Bayesian Analysis of the Doubly Truncated Zubair Weibull Distribution: Parameter Estimation, Reliability, Hazard Rate and Prediction

Abstract

This paper discusses the Bayesian estimation for the unknown parameters, reliability and hazard rate functions of the doubly truncated Zubair Weibull distribution. Informative priors (gamma distribution) for the parameters are used to obtain the posterior distributions. Under the squared-error and linear–exponential loss functions, the Bayes estimators are derived. Credible intervals for the parameters, reliability and hazard rate functions are obtained. Bayesian prediction (point and interval) for the future observation is considered under the two-sample prediction scheme. A simulation study is performed using the Markov Chain Monte Carlo algorithm of simulation for different sample sizes to assess the performance of the estimators. Two real datasets are applied to show the flexibility and applicability of the distribution.

1. Introduction

Truncation in probability distributions may occur in many studies, such as reliability and life testing. The truncated distributions have an important role in several fields, such as agriculture, medicine, physics, engineering etc. Since truncated distribution occurs when there is no ability to predict or record the events above or below a set threshold or inside or outside a certain range, such as the study of plant growth, which cannot be studied before the growth of the plant over the soil.

When the restriction occurs on both sides of the range it is called doubly truncated. If occurrences are limited to values that lie above a given threshold, the upper (right) truncated distribution arises; if occurrences are limited to values that lie below a given threshold, the lower (left) truncated distribution is obtained. Many researchers have discussed truncated distributions, such as [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15].

Recently, many statisticians have been interested in expanding new families in order to increase their flexibility for models and to apply them in a variety of fields; for example, Ref. [16] introduced the properties of a doubly truncated Fréchet distribution, Ref. [17] discussed a new method to develop families of distributions achieving the goals of simplicity and efficiency, Ref. [18] proposed the five-parameter doubly truncated exponentiated inverse Weibull distribution with known truncation points, Ref. [19] discussed truncated log-logistic family of distributions, Ref. [20] presented a truncated Rayleigh Pareto distribution, Ref. [21] proposed the properties of a doubly truncated generalized gamma distribution and a doubly truncated generalized inverse Weibull distribution, Ref. [22] presented a truncated distribution as a sub-model with three parameters called the truncated exponential Topp Leone exponential distribution, Ref. [23] introduced the theory and applications of the truncated Cauchy power odd Fréchet-G family of distributions, Ref. [24] proposed three parameters of truncated distribution called the double-truncated Weibull Pareto distribution and Ref. [25] discussed truncated versions (lower, upper and double) of the X gamma distribution.

Ref. [26] introduced the complementary exponentiated-G Poisson family of distributions with the following probability density function (pdf)

$$f\left(x;\alpha ,\lambda ,\underset{\_}{\xi }\right)={\displaystyle \frac{{\lambda \alpha \mathcal{g}\left(x;\underset{\_}{\xi }\right)G^{\lambda -1}\left(x;\underset{\_}{\xi }\right)e}^{\alpha G^{\lambda }\left(x;\underset{\_}{\xi }\right)}}{e^{\alpha }-1}},\hspace{1em}\hspace{1em}x\mathsf{ϵ}\mathbb{R},\alpha ,\hspace{1em}\hspace{1em}\lambda ,\underset{\_}{\xi }>0,$$

(1)

Ref. [27] proposed the Zubair-G family (Z-G) of distributions as a special case of the complementary exponentiated-G Poisson family of distributions when $\lambda =2$. Hence, the pdf, cumulative distribution function (cdf) and reliability function (rf) of the Z-G family of distributions are given, respectively, as follows:

$$f\left(x;\alpha ,\underset{\_}{\xi }\right)={\displaystyle \frac{2\alpha g(x;\underset{\_}{\xi })G(x;\underset{\_}{\xi })e^{\alpha G^2(x;\underset{\_}{\xi })}}{e^{\alpha }-1}},\hspace{1em}\hspace{1em}\hspace{1em}\alpha ,\underset{\_}{\xi }>0,\hspace{1em}\hspace{1em}\hspace{1em}x\mathsf{ϵ}\mathbb{R},$$

(2)

$$F\left(x;\alpha ,\underset{\_}{\xi }\right)={\displaystyle \frac{e^{\alpha G^2(x;\underset{\_}{\xi })}-1}{e^{\alpha }-1}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha ,\underset{\_}{\xi }>0 ,\hspace{1em}\hspace{1em}\hspace{1em}x \mathsf{ϵ}\mathbb{R},$$

(3)

and

$$R\left(x;\alpha ,\underset{\_}{\xi }\right)={\displaystyle \frac{e^{\alpha }-e^{\alpha G^2(x;\underset{\_}{\xi })}}{e^{\alpha }-1}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha ,\underset{\_}{\xi }>0,\hspace{1em}\hspace{1em}\hspace{1em}x\mathsf{ϵ}\mathbb{R},$$

(4)

where $G\left(x;\underset{\_}{\xi }\right)$ and $g\left(x;\underset{\_}{\xi }\right)$ are the cdf and the pdf of the baseline model.

Also, Ref. [27] introduced the Zubair–Weibull (Z-W) distribution as a sub-model from the proposed family; the authors derived some properties of the distribution and estimated the parameters by using maximum product spacing, ordinary least squares and maximum likelihood (ML) methods. Ref. [28] presented the Zubair Lomax distribution and some properties of it were studied. The author derived ML estimators of the parameters under simple random sample and ranked set sampling. Ref. [29] proposed the Zubair-inverse Lomax distribution with applications; some statistical properties of the distribution were obtained and the ML method was used to estimate the parameters. Ref. [30] introduced the Dagum Zubair-Dagum distribution and some properties were derived. The authors used the ML method to estimate the parameters of the distribution.

The doubly truncated Zubair Weibull (DTZ-W) distribution is very flexible since it has a variety of shapes for the pdf and hazard rate function (hrf). The shapes of the pdf are unimodal, increasing and decreasing; also, the hrf has different shapes, such as increasing, bathtub and unimodal.

Ref. [31] highlights the efficacy of hierarchical Bayesian modeling for uncertainty quantification and reliability updating, demonstrating its power in situations where data truncation complicates direct inference. Similarly, Ref. [32] demonstrates robust inference techniques for dependent tail-weighted degradation data through multivariate Student-t processes, offering insights into handling non-standard data behaviors that can be exacerbated by truncation. Furthermore, advancements in robust prediction, as seen in [33], through adaptive sampling and ensemble techniques, underscore the necessity of sophisticated approaches to ensure reliable estimations under the inherent uncertainties of complex data. These recent contributions collectively validate our decision to employ a Bayesian framework for the DTZ-W distribution, as it provides a naturally flexible and robust mechanism for accurate parameter estimation, reliability, hazard rate analysis and prediction when dealing with the unique characteristics of doubly truncated data.

The accurate estimation of model parameters remains a central challenge in statistical modeling, with various approaches contributing to its advancement. For example, recent work in [34] has investigated the large sample properties of ML estimators, specifically those derived from moving extremes ranked set sampling, which offers valuable insights into estimator performance under different data collection strategies.

This study significantly advances the existing literature by offering a Bayesian analysis of the DTZ-W distribution. While previous research has explored various Zubair-type and truncated distributions, our work introduces a novel application of the DTZ-W distribution that uniquely addresses the complexities inherent in doubly truncated data. This approach not only provides a more accurate representation of the underlying data generation process but also offers a robust framework for parameter estimation, rf, hrf and prediction, particularly for complex data where traditional distribution assumptions are inadequate. In recent years, Bayesian estimation has gained increasing attention due to its ability to combine prior knowledge and effectively manage uncertainty in statistical inference. Unlike non-Bayesian methods that depend only on observed data, the Bayesian approach allows the integration of prior beliefs about the parameters, resulting in posterior distributions that reflect both the data evidence and prior information. This is particularly advantageous in the context of complex lifetime distributions, such as the truncated Zubair Weibull distribution, where standard estimation methods may struggle to provide accurate results. Moreover, Bayesian methods offer flexibility in accommodating various loss functions such as the squared-error (SE) and linear–exponential (LINEX) loss functions, which enhance estimation robustness, especially when the cost of overestimation and underestimation is asymmetric. Therefore, this study adopts the Bayesian framework to derive both point and interval estimations that are better suited for modeling rf and hrf under uncertainty. Specifically, for the DTZ-W distribution, the inherent complexities of its truncated nature often lead to intractable integrals for classical estimation, which are effectively overcome by the computational power of Markov Chain Monte Carlo (MCMC) algorithms employed within the Bayesian framework. This further differentiates the present work from the frequentist approaches typically encountered in the analysis of similar distributions.

