially heterogeneous population. When modelling real-world data, it is typical to find that the data come from a mixed population that might comprise two or more distributions. In terms of applications of finite mixture models, there are plenty of areas, including but not limited to medical, economics, psychology, botany, fisheries research, life testing, and reliability, among others. Cluster analysis, latent structure models, empirical Bayes technique, and nonparametric density estimation are examples of indirect applications of such a process (finite mixture models).

Due to time constraints and several other constraints on data gathering such as constraints on resources, censoring in modeling lifetime data is very useful. Censoring occurs when specific lifetimes are only known for a subset of the individuals or units under study, but information on the remaining lifetimes is incomplete. There are various types of censoring schemes that exist in the literature, among which type II censoring is one of the most popular types. It is observed that type II censorship can be used to save time and money, i.e., it is economically profitable. However, when product lifetimes are extremely lengthy, the experimental period of a type II filtering life test can still be excessively long. Progressive type II censoring is a generalization of type II censoring that is beneficial when the loss of live test units at places other than the termination point is unavoidable. The type II gradually filtering approach has recently piqued the interest of statisticians, for example, see [5,6]. For further information on the theory, methodology, and applications of progressive filtering, see [7,8] for some recent comprehensive work in this context. Reference [9] developed a life test in which the experimenter may choose to organize the test units into various groups, each as an assembly of test units, and then run all of the test units simultaneously until the first failure in each group occurred. In this study, we conjecture that, scenarios regarding the analysis of lifetime data resembles to the case of progressive censoring. Therefore, in this study, we investigate a progressive type II censoring scheme, which is a generalization of the classical type II censoring scheme that allows living items to be removed during the experiment. For more information, see [10,11] and the references cited therein.

In the class of mixing distributions, among popular choices of a two-parameter probability model, the use of Kumaraswamy distribution as a baseline distribution under both the discrete and the continuous set-up have been well established in the literature. For example, the Kumaraswamy-G family by [12]; Kumaraswamy-beta generalized family by [13]; Kumaraswamy-Marshall–Olkin-G family by [14] and estimation of a three-component mixture of distributions via Bayesian and classical approaches is presented [15].

It is worth noting that while the additional parameter(s) increase the flexibility of a parent distribution, they come at a cost. With the addition of a new parameter or group of parameters, efficient estimation, and accurate interpretation of the model parameters for the proposed model become more cumbersome and at times quite difficult, for more information see [16]. In this article, we study a mixture between exponentiated Kumaraswamy Gompertz henceforth, in short (MEKGEKF) and exponentiated Kumaraswamy Fréchet distributions. We conjecture that the proposed mixture model will be useful in modeling behavior of various types of data for which, component-wise, either of the probability models might not provide an adequate fit and the data are obtained because of a progressively type II censoring.

The model described in this study generalizes several well-known distributions, including exponentiated Gompertz, exponentiated Kumaraswamy, exponentiated Fréchet, and exponentiated Fréchet distributions, among others. Furthermore, despite the fact that our MEKGEKF model contains 11 parameters, the density, cumulative, and probability functions, among others, do not have any non-manageable/intractable forms. This is beneficial since it makes it easier to obtain analytical and numerical results. This serves as a major motivation to carry out the present work. In addition, we investigated the MEKGEK model’s structural features and confirmed that all the suggested model’s formulas are basic and manageable given computational resources, for example, statistical computing software such as R/Mathematica/Matlab, etc.

The new MEKGEK model’s hazard function is versatile to suit all traditional forms, including increasing, decreasing, unimodal, and inverted bathtub shapes, among many others. As a result of their wide applicability in real life, these shapes are extremely essential. The actual data sets that are utilized to demonstrate the new MEKGEK model’s goodness-of-fit appears in support of the fact that it is indeed a very useful model. For the reasons stated above, we believe it is critical to investigate the MEKGEK distribution in more detail. We expect that this new mixed distribution will become a part of the applied researchers’ toolkit, and that it will be employed in a variety of contexts. The remainder of this article is structured as follows. Section 2 contains a mathematical description of the suggested model and some interesting structural aspects of the proposed model. Section 3 presents the maximum likelihood function of the suggested model under a progressively type II censoring. In Section 4, we present a general framework for Bayes estimation of the vector of parameters and posterior risk of the proposed model under squared error loss function under a progressively censored type II sampling scheme. In Section 5, we look at how to estimate the MEKGEKF distribution using both the classical and Bayesian paradigms, using a simulated study with various censoring schemes. In Section 6, three applications of the MEKGEKF distribution are given for illustrative purposes by modelling operation data on jobs made of iron sheet, Wire data, and the Australian athletes data set. Finally, some concluding remarks are presented in Section 7.

2. A Finite Mixture of Exponentiated Kumaraswamy Gompertz and Exponentiated Kumaraswamy Fréchet Model

The probability density function (PDF) of the exponentiated Kumaraswamy-G (henceforth, in short, EK) distributions was defined by several authors, such as [12,17,18], and has the following form of the density function

$$f\left(x\right)=abcg\left(x\right)G^{a-1}\left(x\right){\left[1-G^a\left(x\right)\right]}^{b-1}{\left\{{\left[1-G^a\left(x\right)\right]}^b\right\}}^{c-1}$$

(1)

where $a,b,c$ are all positive parameters and x > 0, and G is the baseline distribution function.

The corresponding cumulative distribution function (CDF) is

$$F\left(x\right)={\left\{1-{\left[1-G^a\left(x\right)\right]}^b\right\}}^c, x>0.$$

A mixture model has the advantage of subsuming numerous other existing probability models, including the parent distribution, under specific parametric constraints. This feature is present in the suggested distribution that we have developed and studied in this paper. Taking G as a two parameter Gompertz distribution in (1), the PDF and CDF of exponentiated Kumaraswamy Gompertz (henceforth, in short, EKG) distributions are defined as follows:

$$\begin{array}{c}{f}_1\left(x\right)=abcsexp\left[rx-\left(\frac{s}{r}\right)\left(exp\left(rx\right)-1\right)\right]{\left[\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rx\right)-1\right)\right]\right)\right]}^{a-1}\\ {\left[1-{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rx\right)-1\right)\right]\right)}^a\right]}^{b-1}\\ {\left\{1-{\left[1-{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rx\right)-1\right)\right]\right)}^a\right]}^b\right\}}^{c-1}\\ a, b, c, s, r>0, x>0, \end{array}$$

(2)

and

$$F_1\left(x\right)={\left\{1-{\left[1-{\left[1-exp\left(-\left(\frac{s}{r}\right)\left(exp\left(rx\right)-1\right)\right)\right]}^a\right]}^b\right\}}^c, x>0.$$

where s and r are the shape and scale parameters, respectively.

Similarly, taking G as a two parameter Fréchet distribution in (1), the PDF and CDF of exponentiated Kumaraswamy Fréchet (EKF) distributions is defined as

$$\begin{array}{c}{f}_2\left(x\right)=abcs \left[\frac{r}{s}{\left(\frac{x}{s}\right)}^{-r-1}exp\left[-{\left(\frac{x}{s}\right)}^{-r}\right]\right]{\left[exp\left[-{\left(\frac{x}{s}\right)}^{-r}\right]\right]}^{a-1}\\ {\left[1-{\left(exp\left[-{\left(\frac{x}{s}\right)}^{-r}\right]\right)}^a\right]}^{b-1}{\left\{1-{\left[1-{\left(exp\left[-{\left(\frac{x}{s}\right)}^{-r}\right]\right)}^a\right]}^b\right\}}^{c-1}\end{array}$$

(3)

and

$$F_2\left(x\right)={\left\{1-{\left[1-{\left[1-exp\left(-\left(\frac{s}{r}\right)\left(exp\left(rx\right)-1\right)\right)\right]}^a\right]}^b\right\}}^c, x>0.$$

Next, the PDF of a two-component mixture of EKG and EKF (MEKGEKF) models with mixing proportions, ($p_j, j=1,2)$ is given as follows:

$$f\left(x\right)=p{f}_1\left(x\right)+\left(1-p\right)f_2\left(x\right)$$

(4)

in the mixing proportions must satisfy ${{\displaystyle \sum }}_{i=1}^2{p}_i=1$, and $p_i\ge 0,$ and $f_1\left(x\right),$ and $f_2\left(x\right)$ are defined in (2) and (3), and all the parameters are unknown. The associated CDF of the MEKGEKF model is given by:

$$F\left(x\right)=p{F}_1\left(x\right)+\left(1-p\right)F_2\left(x\right).$$

(5)

For $q=1-p$ of EKG and EKF distributions, a density function of mixture of two component densities with mixing proportions is ($p_j, j=1,2)$, is given as follows:

$$\begin{array}{ll}f\left(x\right)& =p{a}_1{b}_1{c}_1{s}_{1}exp[r_{1}x\\ & -\left(\frac{s_1}{r_1}\right){\left(exp\left(r_{1}x\right)-1\right)}^{}]{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rx\right)-1\right)\right]\right)}^{a_1}[1\\ & {-{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rx\right)-1\right)\right]\right)}^{a_1}]}^{b_1-1}[1\\ & {-{\left[1-{\left[1-exp\left(-\left(\frac{s_1}{r_1}\right)\left(exp\left(r_{1}x\right)-1\right)\right)\right]}^{a_1}\right]}^{b_1}]}^{c_1-1}\\ & +q\frac{a_2{b}_2{c}_2{r}_2}{s_2}{\left(\frac{x}{s_2}\right)}^{-r_2-1}exp\left[-{\left(\frac{x}{s_2}\right)}^{-r_2}\right]{\left[1-exp\left(-{\left(\frac{x}{s_2}\right)}^{-r_2}\right)\right]}^{a_2-1}[1\\ & {-{\left[1-exp\left(-{\left(\frac{x}{s_2}\right)}^{-r_2}\right)\right]}^{a_2}]}^{b_2-1}{\left[1-{\left[1-{\left[1-exp\left(-{\left(\frac{x}{s_2}\right)}^{-r_2}\right)\right]}^{a_2}\right]}^{b_2}\right]}^{c_2-1}, I\left(0<x<\infty \right).\end{array}$$

(6)

The corresponding CDF is given by:

$$F\left(x\right)=p{\left[1-{\left[1-{\left[1-exp\left(-\left(\frac{s_1}{r_1}\right)\left(exp\left(r_{1}x\right)-1\right)\right)\right]}^{a_1}\right]}^{b_1}\right]}^{c_1}+q{\left[1-{\left[1-{\left[exp\left(-{\left(\frac{x}{s_2}\right)}^{-r_2}\right)\right]}^{a_2}\right]}^{b_2}\right]}^{c_2}.$$

(7)

The associated quantile function is given by:

$$Q_j\left(u\right)=p{\left\{{\left[1-{\left(\frac{1+ln\left(\frac{s_1}{r_1}\left(-ln\left(1-u\right)\right)\right)}{r_1}\right)}^{\frac{1}{c_1}}\right]}^{\frac{1}{b_1}}\right\}}^{\frac{1}{a_1}}+q{\left\{{\left[1-{\left(s_2{\left(-ln\left(u\right)\right)}^{\frac{-1}{r_2}}\right)}^{\frac{1}{c_2}}\right]}^{\frac{1}{b_2}}\right\}}^{\frac{1}{a_2}}.$$

(8)

Some representative plots of the PDF and the hrf corresponding to the density in (6) for varying values of the mixing parameter p = $p$ are given in Figure 1.