When the random variable (rv) $X$ has the DTZ-W $\left(\underset{\_}{\psi }\right)$ distribution, then the pdf and cdf are given, respectively, as follows

$$f_{DTZW}\left(x;\underset{\_}{\psi }\right)={\displaystyle \frac{2\alpha \theta \gamma x^{\theta -1}{e}^{-\gamma x^{\theta }}(1-e^{-\gamma x^{\theta }})e^{\alpha {(1-e^{-\gamma x^{\theta }})}^2}}{e^{\alpha {(1-e^{-\gamma d^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}}},\hspace{1em}\hspace{1em}\hspace{1em}0<c<x<d<\infty ,\hspace{1em}\underset{\_}{\psi }>\underset{\_}{0},$$

(5)

and

$$F_{DTZW}\left(x;\underset{\_}{\psi }\right)={\displaystyle \frac{e^{\alpha {(1-e^{-\gamma x^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}}{e^{\alpha {(1-e^{-\gamma d^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}}},\hspace{1em}\hspace{1em}\hspace{1em}0<c<x<d<\infty ,\hspace{1em}\hspace{1em}\hspace{1em}\underset{\_}{\psi }>\underset{\_}{0},$$

(6)

where $\underset{\_}{\psi }=\left(\alpha ,\underset{\_}{\xi },c,d\right) \mathrm{a}\mathrm{n}\mathrm{d} \underset{\_}{\xi }=\left(\theta ,\gamma \right).$

The rf, hrf, reversed hrf (rhrf) and cumulative hrf (chrf) of the DTZ-W distribution are given by

$$R_{DTZW}\left(x;\underset{\_}{\psi }\right)={\displaystyle \frac{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma x^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}},\hspace{1em}\hspace{1em}0<c<x<d<\infty ,\hspace{1em}\underset{\_}{\psi }>\underset{\_}{0},$$

(7)

$$h_{DTZW}\left(x;\underset{\_}{\psi }\right)={\displaystyle \frac{2\alpha \theta \gamma x^{\theta -1}{e}^{-\gamma x^{\theta }}(1-e^{-\gamma x^{\theta }})e^{\alpha {(1-e^{-\gamma x^{\theta }})}^2}}{e^{\alpha {(1-e^{-\gamma d^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma x^{\theta }})}^2}}},\hspace{1em}\hspace{1em}\hspace{1em}0<c<x<d<\infty ,\hspace{1em}\underset{\_}{\psi }>\underset{\_}{0},$$

(8)

$$r_{DTZW}\left(x;\underset{\_}{\psi }\right)={\displaystyle \frac{2\alpha \theta \gamma x^{\theta -1}{e}^{-\gamma x^{\theta }}(1-e^{-\gamma x^{\theta }})e^{\alpha {(1-e^{-\gamma x^{\theta }})}^2}}{e^{\alpha {(1-e^{-\gamma x^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}}},\hspace{1em}0<c<x<d<\infty ,\hspace{1em}\underset{\_}{\psi }>\underset{\_}{0},$$

(9)

and

$$H_{DTZW}\left(x;\underset{\_}{\psi }\right)=-ln⌈{\displaystyle \frac{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma x^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}⌉,\hspace{1em}\hspace{1em}\hspace{1em}0<c<x<d<\infty ,\hspace{1em}\underset{\_}{\psi }>\underset{\_}{0}.$$

(10)

This paper is organized as follows. In Section 2, Bayesian estimation (point and interval) for the unknown parameters, rf and hrf of the DTZ-W $\left(\underset{\_}{\psi }\right)$ distribution is discussed under two types of loss function: the SE loss function as a symmetric loss function and LINEX loss function as an asymmetric loss function. In Section 3, Bayesian prediction (point and interval) is discussed for a future observation of the DTZ-W $\left(\underset{\_}{\psi }\right)$ distribution under two-sample prediction. A simulation study and two applications are presented to illustrate the theoretical results in Section 4.

2. Bayesian Estimation

In recent years, the Bayesian estimation approach has been widely used for analyzing failure time data, which is introduced as an alternative to the non-Bayesian methods. In the Bayesian inference, it is considered the unknown parameters are rvs with a joint prior distribution. The non-informative prior can be used in Bayesian analysis when prior knowledge about parameters is not available. But the informative prior can be used when the prior knowledge is available.

Suppose that $x_{\left(1\right)}{, x}_{\left(2\right) }, \cdots , x_{\left(n\right)}$ is a random sample from a population with the DTZ-W ($\underset{\_}{\psi }$) at the truncation points (c, d) given by (1). Then, the likelihood function (LF) is defined as

$$L\left(\underset{\_}{\psi };\underset{\_}{x}\right)={\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}.$$

(11)

Assuming that the parameters $\underset{\_}{\psi }=\left(\alpha ,\theta ,\gamma ,c,d\right)=\left({\psi }_1,{\psi }_2,{\psi }_3,{\psi }_4,{\psi }_5\right)$ of the DTZ-W ($\underset{\_}{\psi }$) distribution are unknown. The joint prior of the parameters $\alpha$ and $\theta$ is independent of the parameter $\mathsf{\gamma }$ and the joint prior of the parameters $\mathrm{c}$ and $\mathrm{d}$. Suppose that the parameters $\alpha$ and $\theta$ are dependent, then the joint prior distribution for $\alpha$ and $\theta$ is similar to the joint prior which was used by [35]. Hence, the joint prior distribution of $\alpha$ and $\theta$ is given by

$$\pi \left(\alpha ,\theta \right)=g_1\left(\alpha |\theta \right)g_2\left(\theta \right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha ,\theta >0,$$

(12)

where

$$g_1\left(\alpha |\theta \right)={\displaystyle \frac{{\theta }^{a_1}}{\Gamma \left(a_1\right)}}{\alpha }^{a_1-1}{e}^{-\theta \alpha },\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha ,\theta ,a_1>0,$$

(13)

and

$$g_2\left(\theta \right)={\displaystyle \frac{{b_1}^{a_2}}{\Gamma \left(a_2\right)}}{\theta }^{a_2-1}{e}^{-b_1\theta },\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\theta ,a_2,b_1>0.$$

(14)

Substituting (13) and (14) into (12), then

$$\pi \left(\alpha ,\theta \right)\propto {\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\alpha ,\theta ,a_1,a_2,b_1>0.$$

(15)

The prior distribution of $\mathsf{\gamma }$ is defined as a gamma distribution as follows

$$\pi \left(\mathsf{\gamma }\right)={\displaystyle \frac{{b_2}^{a_3}}{\Gamma \left(a_3\right)}}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathsf{\gamma },a_3,b_2>0.$$

(16)

$$\pi \left(\mathsf{\gamma }\right)\propto {\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathsf{\gamma },a_3,b_2>0.$$

(17)

Similarly, by assuming that $\mathrm{c}$ and $\mathrm{d}$ are dependent, the joint prior distribution of $\mathrm{c}$ and $\mathrm{d}$ is given by

$$\pi \left(c,d\right)=g_1\left(c|d\right)g_2\left(d\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}c,d>0,$$

(18)

where

$$g_1\left(c|d\right)={\displaystyle \frac{d^{a_4}}{\Gamma \left(a_4\right)}}{c}^{a_4-1}{e}^{-dc},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}c,d,a_4>0,$$

(19)

$$g_2\left(d\right)={\displaystyle \frac{{b_3}^{a_5}}{\Gamma \left(a_5\right)}}{d}^{a_5-1}{e}^{-b_{3}d},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}d,a_5,b_3>0.$$

(20)

Substituting (19) and (20) into (18), then

$$\pi \left(c,d\right)\propto d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}c,d,a_4,a_5,b_3>0.$$

(21)

The joint prior distribution of $\underset{\_}{\psi }$ can be obtained from (15), (17) and (21) as follows

$$\begin{array}{c}\pi \left(\underset{\_}{\psi }\right)\propto {\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)},\\ \hspace{1em}\hspace{1em}\hspace{1em}\underset{\_}{\psi }>\underset{\_}{0}, a_1,a_2,a_3,a_4,a_5,b_1,b_2,b_3>\underset{\_}{0},\end{array}$$

(22)

where $a_1,a_2,a_3,a_4,a_5,b_1,b_2$ and $b_3$ are the hyperparameters of the joint prior distribution that are selected to indicate the prior information of the unknown parameters.

By combining the LF in (11) and the joint prior distribution in (22), then the joint posterior distribution of $\underset{\_}{\psi }$, denoted by $\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)$, can be obtained as follows:

$$\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)\propto L\left(\underset{\_}{\psi };\underset{\_}{x}\right)\pi \left(\underset{\_}{\psi }\right).$$

(23)

$$\begin{array}{c}\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)=A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\\ \times {\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}},\end{array}$$

(24)

where A is the normalizing constant defined by

$$A^{-1}={\int }_{\underset{\_}{\psi }}L\left(\underset{\_}{\psi };\underset{\_}{x}\right)\pi \left(\underset{\_}{\psi }\right)d\underset{\_}{\psi },$$

(25)

$$\begin{array}{c}{A}^{-1}={\&\int }_{\underset{\_}{\psi }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\\ \times {\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\underset{\_}{\psi },\end{array}$$

(26)

where

${\int }_{\underset{\_}{\psi }}={\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\int }_0^d{\int }_c^{\mathrm{\infty }}$ and $d\underset{\_}{\psi }=d\alpha d\theta d\gamma dc dd.$

The marginal posterior distribution of the parameter ${\psi }_i$ is given by

$$\pi \left({\psi }_i|\underset{\_}{x}\right)={\int }_{{\underset{\_}{\psi }}_j}\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)d{\underset{\_}{\psi }}_j,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i,j=\mathrm{1,2},\mathrm{3,4},5\hspace{1em}and i\ne j.$$

(27)

2.1. Point Estimation

In this subsection, under two different loss functions, the SE and LINEX loss functions, the Bayes estimators of the parameters, rf and hrf of the DTZ-W distribution are obtained.

2.1.1. Bayesian Estimation Under Squared-Error Loss Function

Considering the SE loss function, the Bayes estimators of the parameters, rf and hrf can be derived as follows:

a. 

Bayesian estimation for the parameters

Under the SE loss function, the Bayes estimators of the parameters are the means of their marginal posterior distributions, which can be obtained as follows:

$$\begin{array}{cc}\hfill {\tilde{\psi }}_{i\left(SE\right)}& =E\left({\psi }_i|\underset{\_}{x}\right)=\underset{\underset{\_}{\psi }}{\int }{\psi }_i\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)d\underset{\_}{\psi }\hfill \\ & ={\int }_{\underset{\_}{\psi }}{\psi }_{i}A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times {\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}} d\underset{\_}{\psi },\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{i}=1, 2, 3, 4, 5.\hfill \end{array}$$

(28)

b. 

Bayesian estimation for the reliability and hazard rate functions

Under SE loss function, the Bayes estimators of the rf and hrf can be obtained using (7), (8) and (24), as follows:

$$\begin{array}{cc}\hfill {\tilde{R}}_{SE}\left(x_0\right)& =E\left[R\left(x_0\right)|\underset{\_}{x}\right]=\underset{\underset{\_}{\psi }}{\int }R\left(x_0\right)\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)d\underset{\_}{\psi }\hfill \\ & ={\int }_{\underset{\_}{\psi }}A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times {\displaystyle \frac{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma x_0^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}\hfill \\ & \times {\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\underset{\_}{\psi },\hfill \end{array}$$

(29)

and

$$\begin{array}{cc}\hfill {\tilde{h}}_{SE}\left(x_0\right)& =E\left[h\left(x_0\right)|\underset{\_}{x}\right]=\underset{\underset{\_}{\psi }}{\int }h\left(x_0\right)\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)d\underset{\_}{\psi }\hfill \\ & ={\int }_{\underset{\_}{\psi }}A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times {\displaystyle \frac{2\alpha \theta \gamma x_0^{\theta -1}{e}^{-\gamma x_0^{\theta }}\left(1-e^{-\gamma x_0^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma x_0^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma x_0^{\theta }}\right)}^2}}}\hfill \\ & \times {\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\underset{\_}{\psi }.\hfill \end{array}$$

(30)

2.1.2. Bayesian Estimation Under Linear–Exponential Loss Function

The Bayes estimators of the parameters, rf and hrf considering the LINEX loss function are given, respectively, as follows:

a. 

Bayesian estimation for the parameters

Under the LINEX loss function, the Bayes estimators of the parameter ${\psi }_i$ are obtained as follows:

$$\begin{array}{cc}\hfill {\tilde{\psi }}_{i\left(LINEX\right)}& ={\displaystyle \frac{-1}{\nu }}\ln \left[E\left(e^{-\nu {\psi }_i}\underset{\_}{x}\right)\right]\hfill \\ & ={\displaystyle \frac{-1}{\nu }}\ln \left[{\int }_{\underset{\_}{\psi }}{e}^{-\nu {\psi }_i}\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)d\underset{\_}{\psi }\right]\hfill \\ & ={\displaystyle \frac{-1}{\nu }}\ln \left[{{\int }_{\underset{\_}{\psi }}{e}^{-\nu {\psi }_i}A\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\right.\hfill \\ & \times \left.{\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\underset{\_}{\psi }\right],\hfill \end{array}$$

(31)

where $\nu$ is constant and $\nu \ne 0$.

b. 

Bayesian estimation for the reliability and hazard rate functions

The Bayes estimators of rf and hrf under the LINEX loss function can be derived, respectively, as follows:

$$\begin{array}{cc}\hfill {\tilde{R}}_{\left(LINEX\right)}\left(x_0\right)& ={\displaystyle \frac{-1}{\nu }}\ln \left[E\left(e^{-\nu R\left(\left(x_0\right)\right)}|\underset{\_}{x}\right)\right]={\displaystyle \frac{-1}{\nu }}\ln \left[\underset{\underset{\_}{\psi }}{\int }{e}^{-\nu R\left(\left(x_0\right)\right)}\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)d\underset{\_}{\psi }\right]\hfill \\ & ={\displaystyle \frac{-1}{\nu }}\ln \left[{\int }_{\underset{\_}{\psi }}A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}exp\left[-\nu \left({\displaystyle \frac{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma {\left(x_0\right)}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}\right)\right]\right.\hfill \\ & \times e^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \left.{\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\underset{\_}{\psi }\right],\hfill \end{array}$$

(32)

and

$$\begin{array}{cc}\hfill {\tilde{h}}_{\left(LINEX\right)}\left(x_0\right)& ={\displaystyle \frac{-1}{\nu }}\ln \left[E\left(e^{-\nu h\left(\left(x_0\right)\right)}|\underset{\_}{x}\right)\right]={\displaystyle \frac{-1}{\nu }}\ln \left[\underset{\underset{\_}{\psi }}{\int }{e}^{-\nu h\left(\left(x_0\right)\right)}\pi \left(\underset{\_}{\psi } |\underset{\_}{x}\right)d\underset{\_}{\psi }\right]\hfill \\ & ={\displaystyle \frac{-1}{\nu }}\ln \left[{\int }_{\underset{\_}{\psi }}A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}exp\left[-\nu \left({\displaystyle \frac{2\alpha \theta \gamma {\left(x_0\right)}^{\theta -1}{e}^{-\gamma {\left(x_0\right)}^{\theta }}(1-e^{-\gamma {\left(x_0\right)}^{\theta }})e^{\alpha {(1-e^{-\gamma {\left(x_0\right)}^{\theta }})}^2}}{e^{\alpha {(1-e^{-\gamma d^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma {\left(x_0\right)}^{\theta }})}^2}}}\right)\right]\right.\hfill \\ & \times e^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \left.{\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\underset{\_}{\psi }\right].\hfill \end{array}$$

(33)

To obtain the Bayes estimators of the parameters, rf and hrf based on the SE and LINEX loss functions, Equations (28)–(33) can be solved numerically.