From these plots, one can observe that the PDF of MEKGEKF distribution is unimodal and right skewed. Additionally, from the hrf plots it is observed that the hrf and the MEKGEKF distribution are decreasing, increasing, unimodal, and inverse bathtub shaped.

Laplace Transformation for MEKGEKF Model

Laplace transformation is a useful tool in probability and statistics. In this section, we begin our discussion by driving the Laplace transform for the proposed MEKGEKF model. From (6) of the MEKGEKF density, the associated Laplace transform will be

$$ℒ\left[f^{\ast }\left(u\right)\right]=ℒ\left[p{f}_1\left(u\right)+\left(1-p\right)f_2\left(u\right)\right]=pℒf_1\left(u\right)+\left(1-p\right)ℒf_2\left(u\right),$$

where

$$f^{\ast }\left(u\right)={{\displaystyle \int }}_0^{\infty }{e}^{-ut}f\left(t\right)dt={{\displaystyle \int }}_0^{\infty }{e}^{-ut}f\left(t\right)dt=p{{\displaystyle \int }}_0^{\infty }{e}^{-ut}{f}_1\left(t\right)dt+\left(1-p\right){{\displaystyle \int }}_0^{\infty }{e}^{-ut}{f}_2\left(t\right)dt=p{I}_1+\left(1-p\right)I_2,$$

where

$$\begin{array}{ll}{I}_1=ℒf_1\left(u\right)& ={{\displaystyle \int }}_0^{\infty }{e}^{-ut}{f}_1\left(t\right)dt\\ & ={{\displaystyle \int }}_0^{\infty }{e}^{-ut}\{abcsexp[rt\\ & -\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)]{\left[\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right)\right]}^{a-1}[1\\ & {-{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right)}^a]}^{b-1}\{1\\ & {-{\left[1-{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right)}^a\right]}^b\}}^{c-1}\}dt,\end{array}$$

(9)

where, again,

$$\begin{array}{ll}{I}_2=ℒf_2\left(u\right)& ={{\displaystyle \int }}_0^{\infty }{e}^{-ut}{f}_2\left(t\right)dt\\ & ={\displaystyle {{\displaystyle \int }}_0^{\infty }}{e}^{-ut}\left\{\frac{abcr}{s}{\left(\frac{t}{s}\right)}^{-r-1}exp\left[-{\left(\frac{t}{s}\right)}^{-r}\right]{\left[exp\left(-{\left(\frac{t}{s}\right)}^{-r}\right)\right]}^{a-1}\right[1\\ & {-{\left[1-exp\left(-{\left(\frac{t}{s}\right)}^{{-}_r}\right)\right]}^a]}^{b-1}{\left[1-{\left[1-{\left[1-exp\left(-{\left(\frac{t}{s}\right)}^{-r}\right)\right]}^a\right]}^b\right]}^{c-1}\}dt\\ & ={{\displaystyle \int }}_0^{\infty }{e}^{-ut}\left\{\frac{abcr}{s}{\left(\frac{t}{s}\right)}^{-r-1}exp\left[-{\left(\frac{t}{s}\right)}^{-r}\right]\ast b_1\ast b_2\right\}dt,\end{array}$$

where

$$\begin{array}{lll}{b}_1& =& {\left[exp\left(-{\left(\frac{t}{s}\right)}^{-r}\right)\right]}^{a-1}{\left[1-{\left[1-exp\left(-{\left(\frac{t}{s}\right)}^{{-}_r}\right)\right]}^a\right]}^{b-1}\\ & =& {\left[exp\left(-{\left(\frac{t}{s}\right)}^{-r}\right)\right]}^{a-1}{\displaystyle {\displaystyle \sum }_{j_1=0}^{\infty }}\left(\begin{array}{c}b-1\\ j_1\end{array}\right){\left[exp\left(-{\left(\frac{t}{s}\right)}^{{-}_r}\right)\right]}^{j_{1}a},\end{array}$$

$$b_2={\left[1-{\left[1-{\left[1-exp\left(-{\left(\frac{t}{s}\right)}^{-r}\right)\right]}^a\right]}^b\right]}^{c-1}={\displaystyle \sum }_{j_2=0}^{\infty }{\displaystyle \sum }_{j_3=0}^{\infty }{\left(-1\right)}^{j_2+j_3}\left(\begin{array}{c}c-1\\ j_2\end{array}\right)\left(\begin{array}{c}{j}_{2}b\\ j_3\end{array}\right){\left[exp\left(-{\left(\frac{t}{s}\right)}^{-r}\right)\right]}^{j_{3}a}$$

Combining ($b_1$) and ($b_2$), we get:

$$I_2={\displaystyle {\displaystyle \sum }_{j_1=0}^{\infty }}{\displaystyle {\displaystyle \sum }_{j_2=0}^{\infty }}{\displaystyle {\displaystyle \sum }_{j_3=0}^{\infty }}.{\left(-1\right)}^{j_1+j_2+j_3}\left(\begin{array}{c}b-1\\ j_1\end{array}\right)\left(\begin{array}{c}c-1\\ j_2\end{array}\right)\left(\begin{array}{c}{j}_{2}b\\ j_3\end{array}\right){\left[\left(exp\left[-{\left(\frac{t}{s}\right)}^{-r}\right]\right)\right]}^{j_{3}a}\frac{abcr}{s}{{\displaystyle \int }}_0^{\infty }{\left(\frac{t}{s}\right)}^{-r-1}{e}^{-{\left(\frac{t}{s}\right)}^{-r}}{\left(e^{-{\left(\frac{t}{s}\right)}^{-r}}\right)}^{a\left(j_1+j_3\right)}dt,$$

Next, consider the following:

$$a_1=\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right)={\displaystyle \sum }_{j_1=0}^{\infty }.{\left(-1\right)}^{j_1}\left(\begin{array}{c}a-1\\ j_1\end{array}\right){\left(exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right)}^{j_1},$$

(10)

$$\begin{array}{lll}{a}_2& =& {\left[1-{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right)}^a\right]}^{b-1}\\ & =& {\displaystyle {\displaystyle \sum }_{j_2=0}^{\infty }}.{\left(-1\right)}^{j_2}\left(\begin{array}{c}b-1\\ j_2\end{array}\right){\left[1-\left(exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right)\right]}^{j_{2}a}\\ & =& {\displaystyle {\displaystyle \sum }_{j_2=0}^{\infty }}.{\displaystyle {\displaystyle \sum }_{j_3=0}^{\infty }}\left(\begin{array}{c}b-1\\ j_2\end{array}\right){\left(-1\right)}^{j_2+j_3}\left(\begin{array}{c}{j}_{2}a\\ j_3\end{array}\right){\left[exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right]}^{j_3},\end{array}$$

(11)

$$\begin{array}{lll}{a}_3& =& {\left\{1-{\left[1-{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right)}^a\right]}^b\right\}}^{c-1}\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right)\\ & =& {\displaystyle {\displaystyle \sum }_{j_4=0}^{\infty }}.\left(\begin{array}{c}c-1\\ j_4\end{array}\right){\left(-1\right)}^{j_4}{\left[1-{\left[exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right]}^{{j_3}^a}\right]}^{j_{4}b}\\ & =& {\displaystyle {\displaystyle \sum }_{j_4=0}^{\infty }}.\left(\begin{array}{c}c-1\\ j_4\end{array}\right){\left(-1\right)}^{j_4}{\displaystyle {\displaystyle \sum }_{j_5=0}^{\infty }}.\left(\begin{array}{c}b{j}_4\\ j_5\end{array}\right){\left(-1\right)}^{j_5}{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right)}^{j_{5}a}\\ & =& {\displaystyle {\displaystyle \sum }_{j_4=0}^{\infty }}{\displaystyle {\displaystyle \sum }_{j_5=0}^{\infty }}{\displaystyle {\displaystyle \sum }_{j_6=0}^{\infty }}.{\left(-1\right)}^{j_4+j_5+j_6}\left(\begin{array}{c}c-1\\ j_4\end{array}\right)\left(\begin{array}{c}b{j}_4\\ j_5\end{array}\right)\left(\begin{array}{c}{j}_{5}a\\ j_6\end{array}\right){\left[exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right]}^{j_6},\end{array}$$

(12)

Combing (10), (11), and (12) in (6), we get:

$$\begin{array}{l}{I}_1=abcs{\displaystyle {\displaystyle \sum }_{j_1=0}^{\infty }}{\displaystyle {\displaystyle \sum }_{j_2=0}^{\infty }}{\displaystyle {\displaystyle \sum }_{j_3=0}^{\infty }}.{\displaystyle {\displaystyle \sum }_{j_4=0}^{\infty }}{\displaystyle {\displaystyle \sum }_{j_5=0}^{\infty }}{\displaystyle {\displaystyle \sum }_{j_6=0}^{\infty }}.{\left(-1\right)}^{{{\displaystyle \sum }}_{i=1}^6{j}_i}{}^{.}\left(\begin{array}{c}a-1\\ j_1\end{array}\right)\left(\begin{array}{c}b-1\\ j_2\end{array}\right)\left(\begin{array}{c}{j}_{2}a\\ j_3\end{array}\right)\left(\begin{array}{c}c-1\\ j_4\end{array}\right)\left(\begin{array}{c}{j}_{4}b\\ j_5\end{array}\right)\left(\begin{array}{c}{j}_{5}b\\ j_6\end{array}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ast {{\displaystyle \int }}_0^{\infty }{e}^{-ut}\left[exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right]{\left[exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right]}^{j_1+j_3+j_6}dt,\\ =M_1{{\displaystyle \int }}_0^{\infty }{e}^{-ut+rt}{\left[exp\left[-\left(\frac{s}{r}\right)\left(exp\left(rt\right)-1\right)\right]\right]}^{j_1+j_3+j_6}dt,\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=M_1\left[e^{\frac{s\left(j_1+j_3+j_6\right)}{r}}ExpIntegral\left[\left(\frac{u}{r}\right),\frac{s\left(j_1+j_3+j_6\right)}{r}\right]-{\left(\frac{s\left(j_1+j_3+j_6\right)}{r}\right)}^{-1-\frac{u}{r}}Gamma\left(1-\frac{u}{r}\right)\right]=M_3,\end{array}$$

(13)

on using Mathematica.

Therefore,

$$I_2=M_2{{\displaystyle \int }}_0^{\infty }\frac{r}{s}{\left(\frac{t}{s}\right)}^{r-1}exp\left[-{\left(\frac{t}{s}\right)}^{-r}\left(a{j}_3+a{j}_1\right)\right]dt=M_4$$

(14)

Observe that (13) is not available in closed form.

Therefore, the Laplace transformation for the MEKGEKF model is given by $ℒ\left(\mathrm{MEKGEKF} \right)=p{M}_3+\left(1-p\right)M_4,$ where $M_3$ and $M_4$ are given by (13) and (14), respectively.

3. Maximum Likelihood Estimation of MEKGEKF Distribution under Progressive Type II Censoring

Assume that n units are subjected to a life test at time 0, and that the experimenter determines the quantity $m$, the number of failures to be observed, ahead of time. Next, $R_1$ units are now randomly deleted from the remaining n-1 surviving units at the time of first failure; $R_2$ units from the remaining $n-2-R_1$ units are randomly deleted at the second failure. The test will continue until the mth failure has occurred. The remaining $R_m=n-m-R_1-R_2-\dots -R_{m-1}$ units are deleted at this time. $R_i$ and m are prefixed in this censoring scheme. The m failure times obtained from a progressive type II censoring scheme was arranged. Progressive type II censored ordered statistics are the names given to the values acquired as a result of this type of censoring scheme. This scheme reduces to a classical type II right censoring technique if $R_1=R_2=\dots =R_{m-1}=0,$ so that $R_m=n-m$. Additionally, if $R_1=R_2=\dots =R_{m-1}=0$, the progressively type II censoring method simplifies to the case of a complete sample (m = n, i.e., the case of no censoring).

Let $x_1,x_2,\dots ,x_m$ be a progressively type II censored sample, with the progressive censoring scheme $\left(R_1,R_2,\dots ,R_m\right).$ Based on progressively type II censored samples obtained from the MEKGEKF distributions, from (6), the associated likelihood function will be:

$$\begin{array}{l}L\left(\underset{\_}{x} |a_1,a_2, b_1,b_2,c_1,c_2,s_1,s_2,r_1,r_2,p,q\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\propto {\displaystyle {\displaystyle \prod }_{i=1}^m}\left\{p{a}_1{b}_1{c}_1{s}_{1}exp\right[r_1{x}_{i:m:n}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(\frac{s_1}{r_1}\right)\left(exp\left(r_1{x}_{i:m:n}\right)-1\right)\left]{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(r{x}_{i:m:n}\right)-1\right)\right]\right)}^{a_1}\right[1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{-{\left(1-exp\left[-\left(\frac{s}{r}\right)\left(exp\left(r{x}_{i:m:n}\right)-1\right)\right]\right)}^{a_1}]}^{b_1-1}[1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{-{\left[1-{\left[1-exp\left(-\left(\frac{s_1}{r_1}\right)\left(exp\left(r_1{x}_{i:m:n}\right)-1\right)\right)\right]}^{a_1}\right]}^{b_1}]}^{c_1-1}.\left\{\right(1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{-exp\left[-\left(\frac{s}{r_1}\right)\left(exp\left(r_1{x}_{i:m:n}\right)-1\right)\right])}^{R_i}\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+q\frac{a_2{b}_2{c}_2{r}_2}{s_2}{\left(\frac{x_{i:m:n}}{s_2}\right)}^{-r_2-1}exp\left[-{\left(\frac{x_{i:m:n}}{s_2}\right)}^{-r_2}\right]{\left[1-exp\left(-{\left(\frac{x_{i:m:n}}{s_2}\right)}^{-r_2}\right)\right]}^{a_2}[1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{-{\left[1-exp\left(-{\left(\frac{x_{i:m:n}}{s_2}\right)}^{-r_2}\right)\right]}^{a_2}]}^{b_2-1}[1\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{-{\left[1-{\left[1-exp\left(-{\left(\frac{x_{i:m:n}}{s_2}\right)}^{-r_2}\right)\right]}^{a_2}\right]}^{b_2}]}^{c_2-1}.\left\{{\left(exp\left[-{\left(\frac{x_{i:m:n}}{s_2}\right)}^{-r_2}\right]\right)}^{R_i}\right\}\},i=1,2.\end{array}$$

(15)

where $k=n\left(n-1-R_1\right)\left(n-1-R_2\right)\dots \left(n-m+1-R_1\dots -R_m\right)$. Here, we get the log likelihood function without the constant term. To make things easier, we’ll use the natural logarithm of the likelihood function, $ı$, which is as follows:

$$ı\propto {{\displaystyle \sum }}_{i=1}^{m}log\left[g_j\left(X_{i:m:n}\right)\right]+R_{i}log\left[1-G_j\left(X_{i:m:n}\right)\right],$$

(16)

The maximum-likelihood estimates (MLEs) of the parameter vector $\Omega =\left({\widehat{a}}_1,{\widehat{b}}_1,{\widehat{c}}_1,{\widehat{s}}_1,{\widehat{r}}_1,{\widehat{a}}_2,{\widehat{b}}_2,{\widehat{c}}_2,{\widehat{s}}_2,{\widehat{r}}_2,\widehat{p}\right)$ are calculated by taking partial derivatives of (16) w.r.t. the parameters, and setting them equal to zero. These equations cannot be solved analytically, and we need to consider adopting any iterative techniques, such as simulated annealing, or Newton–Raphson type algorithm. In this case, we adopt the second choice for the iterative algorithm, i.e., the Newton–Raphson method.

Fisher Information Matrix (FIM)

The FIM is important in the calculation of uncertainty and other aspects of estimation for a wide range of statistical methods and applications, including parameter estimation, and experimental design for more details see [19].

The FIM is an excellent indicator of how much information sample data can provide regarding parameters. Assume $f\left(x; \Omega \right)$ is the density function and $ı\left(\underset{\_}{x} |a_j,b_j,c_j,s_j,r_j,p,q\right)=logL\left(\underset{\_}{x} |a_j,b_j,c_j,s_j,r_j,p,q\right)$ is associated with the log-likelihood function. We can define the expected FIM, I, as follows:

$$I=E{\left[\frac{{\partial }^2 ı}{\partial {\Omega }_i\partial {\Omega }_j}\right]|}_{i,j=1,2,3,4,5},$$

We may investigate the global maxima of the log-likelihood by setting different starting values for the parameters. The FIM will be required for interval estimation. The elements of $11\times 11$ observed FIM (since expected values are different to calculate), $J(\overrightarrow{\mathsf{\Omega }})=J\widehat{(\overrightarrow{\mathsf{\Omega }})},$ can be obtained from the authors upon request. The asymptotic distribution of $[\widehat{(\overrightarrow{\mathsf{\Omega }})}-\overrightarrow{\mathsf{\Omega }}]$ is $N_{11}(\overrightarrow{\mathsf{\Omega }} , K{(\overrightarrow{\mathsf{\Omega }})}^{-1})$, under the standard regularity conditions, where $K(\overrightarrow{\mathsf{\Omega }})=E[K(\overrightarrow{\mathsf{\Omega }})],$ is the expected information matrix, and $J{\widehat{(\overrightarrow{\mathsf{\Omega }})}}^{-1}$ is the observed information matrix. The multivariate normal $N_{11}(\overrightarrow{\mathsf{\Omega }} , K{(\overrightarrow{\mathsf{\Omega }})}^{-1})$ distribution can be used to construct approximate confidence intervals for the individual parameters. The asymptotic variance-covariance matrix of the MLE $\widehat{\overrightarrow{\mathsf{\Omega }}}=\left({\widehat{a}}_1,{\widehat{b}}_1,{\widehat{c}}_1,{\widehat{s}}_1,{\widehat{r}}_1,{\widehat{a}}_2,{\widehat{b}}_2,{\widehat{c}}_2,{\widehat{s}}_2,{\widehat{r}}_2,\widehat{p}\right),$ can be obtained from the inverse of the observed FIM as: ${\mathit{V}}_{\mathbf{11}\times \mathbf{11}}={\mathit{K}}^{\mathbf{-}\mathbf{1}}(\widehat{\overrightarrow{\mathsf{\Omega }}})={\left[\begin{array}{cccc}{\mathit{\nu }}_{\mathbf{11}}& {\mathit{\nu }}_{\mathbf{12}}& \dots & {\mathit{\nu }}_{\mathbf{1} \overline{\mathbf{11}}}\\ {\mathit{\nu }}_{\mathbf{21}}& {\mathit{\nu }}_{\mathbf{22}}& \dots & {\mathit{\nu }}_{\mathbf{2} \overline{\mathbf{11}}}\\ \dots & & & \dots \\ {\mathit{\nu }}_{\overline{\mathbf{11}} \mathbf{1}} & {\mathit{\nu }}_{\overline{\mathbf{11}} \mathbf{2}}& \dots & {\mathit{\nu }}_{\overline{\mathbf{11}} \overline{\mathbf{11}}}\end{array}\right]}^{\mathbf{11}\times \mathbf{11}}$.