2.2. Credible Intervals

In this subsection, the credible intervals (CIs) for the parameters of the DTZ-W ($\underset{\_}{\psi }$) distribution are derived. In general, two-sided $100 (1-\tau )\%$ credible intervals of $\underset{\_}{\psi }$ are given by

$$P\left[L_i\left(\underset{\_}{x}\right)<{\psi }_i<U_i\left(\underset{\_}{x}\right)\right]={\int }_{L_i\left(\underset{\_}{x}\right)}^{U_i\left(\underset{\_}{x}\right)}\pi \left({\psi }_i|\underset{\_}{x}\right)d{\psi }_i=1-\tau ,\hspace{1em}\hspace{1em}i=1, 2, 3, 4, 5.$$

(34)

The lower and upper bounds $\left[L\left(\underset{\_}{x}\right), U\left(\underset{\_}{x}\right)\right]$ can be derived as follows:

$$P\left[{\psi }_i>L_i\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]={\int }_{L_i\left(\underset{\_}{x}\right)}^{\infty }\pi \left( {\psi }_i |\underset{\_}{x}\right)d{\psi }_i=1-{\displaystyle \frac{\tau }{2}},\hspace{1em}\hspace{1em}\hspace{1em}i=1, 2, 3, 4, 5.$$

(35)

and

$$P\left[{\psi }_i>U_i\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]={\int }_{U_i\left(\underset{\_}{x}\right)}^{\infty }\pi \left( {\psi }_i |\underset{\_}{x}\right)d{\psi }_i={\displaystyle \frac{\tau }{2}},\hspace{1em}\hspace{1em}\hspace{1em}i=1, 2, 3, 4, 5.$$

(36)

Hence, a $100(1-\tau )\mathrm{\%}$ CI for $\alpha$ is $\left[L_1\left(\underset{\_}{x}\right),U_1\left(\underset{\_}{x}\right)\right]$ and is given below

$$\begin{array}{cc}\hfill P\left[\alpha >L_1\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]& =A\underset{L_1\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{d}{\int }}\underset{c}{\overset{\infty }{\int }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}\hfill \\ & \times d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \prod _{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\theta d\gamma dc dd d\alpha \hfill \\ & =1-{\displaystyle \frac{\tau }{2}},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill P\left[\alpha >U_1\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]& =A\underset{U_1\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{d}{\int }}\underset{c}{\overset{\infty }{\int }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}\hfill \\ & \times d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \prod _{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\theta d\gamma dc ddd\alpha \hfill \\ & ={\displaystyle \frac{\tau }{2}},\hfill \end{array}$$

(38)

a $100(1-\tau )\mathrm{\%}$ CI for $\theta$ is $\left[L_2\left(\underset{\_}{x}\right),U_2\left(\underset{\_}{x}\right)\right]$, as defined as

$$\begin{array}{cc}\hfill P\left[\theta >L_2\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]& =A\underset{L_2\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{d}{\int }}\underset{c}{\overset{\infty }{\int }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}\hfill \\ & \times d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \prod _{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}} d\gamma d\alpha dc dd d\theta \hfill \\ & =1-{\displaystyle \frac{\tau }{2}},\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill P\left[\theta >U_2\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]& =A\underset{U_2\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{d}{\int }}\underset{c}{\overset{\infty }{\int }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}\hfill \\ & \times d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \prod _{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\gamma d\alpha dc dd d\theta \hfill \\ & ={\displaystyle \frac{\tau }{2}},\hfill \end{array}$$

(40)

a $100(1-\tau )\mathrm{\%}$ CI for $\gamma$ is $\left[L_3\left(\underset{\_}{x}\right),U_3\left(\underset{\_}{x}\right)\right]$ and can be obtained as

$$\begin{array}{cc}P\left[\gamma >L_3\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]& =A\underset{L_3\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{d}{\int }}\underset{c}{\overset{\infty }{\int }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}\hfill \\ & \times d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \prod _{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}} d\alpha d\theta dc dd d\gamma \hfill \\ & =1-{\displaystyle \frac{\tau }{2}},\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill P\left[\gamma >U_3\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]& =A\underset{U_3\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{d}{\int }}\underset{c}{\overset{\infty }{\int }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}\hfill \\ & \times d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \prod _{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\alpha d\theta dc dd d\gamma \hfill \\ & ={\displaystyle \frac{\tau }{2}},\hfill \end{array}$$

(42)

a $100(1-\tau )\mathrm{\%}$ CI for $c$ is $\left[L_4\left(\underset{\_}{x}\right),U_4\left(\underset{\_}{x}\right)\right]$ and can be obtained as

$$\begin{array}{cc}P\left[c>L_4\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]& =A\underset{L_4\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{d}{\int }}\underset{c}{\overset{\infty }{\int }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}\hfill \\ & \times d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \prod _{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}} d\alpha d\theta d\gamma dddc\hfill \\ & =1-{\displaystyle \frac{\tau }{2}},\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill P\left[c>U_4\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]& =A\underset{U_4\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{d}{\int }}\underset{c}{\overset{\infty }{\int }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}\hfill \\ & \times d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \prod _{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\alpha d\theta d\gamma dddc\hfill \\ & ={\displaystyle \frac{\tau }{2}}, \hfill \end{array}$$

(44)

a $100(1-\tau )\mathrm{\%}$ CI for $d$ is $\left[L_5\left(\underset{\_}{x}\right),U_5\left(\underset{\_}{x}\right)\right]$ and can be obtained as

$$\begin{array}{cc}\hfill P\left[d>L_5\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]& =A\underset{L_5\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{d}{\int }}\underset{c}{\overset{\infty }{\int }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }} \hfill \\ & \times d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \prod _{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}} d\alpha d\theta d\gamma dcdd \hfill \\ & =1-{\displaystyle \frac{\tau }{2}}, \hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill P\left[d>U_5\left(\underset{\_}{x}\right)|\underset{\_}{x}\right]& =A\underset{U_6\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{d}{\int }}\underset{c}{\overset{\infty }{\int }}{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }} \hfill \\ & \times d^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times \prod _{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\alpha d\theta d\gamma dcdd\hfill \\ & ={\displaystyle \frac{\tau }{2}}.\hfill \end{array}$$

(46)

In addition, $100(1-\tau )\%$ CIs for $R\left(x\right)$ and $h\left(x\right)$ can be obtained numerically as

$$P\left[L\left(\underset{\_}{x}\right)<R\left(x\right)<U\left(\underset{\_}{x}\right)\right]=\underset{L\left(\underset{\_}{x}\right)}{\overset{U\left(\underset{\_}{x}\right)}{\int }}\pi \left( R |\underset{\_}{x}\right)dR=1-\tau ,$$

(47)

and

$$P\left[L\left(\underset{\_}{x}\right)<h\left(x\right)<U\left(\underset{\_}{x}\right)\right]=\underset{L\left(\underset{\_}{x}\right)}{\overset{U\left(\underset{\_}{x}\right)}{\int }}\pi \left( h |\underset{\_}{x}\right)dh=1-\tau ,$$

(48)

where $\pi \left(R|\underset{\_}{x}\right)$ and $\pi \left(h|\underset{\_}{x}\right)$ denoted to the posterior distribution of rf and hrf.

Equations (37)–(48) can be solved numerically to obtain the two-sided $100 (1-\tau )\%$ CIs for the parameters ($\underset{\_}{\psi }$), rf and hrf.

3. Bayesian Prediction Based on Two-Sample Prediction

In this section, two-sample Bayesian predictions (point and interval) for the future observation $Y_{\left(s\right)}, 1\le s\le m$ are derived under two types of loss function: SE and LINEX loss functions.

Suppose that $x_{\left(1\right)}, x_{\left(2\right)},\dots , x_{\left(n\right)}$ are the first ordered lifetimes in a random sample of size n components represented the informative sample and $y_{\left(1\right)}, y_{\left(2\right)},\dots , y_{\left(m\right) }$ are the future independent of the ordered sample of size m from the same distribution. The aim is to predict the $s^{th}$-order statistic in the future sample based on the informative sample. The conditional pdf of $Y_{\left(s\right) }$ given the vector of parameters $\underset{\_}{\psi }$ can be obtained as follows:

$$f_s\left(y_{\left(s\right)}\right|\underset{\_}{\psi })={\displaystyle \frac{m!}{\left(s-1\right)!\left(m-s\right)!}}f(y_{\left(s\right)}\left|\underset{\_}{\psi }\right){\left[F\left(y_{\left(s\right)}\right|\underset{\_}{\psi })\right]}^{s-1}{\left[1-F\left(y_{\left(s\right)}\right|\underset{\_}{\psi })\right]}^{m-s},{ y}_{\left(s\right)}>0,$$

using the binomial expansion theorem for ${\left[1-F\left(y_{\left(s\right)}\right|\underset{\_}{\psi })\right]}^{m-s}$

$${\left[1-F\left(y_{\left(s\right)}\right|\underset{\_}{\psi })\right]}^{m-s}={\sum }_{j=0}^{m-s}\left({\displaystyle \genfrac{}{}{0pt}{}{m-s}{j}}\right){\left(-1\right)}^j{\left[F\left(y_{\left(s\right)}\right|\underset{\_}{\psi })\right]}^j$$