Then, based on the asymptotic normality of the MLE, a $100\left(1-\delta \right)\%$ approximate confidence interval for the parameters (say) ${\widehat{a}}_1$ can be obtained as:

$${\widehat{a}}_1\pm Z_{1-\frac{\delta }{2}}\sqrt{\mathit{v}\mathbf{11}},$$

4. Bayesian Estimation

In recent years, Bayes’ paradigm has gained popularity in a range of fields, including engineering, clinical medicine, biology, and so on. Its ability to analyze earlier (prior) data makes it particularly useful in dependability studies, where data availability is one of the most significant challenges. This section develops the Bayes estimates and corresponding credible intervals of the model parameters, $\mathsf{\Omega }=\left(a_1,b_1,c_1,s_1,r_1, a_2,b_2,c_2,s_2,r_2,p\right), \mathrm{for} \mathrm{the} \mathrm{density} \mathrm{in} \left(6\right).$

4.1. Prior Information and Loss Function

As the gamma distribution can take on a variety of shapes depending on its parameter values, employing independent gamma priors is straightforward and can result in more expressive posterior density estimates. As a result, we consider gamma density priors, which are more flexible and convenient than other less informative independent prior distributions, in order to tailor support for the MEKGEKF distribution parameters. Therefore, the independent gamma priors for the MEKGEKF distribution parameters, $\mathsf{\Omega }=\left(a_1,b_1,c_1,s_1,r_1, a_2,b_2,c_2,s_2,r_2,p\right)$ are assumed to be $Gamma\left({\alpha }_j,{\beta }_j\right);j=1,\dots ,10$, respectively. A logical assumption for an appropriate prior for the mixing weight parameter p could be a uniform (0,1), which we have assumed in the context. The joint prior in this case will be:

$$\pi \left(\mathsf{\Omega }\right)\propto {\displaystyle \prod }_{j=1}^{10}{\mathsf{\Omega }}_j^{{\alpha }_j-1}{e}^{-{\beta }_j{\mathsf{\Omega }}_j},$$

where ${\alpha }_j,{\beta }_j;j=1,\dots ,10$, that reflect the prior knowledge of the unknown parameters $\mathsf{\Omega }=\left(a_1,b_1,c_1,s_1,r_1, a_2,b_2,c_2,s_2,r_2\right)$. It is assumed that ${\alpha }_j \mathrm{and} {\beta }_j$ are known and non-negative. Regarding the use of such priors in this context, see [20,21] and the references cited therein.

Next, we consider an appropriate loss function. According to [22], the choice of the symmetric loss function (SLF) is a critical issue in Bayesian analysis. The squared error loss function (SEL) is the most commonly utilized SLF that is considered in this study for estimating the unknown parameters which is given by:

$$ℒ\left(\mathsf{\Omega },\tilde{\mathsf{\Omega }}\right)={\left(\tilde{\mathsf{\Omega }}-\mathsf{\Omega }\right)}^2,$$

where $\tilde{\mathsf{\Omega }}$ is a close approximation of $\mathsf{\Omega }$. The objective is to use the above function to calculate the Bayesian estimates of the parameters to be obtained as posterior mean of $\mathsf{\Omega }.$ On the other hand, any other loss functions, such as LINEX, composite, precautionary loss functions can be readily added as appropriate.

4.2. Posterior Analysis by SLF

Observing the progressively type II censored sample data from the likelihood function in (15), and combining with the prior knowledge as given earlier, yields the following form of the joint posterior density function. $L\left(\underset{\_}{x}|\mathsf{\Omega }\right)\propto {{\displaystyle \prod }}_{\mathit{j}=\mathbf{1}}^{\mathbf{10}}{\Omega }_{\mathit{j}}^{{\alpha }_j-1}{\mathit{e}}^{-{\beta }_j{\mathsf{\Omega }}_j}{\displaystyle \prod }_{i=1}^{m}g\left(x_{i:m:n};\mathsf{\Omega }\right){\left[1-G\left(x_{i:m:n};\mathsf{\Omega }\right)\right]}^{R_i}$ are the MEKGEKF density and the CDF given in (4) and (5), respectively.

The Bayes estimator of $\mathsf{\Omega }$ (i.e., the parameter vector) will be the posterior expectation of $\mathsf{\Omega }$, denoted by $\tilde{\mathsf{\Omega }},$ under the SEL function. In order to obtain these estimators, the marginal posterior distributions for each of the parameters in $\mathsf{\Omega }$ must be collected. However, explicit expressions for the marginal PDFs for each of the unknown parameters cannot be obtained simultaneously due to interactable and complicated mathematical computation. As a result, we compute Bayesian estimates and credible intervals using simulation methods such as MCMC techniques.

One of the most useful MCMC algorithms is the Metropolis–Hastings (MH) algorithm which is used to generate random samples using the posterior density distribution. Additionally, an independent proposal distribution to approximate Bayesian estimates is applied to create the associated Highest Posterior Density (HPD) credible intervals. Furthermore, this method provides a chain form of the Bayesian estimates that are easy to utilize in practice. For more information on this algorithm, see [23,24] and the references cited therein.

5. Monte Carlo Simulation

To compare the performance of the different estimators mentioned earlier, under both the classical and Bayesian paradigm, we build a Monte Carlo simulation of maximum likelihood estimates for MEKGEKF distribution based on a progressively type II samples obtained under different schemes at first.

5.1. Simulation Study

Several simulation studies were carried out to evaluate the performance of the MLE based on various censoring schemes. We offer a detailed simulation study that compares the performance of the likelihood estimation approach with Bayesian inference assuming gamma prior distributions for each parameter. In simulation studies, the sample size (n), censored progression size (m), replication (N), and different techniques are all changed in a systematic way.

5.2. Simulation Design

A censored sample for the MEKGEKF model in the simulation inquiry was created using a progressively type II censored sample. We define the scheme with replication, N = 10,000, as follows:

-

The total sample size (n) was altered, resulting in two levels of n = 100 and 200.

-

In the presence of $m<n,$ and ${{\displaystyle \sum }}_{i=1}^m{R}_i+m=n$, the size of the progressive censored sample (m) was changed, generating two levels: m = 70, and 85, when n = 100, and m = 150 and 185, when n = 200.

-

The following two censored sample techniques were considered:

-

Scheme I: $R_i=0;i=1,\dots ,m-1$, $R_m=n-m$.

-

Scheme II: $R_i=0;i=2,\dots ,m$, $R_1=n-m$.

-

We create two models, one with parameters $\left(a_1,b_1,c_1,s_1,r_1, a_2,b_2,c_2,s_2,r_2,p\right)$ and the other with parameters $a_j=a,b_j=b,c_j=c;j=1,2.$ The true values of the parameters for the first model are $a_1=1.4, b_1=1.5, c_1=1.35, r_1=1.7, s_1=1.2, a_2=1.5, b_2=0.9, c_2=0.5, r_2=1.5, s_2=1.3$ and $p=0.3$ and 0.7.

-

We determine the length of the confidence interval using a 95% confidence level.

5.3. Simulation Study

For the Monte Carlo simulation, the estimation methods indicated in Section 3 and Section 4 were employed to estimate the model parameters. To acquire the desired MLEs, the iterative NR approach is numerically implemented (as closed form expressions for the MLE are not available in this case) using the ‘maxLik’ package in R. The regularity conditions are satisfied for approximate normality assumption, asymptotic confidence intervals for each of the parameters are computed using the formula as given in Section 3. MCMC techniques and the MH algorithm were used to create Bayesian estimators based on the independent gamma priors as distributed earlier.

5.4. Comment on the Simulation Study

For each predicted model parameter, the bias, the mean square error (MSE), and the length of confidence intervals (L.CI) were calculated.

Table 1. MLE and the Bayesian estimates for eight parameters of MEKGEKF model based on a progressively Type II censoring scheme: scheme II.

a = 1.5, b = 0.9, c = 0.5, r_1 = 1.7, s_1 = 1.2, r_2 = 1.5, s_2 = 1.3
P			0.7						0.3
Scheme II			MLE			Bayesian			MLE			Bayesian
n	m		Bias	MSE	L.CI	Bias	MSE	L.CI	Bias	MSE	L.CI	Bias	MSE	L.CI
100	70	a	1.4548	6.8007	8.4927	0.4668	1.9668	1.4592	1.8214	8.8669	9.2436	0.3883	1.8883	1.3945
		b	−0.0731	0.5318	2.8472	−0.0636	0.4424	0.4254	0.0881	0.7346	3.3453	−0.2926	0.6074	0.4434
		c	2.6442	8.2425	8.0901	1.0169	1.5169	1.1205	1.6371	5.7072	8.5544	0.8821	1.3821	0.9838
		r1	0.4663	0.5138	2.1360	0.3668	0.3682	1.0043	0.6953	1.2613	3.4608	0.5807	1.0281	1.1577
		s1	0.7198	2.3471	5.3068	0.1947	1.3947	0.9199	0.5244	2.9810	6.4548	−0.0788	1.1212	0.9274
		r2	0.1869	1.3567	4.5113	0.0980	1.1598	0.9634	0.3725	1.5640	4.6845	0.1664	1.2666	0.8129
		s2	−1.1514	1.3323	0.3210	−0.8203	0.4797	0.4600	−1.1185	1.2581	0.3265	−0.8400	0.4600	0.3424
		p	0.0785	0.0090	0.2098	0.0771	0.0078	0.1457	0.2033	0.0443	0.2137	0.1207	0.0435	0.1531
	85	a	1.0735	3.4035	7.9220	0.0961	1.5961	0.5930	0.7509	3.2821	6.2301	0.0248	1.5248	0.5898
		b	0.0298	0.4702	2.2867	−0.3147	0.4059	0.3421	0.0739	0.3770	2.3438	−0.2103	0.3569	0.4124
		c	2.1659	6.4644	7.1959	0.3158	0.8158	0.4490	1.0188	3.0638	5.2810	0.1233	0.6233	0.4289
		r1	0.3116	0.4086	2.1901	0.2219	0.2922	0.5581	0.6264	1.1762	3.4740	0.1291	0.8291	0.5640
		s1	0.1782	2.0748	5.0643	−0.0330	1.1670	0.5581	0.1532	1.1607	4.1844	−0.0694	1.1065	0.5794
		r2	0.1721	1.0554	3.9483	−0.0812	1.0376	0.5773	0.0354	0.6646	3.1958	−0.1158	0.3842	0.4881
		s2	−0.9914	1.3158	0.3947	−0.2891	0.4011	0.4262	−1.1133	1.2484	0.3724	−0.3124	0.3988	0.2603
		p	0.0327	0.0037	0.2026	0.0278	0.0027	0.1369	0.1012	0.0135	0.2251	0.0937	0.0114	0.1824
200	150	a	1.2807	5.3217	7.5289	0.6164	2.1164	1.2500	1.3976	5.6277	7.9095	0.5393	2.0393	1.1342
		b	−0.0867	0.4037	2.4697	−0.1329	0.3857	0.3569	0.0721	0.5746	3.3201	−0.2498	0.5016	0.4257
		c	2.6555	6.5439	8.3165	1.2912	1.7912	0.9227	2.6929	12.1891	8.7189	1.1480	1.6480	0.8290
		r1	0.4364	0.4318	1.9280	0.7040	0.4040	0.7817	0.5731	0.9278	3.0378	0.6558	0.8356	0.9415
		s1	0.3695	2.0172	3.1001	0.2994	1.4994	0.7652	0.4159	2.4043	5.8615	−0.0086	1.1914	0.8643
		r2	0.1284	0.9131	3.7156	0.2082	0.7082	0.8265	0.1956	1.0453	3.9376	0.1920	0.9692	0.7567
		s2	−1.1428	1.3146	0.3652	−0.9209	0.3791	0.2273	−1.1268	1.2757	0.3018	−0.9225	0.3775	0.1827
		p	0.0618	0.0057	0.1682	0.0690	0.0054	0.0983	0.1683	0.0302	0.1695	0.1735	0.0305	0.1063
	185	a	0.3712	1.1791	5.8075	0.1286	0.9629	0.5482	1.0336	5.4624	6.2148	0.0475	1.5475	0.6107
		b	0.0636	0.3708	2.2926	−0.1327	0.3573	0.2455	0.0634	0.4340	2.3366	−0.1977	0.4570	0.3614
		c	1.1242	3.0313	4.8262	0.4249	0.9249	0.4452	1.8114	5.8520	6.2913	0.1993	0.6993	0.4343
		r1	0.2521	0.3106	1.9501	0.3504	0.3050	0.4681	0.5507	0.8573	2.9206	0.2079	0.7908	0.5322
		s1	0.3283	1.7236	3.5873	0.0232	1.2232	0.4998	0.1631	1.4317	4.6513	−0.0071	1.0823	0.5209
		r2	0.1184	0.7906	3.3970	−0.1138	0.3286	0.5288	0.0125	0.6278	3.1087	−0.0912	0.5741	0.4166
		s2	−1.1051	1.3032	0.3158	−0.4850	0.2815	0.2048	−1.0926	1.2035	0.3852	−0.4555	0.3284	0.1485
		p	0.0101	0.0022	0.1817	0.0150	0.0020	0.0912	0.0470	0.0043	0.1780	0.0520	0.0035	0.0913

and

Table 2. MLE and the Bayesian estimates for eight parameters of MEKGEKF model based on a progressively type II censoring scheme: scheme I.

a = 1.5, b = 0.9, c = 0.5, r_1 = 1.7, s_1 = 1.2, r_2 = 1.5, s_2 = 1.3
P			0.3						0.7
Scheme I			MLE			Bayesian			MLE			Bayesian
n	m		Bias	MSE	L.CI	Bias	MSE	L.CI	Bias	MSE	L.CI	Bias	MSE	L.CI
100	70	a	1.4812	6.3149	7.9656	0.1960	1.6960	1.2524	0.7604	3.5388	6.7517	0.3284	1.8284	1.2715
		b	−0.1508	0.4706	2.6259	−0.1314	0.4586	0.4919	−0.4730	0.3418	1.3479	−0.4453	0.3345	0.5120
		c	1.5721	3.8398	4.5903	0.5346	1.0346	0.9766	1.0749	3.4595	4.0781	0.8390	1.3390	0.9246
		r1	0.6833	1.0492	2.9946	0.4462	0.9462	1.0826	0.6380	1.1163	3.3045	0.5613	1.0026	1.2121
		s1	0.7531	2.4720	5.4156	0.0847	1.2847	1.0112	1.2079	3.3859	5.4469	0.2683	1.4683	1.0754
		r2	0.1392	0.6678	3.1597	−0.0920	0.5408	0.7215	0.1813	1.9899	5.4894	−0.1313	1.3687	1.1678
		s2	−1.0921	1.2024	0.3869	−0.7200	0.5800	0.5644	−1.1620	1.4099	0.9592	−0.8008	0.4992	0.6737
		p	0.0010	0.0058	0.2980	0.0011	0.0043	0.1928	0.1229	0.0366	0.5747	0.1146	0.0281	0.3735
	85	a	1.1748	4.0035	6.5685	0.0037	1.5037	0.6230	0.5137	3.1754	5.7323	0.0762	1.5762	0.5899
		b	0.0681	0.4630	2.1043	−0.1291	0.4071	0.4652	−0.3068	0.3350	1.2984	−0.3051	0.2549	0.4072
		c	1.4629	3.0898	4.2704	0.0572	0.5572	0.4162	1.0083	3.2852	3.0089	0.2826	0.7826	0.4511
		r1	0.6097	0.9017	2.8564	0.1098	0.8098	0.5643	0.2623	0.4083	2.2863	0.1940	0.3894	0.6012
		s1	0.4566	2.1514	5.4069	−0.0549	1.1451	0.5705	0.4389	2.1187	4.7618	−0.0243	1.1757	0.5946
		r2	0.0905	0.6409	3.1212	−0.0917	0.5334	0.4749	0.0833	0.6379	4.9202	−0.1273	0.5327	0.5981
		s2	−1.0827	1.1816	0.3772	−0.2509	0.5049	0.5743	−0.9167	1.0374	0.4335	−0.2783	0.4022	0.6075
		p	−0.0009	0.0028	0.2085	0.0011	0.0019	0.1861	0.0023	0.0019	0.3635	0.0220	0.0017	0.2700
200	150	a	1.3829	5.1500	7.2363	0.3354	1.8354	1.1577	0.9995	2.9188	5.7686	0.4716	1.9716	1.1515
		b	−0.0177	0.4683	2.6845	−0.3024	0.4598	0.4052	−0.4310	0.3347	1.2514	−0.4111	0.3049	0.4348
		c	1.3678	3.2793	4.3745	0.8015	1.3015	0.8367	1.9678	3.4024	4.2680	1.0799	1.5799	0.9326
		r1	0.4427	0.5808	2.4342	0.5614	0.4261	0.9197	0.5118	0.7459	2.7299	0.5623	0.6323	1.3790
		s1	0.5553	1.9172	4.9771	0.1469	1.3469	0.8726	1.4119	3.2717	5.1476	0.3630	1.5630	1.0576
		r2	−0.0066	0.5290	2.8538	−0.0202	0.4798	0.6039	0.3841	1.6286	5.0738	−0.0233	1.4767	1.0847
		s2	−1.1067	1.1234	0.3784	−0.8566	0.4434	0.3079	−1.1744	1.3408	0.6693	−0.9300	0.3700	0.3243
		p	−0.0028	0.0024	0.1920	0.0075	0.0023	0.1290	0.0881	0.0246	0.5085	0.0853	0.0179	0.3166
	185	a	1.0934	4.6595	6.3212	0.0287	1.5287	0.6073	0.8302	2.5091	4.2423	0.1104	1.6104	0.5314
		b	0.0225	0.4013	4.0693	−0.2002	0.3700	0.3351	−0.1452	0.2952	1.2400	−0.3379	0.2456	0.2406
		c	1.1733	2.6155	4.0981	0.1441	0.6441	0.4089	1.0693	2.6201	2.9631	0.4032	0.9032	0.4197
		r1	0.4078	0.4717	2.3743	0.1892	0.3889	0.5034	0.2704	0.2688	1.7361	0.3418	0.2042	0.4676
		s1	0.2806	1.3223	4.3758	−0.0697	1.1303	0.5238	1.0371	1.7179	4.0285	0.0432	1.2432	0.5002
		r2	0.0020	0.5082	2.6992	−0.0291	0.4371	0.3973	0.0905	0.6067	3.1853	−0.0172	0.4328	0.4848
		s2	−0.8509	0.9002	0.3291	−0.4111	0.3889	0.4614	−1.1593	0.9349	0.2850	−0.4671	0.2833	0.2518
		p	−0.0024	0.0021	0.1782	0.0040	0.0019	0.1234	−0.0137	0.0013	0.1905	−0.0062	0.0017	0.1343

show the estimated values for the eight parameters of MEKGEKF model. The predicted results for the 11 parameters are shown in

Table 3. MLE and the Bayesian estimates for 11 parameters of MEKGEKF model based on a progressively type II censoring scheme: scheme II.