Then

$$\begin{array}{cc}\hfill f_s\left(y_{\left(s\right)}|\underset{\_}{\psi }\right)& =f\left(y_{\left(s\right)}|\underset{\_}{\psi }\right){\sum }_{j=0}^{m-s}{\displaystyle \frac{m!}{\left(s-1\right)!\left(m-s\right)!}}\left({\displaystyle \genfrac{}{}{0pt}{}{m-s}{j}}\right){\left(-1\right)}^j{\left[F\left(y_{\left(s\right)}|\underset{\_}{\psi }\right)\right]}^{s+j-1}\hfill \\ & =f\left(y_{\left(s\right)}\right|\underset{\_}{\psi }){\sum }_{j=0}^{m-s}{\displaystyle \frac{m!}{\left(s-1\right)!\left(m-s\right)!}}\left({\displaystyle \frac{\left(m-s\right)!}{j!\left(m-s-j\right)!}}\right){\left(-1\right)}^j{\left[F\left(y_{\left(s\right)}\right|\underset{\_}{\psi })\right]}^{s+j-1}\hfill \\ & =f\left(y_{\left(s\right)}\right|\underset{\_}{\psi }){\sum }_{j=0}^{m-s}{\displaystyle \frac{m!}{\left(s-1\right)!j!\left(m-s-j\right)!}}{\left(-1\right)}^j{\left[F\left(y_{\left(s\right)}\right|\underset{\_}{\psi })\right]}^{s+j-1}\hfill \\ & =f\left(y_{\left(s\right)}\right|\underset{\_}{\psi }){\sum }_{j=0}^{m-s}{c}_{m,s,j}{\left[F\left(y_{\left(s\right)}\right|\underset{\_}{\psi })\right]}^{s+j-1},\hfill \end{array}$$

(49)

where

$$c_{m,s,j}={\displaystyle \frac{m!}{\left(s-1\right)!j!\left(m-s-j\right)!}}{\left(-1\right)}^j.$$

By substituting (5) and (6) into (49), the conditional pdf of $s^{th}$ -order statistic is given by

$$\begin{array}{cc}\hfill f_s\left(y_{\left(s\right)}|\underset{\_}{\psi }\right)& ={\displaystyle \frac{2\alpha \theta \gamma {y_{\left(s\right)}}^{\theta -1}{e}^{-\gamma {y_{\left(s\right)}}^{\theta }}\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}\hfill \\ & \times {\sum }_{j=0}^{m-s}{c}_{m,s,j}{\left[{\displaystyle \frac{e^{\alpha {(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}}{e^{\alpha {(1-e^{-\gamma d^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}}}\right]}^{s+j-1}\hfill \\ \hfill f_s\left(y_{\left(s\right)}|\underset{\_}{\psi }\right)& ={\displaystyle \frac{2\alpha \theta \gamma {y_{\left(s\right)}}^{\theta -1}{e}^{-\gamma {y_{\left(s\right)}}^{\theta }}\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)}^2}}{{\left[e^{\alpha {(1-e^{-\gamma d^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}\right]}^{s+j}}}\hfill \\ & \times {\sum }_{j=0}^{m-s}{c}_{m,s,j}{\left[e^{\alpha {(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}\right]}^{s+j-1},{ y}_{\left(s\right)}>0, \underset{\_}{\psi }>\underset{\_}{0}.\hfill \end{array}$$

(50)

The Bayesian predictive density (BPD) function of $Y_{\left(s\right)}$ given $\underset{\_}{x}$ is given as

$$h\left(y_{\left(s\right)}|\underset{\_}{x}\right)={\int }_{\underset{\_}{\psi }}{f}_s\left(y_{\left(s\right)}|\underset{\_}{\psi }\right)\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)d\underset{\_}{\psi }, { y}_{\left(s\right)}>0,$$

(51)

where $\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)$ is the posterior distribution of $\underset{\_}{\psi }$ and $f_s\left(y_{\left(s\right)}|\underset{\_}{\psi }\right)$ is the $s^{th}$-order statistic of $y_{\left(s\right)}$.

The BPD of $Y_{\left(s\right)}$ given $\underset{\_}{x}$ can be obtained by substituting (24) and (50) into (51), as follows:

$$\begin{array}{cc}\hfill h\left(y_{\left(s\right)}|\underset{\_}{x}\right)& =\underset{\underset{\_}{\psi }}{\int }{\displaystyle \frac{2\alpha \theta \gamma {y_{\left(s\right)}}^{\theta -1}{e}^{-\gamma {y_{\left(s\right)}}^{\theta }}\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)}^2}}{{\left[e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}\right]}^{s+j}}} \hfill \\ & \times {\sum }_{j=0}^{m-s}{c}_{m,s,j}{\left[e^{\alpha {(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}\right]}^{s+j-1}\hfill \\ & \times A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times {\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d\underset{\_}{\psi },\hfill \\ & \hfill \hspace{1em}{ y}_{\left(s\right)}>0,\underset{\_}{\psi }>0,\end{array}$$

(52)

where $c_{m,s,j}$ is given by (50).

3.1. Point Prediction

Bayesian point prediction for $Y_{\left(s\right)}$ is considered under two types of loss functions: the SE loss function as a symmetric loss function and the LINEX loss function as an asymmetric loss function.

I. 

Squared-error loss function

The Bayes predictor (BP) for the future observation $Y_{\left(s\right)}$ can be derived under the SE loss function as follows:

$${\tilde{y}}_{\left(s\right)\left(SE\right)}=E\left(y_{\left(s\right)}|\underset{\_}{x}\right)={\int }_{y_{\left(s\right)}}{y}_{\left(s\right)}h\left(y_{\left(s\right)}|\underset{\_}{\mathrm{x}}\right)d{y}_{\left(s\right)}.$$

(53)

Using the BPD of $Y_{\left(s\right)}$ in (52) and substituting it into (53) gives

$$\begin{array}{cc}\hfill {\tilde{y}}_{\left(s\right)\left(SE\right)}& =\underset{y_{\left(s\right)}}{\int }\underset{\underset{\_}{\psi },}{\int }{y}_{\left(s\right)}{\displaystyle \frac{2\alpha \theta \gamma {y_{\left(s\right)}}^{\theta -1}{e}^{-\gamma {y_{\left(s\right)}}^{\theta }}\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)}^2}}{{\left[e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}\right]}^{s+j}}}\hfill \\ & \times {\sum }_{j=0}^{m-s}{c}_{m,s,j}{\left[e^{\alpha {(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}\right]}^{s+j-1}\hfill \\ & \times A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times {\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d{y}_{\left(s\right)}d\underset{\_}{\psi }.\hfill \end{array}$$

(54)

II. 

Linear–exponential loss function

The BP for the future observation $Y_{\left(s\right)}$ can be derived under the LINEX loss function as follows:

$${\tilde{y}}_{\left(s\right)\left(\mathrm{L}\mathrm{I}\mathrm{N}\mathrm{E}\mathrm{X}\right)}={\displaystyle \frac{-1}{\nu }}\ln \left[E\left(e^{-\nu y_{\left(s\right)}} |\underset{\_}{x}\right)\right]={\displaystyle \frac{-1}{\nu }}\ln {\int }_{y_{\left(s\right)}}{e}^{-\nu y_{\left(s\right)}}h\left(y_{\left(s\right)}|\underset{\_}{\mathrm{x}}\right)d{y}_{\left(s\right)}.$$

(55)

Using the BPD of $Y_{\left(s\right)}$ in (52) and substituting it into (55), then

$$\begin{array}{cc}\hfill {\tilde{y}}_{\left(s\right)\left(\mathrm{L}\mathrm{I}\mathrm{N}\mathrm{E}\mathrm{X}\right)}& ={\displaystyle \frac{-1}{\nu }}\ln \underset{y_{\left(s\right)}}{\int }\underset{\underset{\_}{\psi }}{\int }{e}^{-\nu y_{\left(s\right)}}{\displaystyle \frac{2\alpha \theta \gamma {y_{\left(s\right)}}^{\theta -1}{e}^{-\gamma {y_{\left(s\right)}}^{\theta }}\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)}^2}}{{\left[e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}\right]}^{s+j}}}\hfill \\ & \times {\sum }_{j=0}^{m-s}{c}_{m,s,j}{\left[e^{\alpha {(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}\right]}^{s+j-1}\hfill \\ & \times A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times {\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d{y}_{\left(s\right)}d\underset{\_}{\psi }.\hfill \end{array}$$

(56)

3.2. Interval Prediction

The $100(1-\tau )\%$ Bayesian predictive intervals (BPIs) for the future observation $Y_{\left(s\right)}$ can be obtained as follows:

$$P\left(L_{\left(s\right)}\left(\underset{\_}{x}\right)<y_{\left(s\right)}<U_{\left(s\right)}\left(\underset{\_}{x}\right)|\underset{\_}{x}\right)={\int }_{L_{\left(s\right)}\left(\underset{\_}{x}\right)}^{U_{\left(s\right)}\left(\underset{\_}{x}\right)}h\left(y_{\left(s\right)}|\underset{\_}{x}\right)d{y}_{\left(s\right)}=1-\tau .$$