a1 = 1.4, b1 = 1.5, c1 = 1.35, r1 = 1.7, s1 = 1.2, a2 = 1.5, b2 = 0.9, c2 = 0.5, r2 = 1.5, s2 = 1.3
P			0.3						0.7
Scheme II			MLE			Bayesian			MLE			Bayesian
n	m		Bias	MSE	L.CI	Bias	MSE	L.CI	Bias	MSE	L.CI	Bias	MSE	L.CI
100	70	a1	0.4485	1.3383	4.1843	0.8174	1.2174	1.2510	0.9093	3.8809	4.3964	0.9390	2.3390	1.2002
		b1	−1.1070	1.2722	0.8496	−0.5150	0.9850	0.5027	−1.2796	1.5234	1.0292	−0.5234	0.9766	0.5162
		c1	1.3367	3.9992	5.8367	0.6282	1.9782	0.9096	1.2355	2.9445	4.6726	1.0508	2.4008	1.1619
		r1	1.6772	3.3734	2.9377	0.5953	2.2953	1.0590	1.4994	2.5892	2.2916	0.7760	2.4760	0.8898
		s1	0.6009	0.9859	3.1016	−0.2872	0.9128	0.6180	0.8792	1.4961	2.5483	−0.1838	1.0162	0.6048
		a2	−0.2485	0.6120	2.9109	0.2737	0.5774	1.0895	−0.4913	0.8773	2.8624	0.0242	0.7524	1.3380
		b2	0.6586	1.3843	3.8256	−0.1264	0.7736	0.8745	0.1640	0.7182	3.3003	0.1101	0.6010	1.0924
		c2	0.3909	0.4746	2.2259	0.4365	0.3650	0.6701	0.4269	0.4320	1.9541	0.0824	0.3582	0.7868
		r2	−0.2469	0.1625	1.2503	−0.0359	0.1464	0.3180	−0.2024	0.3699	1.9086	−0.0322	0.3047	0.8715
		s2	−0.1125	0.3777	2.3709	−0.8256	0.3474	0.5580	−0.1585	0.4344	2.5441	−0.3162	0.3984	1.0596
		p	0.1988	0.0433	0.2401	0.1994	0.0410	0.1809	0.0816	0.0087	0.1784	0.0777	0.0078	0.1363
	85	a1	0.3065	1.0730	3.8826	0.2303	1.0063	0.5693	0.8141	2.4205	4.2024	0.3239	1.7239	0.5744
		b1	−1.0140	1.2339	0.7704	−0.3519	0.9148	0.4980	−1.0948	1.2360	0.7595	−0.4637	0.8036	0.4194
		c1	0.8985	1.7866	3.8831	0.1837	1.5337	0.5331	1.1830	2.2545	3.6284	0.3017	1.6517	0.5316
		r1	1.5991	3.0612	2.8145	0.1344	1.8344	0.5542	1.4235	2.3458	2.2174	0.2364	1.9364	0.5461
		s1	0.5637	0.8687	2.6718	−0.2313	0.7969	0.5553	0.8450	1.3584	2.0761	−0.1624	0.9616	0.4623
		a2	−0.2494	0.6058	2.8028	−0.0166	0.4834	0.5798	−0.3886	0.8092	2.3781	−0.0359	0.7464	0.6107
		b2	0.5745	1.3758	3.5545	0.0447	0.6945	0.5135	0.1401	0.4258	2.5009	0.0275	0.3928	0.5811
		c2	0.3187	0.2986	1.7418	−0.0194	0.2481	0.3658	0.3040	0.3637	1.0437	−0.0826	0.2417	0.3953
		r2	−0.2091	0.1530	1.0279	−0.0433	0.1457	0.3050	−0.0207	0.1940	1.7265	−0.0499	0.1450	0.5383
		s2	0.0057	0.3710	2.3901	−0.3031	0.2997	0.5172	−0.1440	0.4047	2.2562	−0.1094	0.2191	0.5856
		p	0.0952	0.0121	0.2147	0.0915	0.0131	0.1721	0.0355	0.0035	0.1684	0.0298	0.0027	0.1270
200	150	a1	0.9470	2.2872	4.6271	0.9128	2.1277	1.3001	0.4114	1.1061	3.7981	0.9555	1.0355	1.1540
		b1	−1.0714	1.1984	0.8828	−0.8215	0.6785	0.4174	−1.1240	1.3073	0.8212	−0.5984	0.9016	0.4386
		c1	1.1953	2.5050	4.0709	1.0243	2.3743	1.0803	1.0262	2.1278	4.0677	0.8228	1.1728	0.9477
		r1	1.4183	2.2867	2.0585	0.9411	2.0641	0.7755	1.6281	1.9823	2.6246	0.7962	1.4962	0.9049
		s1	0.8392	1.2003	2.7634	0.2245	1.0242	0.8303	0.6319	0.9007	2.7787	−0.1807	0.7019	0.6244
		a2	−0.4219	0.7772	3.0374	−0.0040	0.6050	1.2278	−0.3176	0.6521	2.9135	0.2909	0.4791	1.0640
		b2	0.1270	0.5165	2.7757	0.1388	0.4039	0.9506	0.5599	0.8457	2.8627	0.0441	0.4412	0.5725
		c2	0.2976	0.3911	2.1584	0.0945	0.2595	0.7621	0.4042	0.4092	1.9457	0.4193	0.3919	0.6277
		r2	−0.1037	0.1679	1.5555	−0.1061	0.1394	0.6315	−0.2769	0.1421	1.0037	−0.0619	0.1144	0.3238
		s2	−0.1233	0.4092	2.4630	−0.2884	0.3501	1.0169	−0.0964	0.3517	2.2962	−0.7122	0.2588	0.3846
		p	0.0640	0.0054	0.1414	0.0649	0.0048	0.1131	0.1716	0.0308	0.1437	0.1748	0.0305	0.1083
	185	a1	0.6999	1.7976	4.4871	0.4091	1.2809	0.5714	0.3199	0.8713	3.4410	0.3034	0.7034	0.5722
		b1	−0.9098	1.0241	0.7362	−0.5981	0.5902	0.3495	−1.1333	1.3208	0.7489	−0.4507	0.4933	0.4471
		c1	1.0293	2.1656	3.8915	0.3931	1.7431	0.4848	0.9330	1.9206	4.0211	0.2792	1.0629	0.4622
		r1	1.3227	1.9678	1.8336	0.3660	1.2066	0.4531	0.7003	1.3164	2.5592	0.2341	0.9341	0.5152
		s1	0.8191	1.0731	2.6907	−0.1627	0.9504	0.4453	0.5712	0.8009	2.7834	−0.2330	0.6967	0.4539
		a2	−0.3538	0.7252	3.0395	−0.0035	0.4645	0.5451	−0.1551	0.5591	2.8702	−0.0169	0.4831	0.5817
		b2	0.1217	0.4538	2.7494	0.0526	0.2953	0.4958	0.4873	0.0654	1.7068	0.0827	0.0598	0.4627
		c2	0.2832	0.2799	1.7535	−0.0877	0.2412	0.3525	0.2762	0.2598	1.6809	−0.0267	0.2047	0.3522
		r2	−0.0974	0.1475	1.4579	−0.0673	0.1243	0.4490	−0.2340	0.1258	0.8074	−0.0682	0.1143	0.2717
		s2	−0.1223	0.3528	2.1519	−0.1073	0.3193	0.5755	−0.0068	0.3301	2.2542	−0.2724	0.2303	0.2507
		p	0.0158	0.0017	0.1451	0.0168	0.0014	0.1018	0.0464	0.0035	0.1435	0.0520	0.0031	0.1015

and

Table 4. MLE and the Bayesian estimates for 11 parameters of MEKGEKF model based on a progressively type II censoring. scheme: scheme I.