(57)

The lower and upper limits, $\left[L_{\left(s\right)}\left(\underset{\_}{x}\right), U_{\left(s\right)}\left(\underset{\_}{x}\right)\right]$, can be derived by

$$P\left(Y_{\left(s\right)}>L_{\left(s\right)}\left(\underset{\_}{x}\right)|\underset{\_}{x}\right)={\int }_{L_{\left(s\right)}\left(\underset{\_}{x}\right)}^{\infty }h\left(y_{\left(s\right)}|\underset{\_}{x}\right)d{y}_{\left(s\right)}=1-{\displaystyle \frac{\tau }{2}},$$

(58)

and

$$P\left(Y_{\left(s\right)}>U_{\left(s\right)}\left(\underset{\_}{x}\right)|\underset{\_}{x}\right)={\int }_{U_{\left(s\right)}\left(\underset{\_}{x}\right)}^{\infty }h\left(y_{\left(s\right)}|\underset{\_}{x}\right)d{y}_{\left(s\right)}={\displaystyle \frac{\tau }{2}}.$$

(59)

Substituting (52) into (58) and (59), then

$$\begin{array}{cc}\hfill P\left(Y_{\left(s\right)}>L_{\left(s\right)}\left(\underset{\_}{x}\right)|\underset{\_}{x}\right)& =\underset{L_{\left(s\right)}\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}{\displaystyle \frac{2\alpha \theta \gamma {y_{\left(s\right)}}^{\theta -1}{e}^{-\gamma {y_{\left(s\right)}}^{\theta }}\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)}^2}}{{\left[e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}\right]}^{s+j}}}\hfill \\ & \times {\sum }_{j=0}^{m-s}{c}_{m,s,j}{\left[e^{\alpha {(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}\right]}^{s+j-1}\hfill \\ & \times A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times {\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d{y}_{\left(s\right)}=1-{\displaystyle \frac{\tau }{2}},\hfill \end{array}$$

(60)

and

$$\begin{array}{cc}\hfill P\left(Y_{\left(s\right)}>U_{\left(s\right)}\left(\underset{\_}{x}\right)|\underset{\_}{x}\right)& =\underset{U_{\left(s\right)}\left(\underset{\_}{x}\right)}{\overset{\infty }{\int }}{\displaystyle \frac{2\alpha \theta \gamma {y_{\left(s\right)}}^{\theta -1}{e}^{-\gamma {y_{\left(s\right)}}^{\theta }}\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }}\right)}^2}}{{\left[e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}\right]}^{s+j}}}\hfill \\ & \times {\sum }_{j=0}^{m-s}{c}_{m,s,j}{\left[e^{\alpha {(1-e^{-\gamma {y_{\left(s\right)}}^{\theta }})}^2}-e^{\alpha {(1-e^{-\gamma c^{\theta }})}^2}\right]}^{s+j-1}\hfill \\ & \times A{\theta }^{a_1+a_2-1}{\alpha }^{a_1-1}{e}^{-\theta \left(\alpha +b_1\right)}{\mathsf{\gamma }}^{a_3-1}{e}^{-b_2\mathsf{\gamma }}{d}^{a_4+a_5-1}{c}^{a_4-1}{e}^{-d\left(c+b_3\right)}\hfill \\ & \times {\prod }_{i=1}^n{\displaystyle \frac{2\alpha \theta \gamma {x_{\left(i\right)}}^{\theta -1}{e}^{-\gamma {x_{\left(i\right)}}^{\theta }}\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)e^{\alpha {\left(1-e^{-\gamma {x_{\left(i\right)}}^{\theta }}\right)}^2}}{e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}}}d{y}_{\left(s\right)}={\displaystyle \frac{\tau }{2}}.\hfill \end{array}$$

(61)

Equations (54)–(61) can be solved numerically.

4. Numerical Study

The aim of this section is to investigate the precision of the theoretical results of Bayesian estimation and prediction based on simulated and real datasets.

4.1. Simulation Study

In this subsection, a simulation study is conducted to demonstrate the performance of the presented Bayes estimators and Bayes predictor based on generated data from the DTZ-W ($\underset{\_}{\psi }$) distribution. Bayes averages, relative absolute biases (RABs), relative errors (REs) and 95% CIs for the parameters, rf and hrf are calculated. Also, two-sample Bayes predictors (point and interval) for a future observation from the DTZ-W ($\underset{\_}{\psi }$) distribution based on complete sampling are computed.

The Adaptive Metropolis (AM) algorithm, a specialized MCMC method, extends the traditional Metropolis–Hastings algorithm. Developed in [36], it enhances sampling efficiency by dynamically adjusting its proposal distribution based on the chain’s past performance. All simulation studies, which are crucial for visualizing and clarifying our findings, were conducted using the R package version 4.5.0

The steps of the AM algorithm are outlined below:

Step 1. Choose an initial $\left(l\times 1\right)$ vector of values for ${\underset{\_}{\psi }}^{\left(0\right)}$.

Step 2. At each iteration $\xi ,$ where $\xi =1, 2, \cdots , h$, a proposed value ${\underset{\_}{\psi }}^{*}$ is sampled from the candidate distribution $j_{\xi }\left({\underset{\_}{\psi }}^{*}|{\underset{\_}{\psi }}^{\left(0\right)}, {\underset{\_}{\psi }}^{\left(1\right)}, \cdots , {\underset{\_}{\psi }}^{\left(\xi -1\right)}\right)$. The algorithm utilizes a Gaussian candidate distribution with the current point as its mean ${\underset{\_}{\psi }}^{\left(\xi -1\right)}$, with a covariance matrix $C_{\xi }$ that depends on the sequence of previous points $C_{\xi }\left({\underset{\_}{\psi }}^{\left(0\right)}, {\underset{\_}{\psi }}^{\left(1\right)}, \cdots , {\underset{\_}{\psi }}^{\left(\xi -1\right)}\right)$.

Step 3. Evaluate the acceptance rate:

$$\mathrm{A}\mathrm{R}={\displaystyle \frac{\pi \left({\underset{\_}{\psi }}^{\mathrm{*}}|\underset{\_}{x}\right)}{\pi \left({\underset{\_}{\psi }}^{\left(\xi -1\right)}|\underset{\_}{x}\right)}},$$

the posterior distribution $\pi \left(\underset{\_}{\psi }|\underset{\_}{x}\right)$ is considered without including the normalization constant.

Step 4. Retain ${\underset{\_}{\psi }}^{\mathrm{*}}$ as ${\underset{\_}{\psi }}^{\left(\xi \right)}$ with a probability $\min \left(AR, 1\right)$. If ${\underset{\_}{\psi }}^{\mathrm{*}}$ is rejected, then set ${\underset{\_}{\psi }}^{\left(\xi \right)}={\underset{\_}{\psi }}^{\left(\xi -1\right)}$. This acceptance decision can be implemented through simulating a random variable $U$ from a uniform distribution. If the value $u$ is less than or equal to $AR$, then ${\underset{\_}{\psi }}^{\left(\xi \right)}$ at iteration $\xi$ is updated to ${\underset{\_}{\psi }}^{*}$; otherwise, it retains the value from the previous iteration, ${\underset{\_}{\psi }}^{\left(\xi \right)}={\underset{\_}{\psi }}^{\left(\xi -1\right)}$.

Step 5. Execute Steps 2–4 repeatedly for $h$ iterations, where $h$ should be substantial enough to ensure the stability of the results.