a1 = 1.4, b1 = 1.5, c1 = 1.35, r1 = 1.7, s1 = 1.2, a2 = 1.5, b2 = 0.9, c2 = 0.5, r2 = 1.5, s2 = 1.3
P			0.3						0.7
Scheme I			MLE			Bayesian			MLE			Bayesian
n	m		Bias	MSE	L.CI	Bias	MSE	L.CI	Bias	MSE	L.CI	Bias	MSE	L.CI
100	70	a1	0.5169	0.9005	3.1227	0.5818	0.8182	1.3106	0.9291	0.8008	3.7994	0.6531	0.5315	1.2357
		b1	−1.0868	1.3210	0.6667	−0.4308	1.0692	0.5615	−1.0763	1.1935	0.7364	−0.6317	0.8683	0.6844
		c1	0.9621	1.5594	3.1238	0.4656	1.4816	0.9816	1.0894	1.2896	3.3035	0.5848	1.0935	0.9631
		r1	1.5841	2.9146	2.4982	0.4781	2.1781	1.1460	1.1802	1.9416	2.9064	0.4502	1.1502	1.2367
		s1	0.7478	0.9938	2.5870	−0.2292	0.9708	0.6886	0.6127	0.6343	1.9965	−0.2632	0.5368	0.8088
		a2	−0.3018	0.5385	2.6247	0.2164	0.4716	1.1929	−0.3651	0.5180	3.2154	0.0158	0.4516	1.2552
		b2	−0.1902	0.6458	2.3351	−0.4175	0.4825	0.6701	−0.2009	0.5633	2.8375	−0.1374	0.4763	1.1678
		c2	0.3815	0.1765	1.4198	0.4604	0.1604	0.6813	−0.0964	0.1220	1.3172	0.0920	0.1159	0.7786
		r2	−0.3897	0.1387	1.1633	−0.0728	0.1272	0.4186	0.1480	0.3581	2.2753	−0.0450	0.3455	0.6009
		s2	−0.0800	0.3166	2.1854	−0.9277	0.2937	0.4913	−0.3843	0.3037	1.7090	−1.0297	0.2703	0.3944
		p	−0.0047	0.0028	0.2060	0.0083	0.0022	0.1957	0.1430	0.0024	0.5312	0.1234	0.0018	0.3952
	85	a1	0.4648	0.8902	3.0521	0.1869	0.7587	0.5695	0.8579	0.7902	3.2367	0.3112	0.4711	0.5680
		b1	−1.0103	1.3250	0.6171	−0.2938	1.0162	0.4734	−1.0878	0.9211	0.6491	−0.4654	0.3455	0.4141
		c1	0.8610	1.4713	3.0353	0.1511	1.3501	0.5421	1.1263	0.9323	3.3493	0.2794	0.6294	0.5311
		r1	1.0615	2.8012	2.4989	0.1316	1.8316	0.5700	1.1411	1.6850	2.4281	0.2193	0.9592	0.5909
		s1	0.7070	0.9360	2.5915	−0.1975	0.9002	0.5249	0.6032	0.6055	1.7194	−0.2520	0.4595	0.4923
		a2	−0.2724	0.5073	2.5823	−0.0070	0.4193	0.5672	−0.3098	0.4768	3.0619	−0.0150	0.4085	0.5739
		b2	0.7137	0.6078	2.1882	−0.0177	0.4588	0.5102	−0.2674	0.4620	2.4521	−0.0207	0.3879	0.6074
		c2	0.2379	0.1722	1.3590	−0.0216	0.1478	0.3903	0.0915	0.1206	1.2677	−0.0408	0.1046	0.4210
		r2	−0.2922	0.1294	0.9379	−0.0687	0.1143	0.3604	0.0248	0.1299	1.8790	−0.0443	0.1046	0.4184
		s2	−0.0270	0.3031	2.1257	−0.3122	0.2899	0.4602	−0.2504	0.2372	1.6183	−0.2982	0.2002	0.2621
		p	−0.0043	0.0021	0.1798	0.0080	0.0013	0.1751	0.0245	0.0011	0.3408	0.0157	0.0011	0.2787
200	150	a1	0.4924	0.8209	2.9843	0.8163	0.7216	1.1666	0.4070	0.8144	2.0689	0.3924	0.6324	1.1376
		b1	−1.1025	1.2418	0.6369	−0.5172	0.9828	0.4745	−1.0867	1.2104	0.6747	−0.7020	0.7980	0.5389
		c1	0.9006	1.3396	2.8524	0.6858	1.0358	0.9191	1.1891	2.4467	3.9874	0.8328	2.1828	0.9474
		r1	1.4927	2.5105	2.0848	0.6926	2.3926	0.9399	0.9981	1.5157	2.8279	0.6886	1.3886	1.2945
		s1	0.7647	0.9864	2.4866	−0.1765	1.0235	0.5930	0.7926	1.0317	2.4925	−0.1262	1.0074	0.7845
		a2	−0.1830	0.4411	2.5052	0.2384	0.3738	1.0668	−0.3452	0.6229	2.7849	0.0898	0.5898	1.1480
		b2	0.0844	0.3346	2.2454	−0.2343	0.2666	0.4900	−0.1491	0.5143	2.7525	−0.1735	0.4726	1.1703
		c2	0.1630	0.1754	1.5138	0.4305	0.1293	0.6352	−0.0214	0.1848	1.6848	0.2158	0.1716	0.7023
		r2	−0.2889	0.1328	0.8715	−0.0581	0.1442	0.3293	0.1260	0.2577	1.9296	−0.0364	0.2464	0.5144
		s2	−0.0763	0.2926	2.1012	−0.7679	0.2532	0.4059	−0.3709	0.3822	1.9407	−0.9083	0.3592	0.4977
		p	−0.0037	0.0015	0.1509	0.0044	0.0013	0.1273	0.1178	0.0286	0.4754	0.1103	0.0181	0.3326
	185	a1	0.3211	0.8198	2.3220	0.2876	0.6876	0.5772	0.3851	0.4211	2.0111	0.3422	0.3822	0.5432
		b1	−0.9123	1.2290	0.6076	−0.4146	0.8536	0.4177	−1.0602	1.1631	0.5776	−0.5455	0.6955	0.3386
		c1	0.8563	1.3047	2.3635	0.2471	1.0597	0.5439	1.0288	1.5699	3.7459	0.3977	1.3748	0.5035
		r1	0.9588	2.3823	1.9153	0.2155	1.9155	0.5365	0.8268	1.2794	1.6905	0.3665	1.0665	0.4808
		s1	0.6754	0.3958	2.2693	−0.2223	0.2978	0.4440	0.5979	0.9426	2.2685	−0.1179	0.9021	0.4345
		a2	−0.1667	0.4077	2.2629	−0.0100	0.3490	0.5817	−0.3227	0.5803	2.2811	−0.0238	0.4762	0.5296
		b2	0.0691	0.3153	2.2246	0.0316	0.2193	0.4818	−0.1125	0.4567	2.6146	−0.0005	0.3900	0.5644
		c2	0.1597	0.1756	1.4510	−0.0270	0.1147	0.3524	0.0235	0.1313	1.2992	−0.0381	0.1246	0.3373
		r2	−0.2354	0.1306	0.7670	−0.0849	0.1242	0.2977	−0.0872	0.1562	1.5124	−0.0260	0.1440	0.3774
		s2	0.0259	0.2830	2.0265	−0.2729	0.2709	0.3511	−0.1701	0.2421	1.8152	−0.2502	0.1498	0.4551
		p	−0.0018	0.0013	0.1387	0.0037	0.0011	0.1257	−0.0072	0.0018	0.1644	−0.0043	0.0017	0.1477

. There is some interesting information that can be observed from these tables. As the sample size n and m increases, the estimates become more accurate, implying that they are asymptotically unbiased which a desirable property. When we increase the value of parameter $p$, the mixing component to 1, the bias, MSE and L.CI decrease. Furthermore, the MSE decreases as the sample size increases in all situations, implying that these estimates are consistent. When comparing between the two estimates, we observe that the Bayes estimates have the lowest MSE in the vast majority of cases. The L.CI for estimates approaches 0 as it grows, indicating that the CI is the shortest. However, we cannot say that the Bayesian method will be uniformly better in all situations.

6. Real Data Applications

We consider several lifetime data sets in this section to demonstrate how well this new finite mixture distribution fits for illustrative purposes. For the components (of the mixture) of the distribution’s efficacy, we calculated the Akaikes information criterion (AIC), the Bayesian information criterion (BIC), the Hannan Quinn information criterion (HQIC), and the Kolmogorov–Smirnov (K-S) values. The parameters were estimated using the MLE method, and the MLE optimizations are done using the Nelder–Mead technique, which is a numerical approach for finding the global maximum of an objective function in a multidimensional space, and the R software is utilized.

6.1. Operation Data on Jobs Made of Iron Sheet

The data set consists of 50 observations (in millimeters), the hole diameter is 12 mm, and the sheet thickness is 3.15 mm, as reported in [25]. The data values are: 0.04, 0.02, 0.06, 0.12, 0.14, 0.08, 0.22, 0.12, 0.08, 0.26, 0.24, 0.04, 0.14, 0.16, 0.08, 0.26, 0.32, 0.28, 0.14, 0.16, 0.24, 0.22, 0.12, 0.18, 0.24, 0.32, 0.16, 0.14, 0.08, 0.16, 0.24, 0.16, 0.32, 0.18, 0.24, 0.22, 0.16, 0.12, 0.24, 0.06, 0.02, 0.18, 0.22, 0.14, 0.06, 0.04, 0.14, 0.26, 0.18, 0.16.

We conjecture that the data have interesting characteristics if we divide it into two parts and also conjecture that a mixing distribution (finite mixture) will be adequate to explain the hidden characteristics for the data. The two-component mixture of EKF and EKG distribution was applied to the data set given as.

Subpopulation I: 0.04, 0.06, 0.12, 0.22, 0.08, 0.26, 0.14, 0.08, 0.32, 0.14, 0.16, 0.12, 0.24, 0.16, 0.08, 0.16, 0.32, 0.24, 0.22, 0.24, 0.02, 0.22, 0.06, 0.14, 0.26; these data are applied to EKF distribution.

Subpopulation II: 0.02, 0.14, 0.08, 0.12, 0.24, 0.04, 0.16, 0.26, 0.28, 0.24, 0.22, 0.18, 0.32, 0.14, 0.24, 0.16, 0.18, 0.16, 0.12, 0.06, 0.18, 0.14, 0.04, 0.18, 0.16; these data are applied to EKG distribution.

From

Table 5. MLE, stander error (SE), KS test, and p-value for different sets for the hole data.

Set			a	b	c	r	s	KS	p-Value
I	EKG	estimates	11.0300	5.4197	14.0539	2.3218	2.5587	0.1515	0.6142
		SE	11.0044	0.0318	0.3192	1.5341	0.0018
	EKF	estimates	0.1642	2836.5002	0.1866	0.5723	250.1562	0.1703	0.4634
		SE	0.0030	0.1182	0.0382	0.0024	0.0027
II	EKF	estimates	3.1161	232708.150	0.2434	0.3812	9.6146	0.1692	0.4716
		SE	0.0166	7.8429	0.0585	0.0081	0.0163
	EKG	estimates	8.7129	1.8848	7.8261	3.8506	4.8280	0.1248	0.8309
		SE	7.9166	1.1759	0.9106	2.4614	3.1271

and

Table 6. Goodness-of-fit measures, AIC, BIC, CAIC, and HQIC for the hole diameter data.

		AIC	CAIC	BIC	HQIC
I	EKF	−45.4153	−42.2575	−39.3210	−43.7250
	EKG	−42.6578	−39.4999	−36.5634	−40.9674
II	EKF	−48.6074	−45.4495	−42.5130	−46.9170
	EKG	−47.9072	−44.7493	−41.8128	−46.2168

, we note that EKF is the best model for each data, but the p-value for EKG is larger than EKF for data set II. Therefore, the first distribution EKF is good for data set I, and the second distribution EKG is good for data set II in the mixture model. However, either of the distributions cannot provide a good fit for the entire data set.

Table 6 shows the different goodness-of-fit measures values as AIC, BIC, CAIC, and HQIC for EKG and EKF distribution for the hole diameter data. Here, we conclude that the EKF model is better than the EKG model.

Figure 2 and Figure 3 show the Estimated CDF and PDF for EKG and EKF distribution for Subpopulation I and estimated CDF and PDF for EKG and EKF distribution for Subpopulation II. While, the PP-plot for EKG and EKF distribution for Subpopulation I and the PP-plot for EKG and EGF distribution for Subpopulation II are indicated in Figure 4 and Figure 5, respectively.

Table 7. MLE and the Bayesian estimates for the fitted MEKGEKF model.

	MLE				Bayesian
	Estimates	SE	Lower	Upper	Estimates	SE	Lower	Upper
a1	8.77746	3.08564	3.5401	15.6358	8.27457	0.65807	7.15824	9.32375
b1	1.88648	1.28460	0	4.3090	3.06130	0.57748	2.06183	4.26348
c1	7.84970	0,37569	11.6612	11.6612	6.88541	0.42855	6.08697	7.70303
r1	3.78019	0.91045	0	736361.4	5.14365	0.96433	3.61036	6.73922
s1	4.81577	1.40953	3.4584	7.0274	3.94590	0.57362	3.00186	5.08593
a2	0.19847	0.00540	0	2.9725	0.89840	0.35695	0.17860	1.53123
b2	3120.229	71.0643	6415.481	6415.502	2836.4548	0.27465	2835.979	2836.886
c2	0.20924	0.03774	0	139.4791	0.74696	0.52516	0.0030	1.5735
r2	0.54347	0.00409	0.4678	0.6157	0.34097	0.07058	0.21109	0.48834
s2	250.3161	3.95647	271.0581	271.0741	207.96673	24.56475	169.3517	249.5836
P	0.49973	0.05001	0	8.2547	0.50211	0.22917	0.06905	0.87604

report the MLE and Bayesian estimation parameters of mixture of exponentiated Kumaraswamy Gompertz and exponentiated Kumaraswamy Fréchet distribution. From the reported results, we conclude that the Bayesian estimation is preforming better than the MLE where the standard error (SE) values are smaller compared the SE values obtained under the MLE method.