Step 6. A warm-up phase is applied to mitigate the influence of initial values, during which the first $M$ simulated $\underset{\_}{\psi }$ values are rejected. Using the AM algorithm, the Bayes estimates of ${\psi }_j$, $j=1,2,3,4,5$, can be derived in terms of both the SE and LINEX loss functions as follows:

Where $\left({\psi }_{1l},{\psi }_{2l},{\psi }_{3l},{\psi }_{4l},{\psi }_{5l}\right)=\left({\alpha }_l,{\mathsf{\gamma }}_l,{\mathsf{\theta }}_l,c_l,d_l\right),l=1,2,\cdots ,h$ are drawn from the posterior distribution, with $M$ denoting the warm-up phase

The steps of the simulation procedure are

• Generate random samples from the DTZ-W ($\underset{\_}{\psi }$) distribution using the following transformation

  $$x_u={\left\{{\displaystyle \frac{-1}{\gamma }}\ln \left[1-{\left({\displaystyle \frac{\ln \left\{u\left[e^{\alpha {\left(1-e^{-\gamma d^{\theta }}\right)}^2}-e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}\right]+e^{\alpha {\left(1-e^{-\gamma c^{\theta }}\right)}^2}\right\}}{\alpha }}\right)}^{{\displaystyle \frac{1}{2}}}\right]\right\}}^{{\displaystyle \frac{1}{\theta }}},0<u<1,$$

  where $u$ are random samples from the uniform distribution.
• Two datasets are generated from the DTZ-W ($\underset{\_}{\psi }$) distribution using two different combinations of population parameter values
  Ι:$(\mathsf{\alpha }=2,\mathsf{\gamma }=0.5,\mathsf{\theta }=0.3,c=0.2\mathrm{a}\mathrm{n}\mathrm{d}d=2.5),$
  and
  Π: $\left(\mathsf{\alpha }=3,\mathsf{\gamma }=0.3,\mathsf{\theta }=0.2,c=0.1\mathrm{a}\mathrm{n}\mathrm{d}d=4\right),$
  where the samples of sizes are ($n=$30, 60 and 100) using number of replications (NR) = 10,000 for each sample size.
• Computing the Bayes averages, RABs and REs of the Bayes estimates of the parameters, rf and hrf as follows:

  $$\mathrm{A}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}={\displaystyle \frac{{\sum }_{j=1}^{NR}estimator}{NR}},$$

  $$\mathrm{R}\mathrm{A}\mathrm{B}\mathrm{s}\left(\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\right)={\displaystyle \frac{\left|estimatedvalue-true value\right|}{true value}},$$

  $$ER={\displaystyle \frac{{\sum }_{j=1}^{NR}{\left(estimated value-true value\right)}^2}{NR}},$$

  and

  $$\mathrm{R}\mathrm{E}={\displaystyle \frac{\sqrt{ER (estimated value)}}{true value}}.$$
• The Bayes predictors (point and interval) for a future observation from the DTZ-W ($\underset{\_}{\psi }$) distribution are computed for the two-sample case.

Table 1. Bayes averages, relative absolute biases, relative errors and 95% credible intervals of the parameters using a non-informative prior under the SE and LINEX loss functions for different samples of size n, $(NR=\mathrm{10,000},\mathsf{\alpha }=2,\mathsf{\gamma }=0.5,\mathsf{\theta }=0.3,\mathrm{c}=0.2 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}=2.5)$.

n	Parameters	SE						$\mathbf{LINEX} \left(\mathit{\nu }=0.1\right)$
		Average	RABs	REs	UL	LL	Length	Average	RABs	REs	UL	LL	Length
30	$\mathit{\alpha }$	2.0331	0.0165	0.1096	2.0659	1.9983	0.0676	2.0069	0.0035	0.0048	2.0118	2.0023	0.0095
	$\mathit{\gamma }$	0.5286	0.0572	0.3278	0.5737	0.4938	0.0799	0.4944	0.0112	0.0126	0.5010	0.4883	0.0127
	$\mathit{\theta }$	0.3200	0.0666	0.2666	0.3421	0.2992	0.0429	0.2980	0.0067	0.0016	0.3014	0.2938	0.0076
	$\mathit{c}$	0.1829	0.0856	0.2932	0.2028	0.1674	0.0354	0.2053	0.0265	0.0282	0.2095	0.2009	0.0086
	$\mathit{d}$	2.5123	0.0049	0.0120	2.5388	2.4827	0.0561	2.5038	0.0015	0.0012	2.5083	2.4988	0.0095
60	$\mathit{\alpha }$	2.0281	0.0140	0.0788	2.0470	2.0002	0.0468	1.9987	0.0006	0.0002	2.0022	1.9954	0.0068
	$\mathit{\gamma }$	0.5211	0.0423	0.1792	0.5504	0.4913	0.0591	0.5022	0.0045	0.0020	0.5037	0.4999	0.0038
	$\mathit{\theta }$	0.3094	0.0315	0.0595	0.3238	0.2947	0.0291	0.3009	0.0031	0.0003	0.3027	0.2993	0.0034
	$\mathit{c}$	0.2057	0.0284	0.0324	0.2179	0.1904	0.0275	0.1989	0.0055	0.0012	0.2013	0.1969	0.0044
	$\mathit{d}$	2.5022	0.0009	0.0004	2.5179	2.4882	0.0297	2.4980	0.0008	0.0003	2.5003	2.4961	0.0042
100	$\mathit{\alpha }$	2.0208	0.0104	0.0433	2.0305	2.0032	0.0273	1.9991	0.0004	0.0001	2.0004	1.9970	0.0034
	$\mathit{\gamma }$	0.5039	0.0078	0.0062	0.5130	0.4941	0.0189	0.4988	0.0023	0.0005	0.5004	0.4968	0.0036
	$\mathit{\theta }$	0.3070	0.0233	0.0325	0.3160	0.2979	0.0181	0.2992	0.0026	0.0002	0.3007	0.2981	0.0026
	$\mathit{c}$	0.2029	0.0146	0.0085	0.2097	0.1960	0.0137	0.1993	0.0036	0.0005	0.2003	0.1985	0.0018
	$\mathit{d}$	2.4985	0.0006	0.0002	2.5046	2.4922	0.0124	2.4990	0.0004	0.0001	2.5001	2.4980	0.0021

and

Table 2. Bayes averages, relative absolute biases, relative errors and 95% credible intervals of the parameters, using non-informative prior under the SE and LINEX loss functions for different samples of size n, $(NR=\mathrm{10,000},\mathsf{\alpha }=3,\mathsf{\gamma }=0.3,\mathsf{\theta }=0.2,\mathrm{c}=0.1 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}=4)$.

n	Parameters	SE						$\mathbf{LINEX} \left(\mathit{\nu }=0.1\right)$
		Average	RABs	REs	UL	LL	Length	Average	RABs	REs	UL	LL	Length
30	$\mathit{\alpha }$	2.9591	0.0136	0.1116	3.0116	2.9079	0.1037	2.9958	0.0014	0.0011	3.0002	2.9913	0.0089
	$\mathit{\gamma }$	0.3241	0.0803	0.3872	0.3527	0.2997	0.0530	0.2937	0.0208	0.0259	0.2978	0.2890	0.0088
	$\mathit{\theta }$	0.2116	0.0581	0.1353	0.2571	0.1789	0.0782	0.1940	0.0301	0.0241	0.2001	0.1883	0.0118
	$\mathit{c}$	0.1211	0.2112	0.8924	0.1588	0.0848	0.0740	0.0974	0.0254	0.0129	0.1026	0.0930	0.0096
	$\mathit{d}$	3.9574	0.0106	0.0906	4.0066	3.9288	0.0778	3.9954	0.0011	0.0010	4.0000	3.9930	0.0070
60	$\mathit{\alpha }$	3.0349	0.0116	0.0811	3.0630	2.9898	0.0732	2.9983	0.0005	0.0002	3.0023	2.9950	0.0073
	$\mathit{\gamma }$	0.3024	0.0081	0.0039	0.3218	0.2856	0.0362	0.2991	0.0029	0.0005	0.3017	0.2957	0.0060
	$\mathit{\theta }$	0.2095	0.0477	0.0911	0.2343	0.1836	0.0506	0.1980	0.0097	0.0025	0.2015	0.1941	0.0074
	$\mathit{c}$	0.1181	0.1816	0.6594	0.1520	0.0908	0.0612	0.1025	0.0253	0.0128	0.1053	0.0997	0.0056
	$\mathit{d}$	4.0081	0.0020	0.0033	4.0319	3.9873	0.0446	3.9975	0.0006	0.0003	4.0001	3.9942	0.0059
100	$\mathit{\alpha }$	3.0154	0.0051	0.0158	3.0366	3.0007	0.0359	2.9986	0.0004	0.0001	3.0003	2.9974	0.0029
	$\mathit{\gamma }$	0.3012	0.0042	0.0010	0.3121	0.2919	0.0202	0.3007	0.0025	0.0004	0.3019	0.2999	0.0020
	$\mathit{\theta }$	0.2070	0.0353	0.0498	0.2176	0.1972	0.0204	0.1992	0.0040	0.0004	0.2001	0.1985	0.0016
	$\mathit{c}$	0.0880	0.1198	0.2869	0.1032	0.0714	0.0318	0.1002	0.0023	0.0001	0.1013	0.0995	0.0018
	$\mathit{d}$	3.9928	0.0018	0.0026	4.0057	3.9777	0.0280	4.0017	0.0004	0.0002	4.0031	4.0002	0.0029

show the Bayes averages, RABs, REs and 95% CIs of the unknown parameters $\mathsf{\alpha },\mathsf{\gamma },\mathsf{\theta },c \mathrm{a}\mathrm{n}\mathrm{d} d$ under the SE and LINEX loss functions for two different combinations of the parameters.