From the above summary measures, it appears that the MEKGEKF model might be a reasonably good fit for the data.

6.2. Wire Data

In the Wire Fatigue Experiment [26], a 48-stranded stainless-steel wire was ruptured by clamping it in needle-nose pliers and hanging a 1.65-pound weight on it while using 3/4 of a liter of water, then a 2.2-pound weight while using 1 liter of water. The pliers were spun clockwise and counterclockwise by 180 degrees. The number of half twists that resulted in total rupture (failure) was counted. For the Wire Fatigue Experiment, the wire data represent the number of half twists to total rupture as follows: 37, 30, 27, 51, 10, 24, 15, 14, 34, 34, 42, 25, 15, 13, 16, 12, 27, 21, 37, 35, 18, 17, 17, 13, 35, 27, 41, 41, 14, 17, 20, 16, 28, 24, 32, 24, 17, 19, 15, 20, 45, 39, 27, 33, 18, 13, 11, 13.

From

Table 8. MLE with standard error (SE) and KS test for different sets and different model for Wire data.

		a	b	c	r	s	KS	p-Value
EKG	estimates	37.2316	10.5566	33.4543	0.0084	0.0170	0.1391	0.3108
	SE	12.0742	2.9050	17.2727	0.0001	0.0011
EKF	estimates	23.1579	8.5000	0.1136	1.8776	12.7105	0.0971	0.7557
	SE	0.0028	0.0085	0.0164	0.0028	0.0028

, we note that EKF is the fitting model for Wire data, but the p-value of EKF is larger than EKG for Wire data. Figure 6 and Figure 7 shows that the shape of the probability density function is bimodal for both EKG and EKF distribution but the original data shape is uni-modal.

Table 9. AIC, BIC, CAIC, and HQIC for different sets and different model for Wire data.

	AIC	CAIC	BIC	HQIC
EKG	367.803	369.2316	377.159	371.3387
EKF	358.612	360.0406	367.968	362.1476

shows different goodness-of-fit measures values such as the AIC, BIC, CAIC, and the HQIC for EKG and EKF distribution for the hole diameter data.

Table 10. MLE and the Bayesian estimates for the fitted MEKGEF model for the Wire data.

	MLE				Bayesian
	Estimates	Lower	Upper	SE	Estimates	Lower	Upper	SE
a1	37.1677	3.6785	70.6569	17.0863	36.8687	36.4984	36.9129	0.1100
b1	10.7300	4.1118	17.3482	3.3766	10.2542	10.1217	10.8638	0.2328
c1	35.9627	24.4424	47.4829	5.8777	34.5129	34.3080	34.9799	0.4061
r1	0.0084	0.0079	0.0089	0.0003	0.0281	0.0264	0.0307	0.0002
s1	0.0170	0.0148	0.0193	0.0012	0.1695	0.1491	0.1923	0.0017
a2	23.1584	23.1530	23.1638	0.0027	23.5680	23.5412	23.5967	0.0019
b2	9.4998	9.4829	9.5167	0.0086	9.1781	9.0840	9.3653	0.0082
c2	0.1152	0.0826	0.1478	0.0166	0.1350	0.0796	0.1923	0.0066
r2	1.8759	1.8705	1.8813	0.0027	2.4520	2.2982	2.6348	0.0015
s2	12.7133	12.7079	12.7186	0.0027	1.4324	1.4000	1.4488	0.0019
p	0.5001	0.4001	0.6001	0.0510	0.0828	0.0102	0.1467	0.0388

illustrate the MLE and the Bayesian estimates for the fitted MEKGEF model for the Wire data. Consequently, we conclude that the EKF is petter than EKG model. However, from the above summary, it can be said that component-wise, either of the EKG and the EKF might not adequately fit the data. Consequently, we conjecture that the 11-parameter MEKGEKF distribution in (6) will be a good fit and in the next, we fit this distribution to this data set.

6.3. Australian Athletes Data Set

These data were gathered as part of a study that looked at how data on various blood parameters differed according to the athlete’s sport body size and gender. There are 202 observations on 13 variables in this data set (for more details see [27]). The variables rcc (red blood cell count), wcc (white blood cell count), and bmi (body mass index) were investigated (body mass index).

The data values are 3.80 3.90 3.90 3.91 3.95 3.95 3.96 3.96 4.00 4.02 4.03 4.06 4.07 4.08 4.09 4.09 4.10 4.11 4.11 4.12 4.13 4.13 4.14 4.15 4.16 4.16 4.17 4.17 4.19 4.20 4.20 4.21 4.23 4.23 4.24 4.24 4.25 4.26 4.26 4.27 4.27 4.30 4.31 4.31 4.32 4.32 4.32 4.35 4.36 4.36 4.37 4.38 4.38 4.39 4.40 4.40 4.40 4.41 4.41 4.41 4.42 4.42 4.44 4.44 4.44 4.45 4.45 4.46 4.46 4.46 4.46 4.46 4.48 4.49 4.50 4.50 4.51 4.51 4.51 4.51 4.52 4.53 4.54 4.55 4.56 4.57 4.58 4.62 4.63 4.63 4.63 4.64 4.66 4.68 4.71 4.71 4.71 4.71 4.73 4.75 4.75 4.76 4.77 4.77 4.78 4.81 4.81 4.82 4.82 4.83 4.83 4.83 4.83 4.84 4.86 4.86 4.87 4.87 4.87 4.87 4.87 4.87 4.88 4.89 4.89 4.90 4.90 4.91 4.91 4.92 4.93 4.93 4.94 4.95 4.95 4.96 4.97 4.97 4.98 4.99 5.00 5.00 5.00 5.01 5.01 5.01 5.02 5.02 5.03 5.03 5.03 5.03 5.04 5.04 5.08 5.09 5.09 5.09 5.10 5.11 5.11 5.11 5.11 5.11 5.13 5.13 5.13 5.13 5.16 5.16 5.16 5.16 5.17 5.17 5.18 5.21 5.21 5.22 5.22 5.24 5.24 5.25 5.29 5.31 5.32 5.33 5.33 5.34 5.34 5.34 5.34 5.38 5.40 5.48 5.48 5.49 5.50 5.59 5.66 5.69 5.93 6.72.

From Figure 8 and Figure 9, it appears that, component-wise, EKG and EKF will not provide an adequate fit to the given data. The p-value in

Table 11. MLE with standard error (SE) and KS test for different sets and different model for Australian athletes data set.

		a	b	c	r	s	KS	p-Value
EKG	estimates	26.8791	23.1488	25.1263	0.1938	0.1300	0.0791	0.1595
	SE	10.6790	5.6546	16.5222	0.0370	0.0149
EKF	estimates	1.9174	6.2048	0.1795	7.8763	5.3766	0.0840	0.1152
	SE	0.9154	3.7807	0.3686	1.1437	0.4242

also supports this assessment. As before, we conjecture that the 11 parameter MEKGEKF distribution might provide a good fit to the data. We fit the proposed distribution and the goodness-of-fit measures are reported in

Table 12. AIC, BIC, CAIC, and HQIC for different sets and different model for Australian athletes data set.

	AIC	CAIC	BIC	HQIC
EKG	263.2465	263.5526	279.7878	269.9391
EKF	257.1600	257.4661	273.7014	263.8527

. The MLE and the Bayesian estimates for the fitted MEKGEF model for the Australian athletes data are shown in

Table 13. MLE and the Bayesian estimates for the fitted MEKGEF model for the Australian athletes data.

	MLE				Bayesian
	Estimates	Lower	Upper	SE	Estimates	Lower	Upper	SE
a1	27.0339	19.2220	34.8458	3.9857	27.4058	27.0145	27.5292	0.1287
b1	20.8161	16.5884	25.0437	2.1570	20.4739	20.2677	20.6272	0.1151
c1	49.0710	37.4738	60.6683	5.9170	49.0488	48.9128	49.2190	0.0939
r1	0.1710	0.0688	0.2732	0.0521	0.2407	0.1601	0.3314	0.0500
s1	0.1385	0.0916	0.1855	0.0240	0.1472	0.1196	0.1913	0.0188
a2	2.0455	1.9697	2.1212	0.0386	2.5514	2.5024	2.5820	0.0158
b2	6.4045	6.2346	6.5744	0.0867	6.0228	5.9922	6.0930	0.0711
c2	0.1683	0.1395	0.1971	0.0147	0.4713	0.4327	0.5161	0.0108
r2	7.9388	7.8751	8.0025	0.0325	7.1916	7.0316	7.2929	0.0315
s2	5.3612	5.3021	5.4204	0.0302	0.3103	0.0181	0.5202	0.0298
p	0.5002	0.4515	0.5490	0.0249	0.3212	0.3168	0.5486	0.0194

.

From the above summary measures, it appears that the proposed distribution given in (6) is a reasonable model to fit the above data.

7. Concluding Remarks

Classical and Bayesian inferences for finite mixture models in the case of absolutely continuous probability models as subpopulations is not new in the literature except for the fact that probability distribution(s) defined on the unit interval (0,1) has not been considered effectively earlier. In this paper, we discuss the utility of a finite mixture model for which the components are also some mixtures with Kumaraswamy distribution as one of the mixing components under the frequentist approach, the method of the maximum likelihood is adopted while for the Bayesian estimation, independent gamma priors are considered for all the model parameters of the derived model which we call the MEKGEKF distribution. The performance of both these estimation methods are assessed in the light of a progressively censored Type II random samples drawn from the newly developed probability model. From the simulation study (under various parameter settings and various censoring scheme), it appears that this MEKGEKF distribution might be useful in analyzing certain dataset(s) for which either or both of its component distributions will be inadequate to completely explain the data.