Table 3. Bayes averages, relative absolute biases, relative errors and 95% confidence intervals of the reliability and hazard rate functions at ($x_0=0.4$) from the DTZ-W distribution using the SE and LINEX loss functions for different samples of size n and replications. $(NR=\mathrm{10,000},\mathsf{\alpha }=2,\mathsf{\gamma }=0.5,\mathsf{\theta }=0.3,\mathrm{c}=0.2 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}=2.5)$.

n	rf and hrf	SE						$\mathbf{LINEX} \left(\mathit{\nu }=0.1\right)$
		Average	RABs	REs	UL	LL	Length	Average	RABs	REs	UL	LL	Length
30	$\mathit{R}\left({\mathit{x}}_0\right)$	0.8788	0.0440	0.3260	0.1903	0.0971	0.0932	0.8352	0.0079	0.0104	0.2093	0.2005	0.0088
	$\mathit{h}\left({\mathit{x}}_0\right)$	0.7750	0.0439	0.3131	2.5433	2.4873	0.0560	0.8065	0.0051	0.0041	2.5126	2.4990	0.0136
60	$\mathit{R}\left({\mathit{x}}_0\right)$	0.8533	0.0136	0.0314	0.2132	0.1759	0.0373	0.8444	0.0031	0.0016	0.2027	0.1975	0.0052
	$\mathit{h}\left({\mathit{x}}_0\right)$	0.8272	0.0204	0.0679	2.5190	2.4899	0.0291	0.8082	0.0029	0.0014	2.5025	2.4997	0.0028
100	$\mathit{R}\left({\mathit{x}}_0\right)$	0.8352	0.0078	0.0103	0.1975	0.1725	0.0250	0.8433	0.0018	0.0005	0.1996	0.1975	0.0021
	$\mathit{h}\left({\mathit{x}}_0\right)$	0.8267	0.0198	0.0638	2.5068	2.4838	0.0230	0.8083	0.0028	0.0012	2.5006	2.4991	0.0015

and

Table 4. Bayes averages, relative absolute biases, relative errors and 95% confidence intervals of the reliability and hazard rate functions at ($x_0=0.4$) from the DTZ-W distribution using the SE and LINEX loss functions for different samples of size n and replications. NR = 10,000, $(\mathsf{\alpha }=3,\mathsf{\gamma }=0.3,\mathsf{\theta }=0.2,\mathrm{c}=0.1 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}=4)$.

n	rf and hrf	SE						$\mathbf{LINEX} \left(\mathit{\nu }=0.1\right)$
		Average	RABs	REs	UL	LL	Length	Average	RABs	REs	UL	LL	Length
30	$\mathit{R}\left({\mathit{x}}_0\right)$	0.7472	0.0338	0.1767	0.1236	0.0108	0.1128	0.7709	0.0031	0.0015	0.1008	0.0913	0.0095
	$\mathit{h}\left({\mathit{x}}_0\right)$	0.6334	0.0667	0.6040	4.0105	3.9206	0.0899	0.6825	0.0057	0.0044	4.0082	3.9986	0.0096
60	$\mathit{R}\left({\mathit{x}}_0\right)$	0.7566	0.0216	0.0720	0.1256	0.0490	0.0766	0.7753	0.0026	0.0010	0.1008	0.0941	0.0067
	$\mathit{h}\left({\mathit{x}}_0\right)$	0.6843	0.0083	0.0094	4.0464	3.9596	0.0868	0.6811	0.0037	0.0018	3.9999	3.9940	0.0059
100	$\mathit{R}\left({\mathit{x}}_0\right)$	0.7584	0.0192	0.0573	0.1159	0.1001	0.0158	0.7716	0.0022	0.0007	0.1026	0.0990	0.0036
	$\mathit{h}\left({\mathit{x}}_0\right)$	0.6747	0.0058	0.0046	4.0261	3.9989	0.0272	0.6798	0.0018	0.0004	4.0005	3.9982	0.0023

show the same calculations for rf and hrf of the DTZ-W ($\underset{\_}{\psi }$) distribution.

Table 5. Two-sample Bayesian prediction and 95% confidence intervals for the future observation from the DTZ-W distribution for sample size $n=100, m=25, \mathrm{N}\mathrm{R}=\mathrm{10,000},$ and $(\mathsf{\alpha }=2,\mathsf{\gamma }=0.5,\mathsf{\theta }=0.3,\mathrm{c}=0.2 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}=2.5)$.

s	SE				$\mathbf{LINEX} \left(\mathit{\nu }=0.1\right)$
	${\tilde{\mathit{y}}}_{\left(\mathit{s}\right)}$	UL	LL	Length	${\tilde{\mathit{y}}}_{\left(\mathit{s}\right)}$	UL	LL	Length
1	0.9991	0.9999	0.9979	0.0020	1.0002	1.0011	0.9992	0.0019
8	3.0000	3.0014	2.9988	0.0026	3.0003	3.0013	2.9990	0.0023
15	6.0013	0.0026	5.9999	0.0027	5.9989	6.0002	5.9978	0.0024
25	14.999	15.0007	14.9975	0.0032	14.9988	15.0004	14.9974	0.0030

and

Table 6. Two-sample Bayesian prediction and 95% confidence intervals for the future observation from the DTZ-W distribution for sample size $n=100, m=25, \mathrm{N}\mathrm{R}=\mathrm{10,000},$ and $(\mathsf{\alpha }=3,\mathsf{\gamma }=0.3,\mathsf{\theta }=0.3,\mathrm{c}=0.2 \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{d}=4)$.

s	SE				$\mathbf{LINEX} \left(\mathit{\nu }=0.1\right)$
	${\tilde{\mathit{y}}}_{\left(\mathit{s}\right)}$	UL	LL	Length	${\tilde{\mathit{y}}}_{\left(\mathit{s}\right)}$	UL	LL	Length
1	0.9991	1.0005	0.9980	0.0025	1.0004	1.0015	0.9993	0.0022
8	2.9993	3.0006	2.9972	0.0034	2.9981	2.9998	2.9966	0.0032
15	6.0008	6.0031	5.9991	0.0040	5.9975	5.9996	5.9959	0.0037
25	14.9976	15.0003	14.9960	0.0043	15.0020	15.0034	14.9995	0.0039

show the two-sample Bayesian prediction and 95% CIs for the future observation from the DTZ-W ($\underset{\_}{\psi }$) distribution under the SE and LINEX loss functions.

4.2. Applications

This subsection is devoted to demonstrating how the proposed methods can be used in practice. Two real-life datasets are used for this purpose. The Kolmogorov–Smirnov (K-S) goodness-of-fit test is applied to demonstrate that the DTZ-W distribution is fitted to the two real datasets through Mathematica11.

Application I:

The first application utilizes a well-known dataset originally presented in [37]. This dataset comprises time-between-failure observations for a repairable item. Such data are fundamental in reliability engineering, as they capture the operational lifespan and failure patterns of components or systems that can be restored to service after a failure event. The context of this dataset is thus directly relevant to the assessment of system longevity, maintenance scheduling, and the overall reliability performance of industrial equipment. Analyzing this type of data with the DTZ-W distribution allows for a more detailed and accurate understanding of reliability, particularly if there are inherent truncation points or operational limits on the observable failure times.

The data are 1.43, 0.11, 0.71, 0.77, 2.63, 1.49, 3.46, 2.46, 0.59, 0.74, 1.23, 0.94, 4.36, 0.40, 1.74, 4.73, 2.23, 0.45, 0.70, 1.06, 1.46, 0.30, 1.82, 2.37, 0.63, 1.23, 1.24, 1.97, 1.86 and 1.17.

Application II:

The second application employs a dataset from [38], a foundational text in reliability and life data analysis. This dataset comprises the number of cycles (divided by 1000) accumulated until failure for 60 electrical appliances subjected to a life test. In reliability engineering, data from life tests, especially cycle-to-failure data, are critical for assessing product durability, predicting lifespan under operational stress, and optimizing design. This dataset offers valuable insights into the performance and failure characteristics of electrical components, directly informing us about issues of product reliability and warranty analysis. Applying the DTZ-W distribution to such data enables a more precise understanding of failure patterns, particularly if there are inherent truncation points or specific operational windows within which the appliance failures are observed or reported. This context is crucial for validating the flexibility and applicability of the proposed distribution in real-world engineering scenarios.

The data are 0.014, 0.034, 0.059, 0.061, 0.069, 0.080, 0.123, 0.142, 0.165, 0.210, 0.381, 0.464, 0.479, 0.556, 0.574, 0.839, 0.917, 0.969, 0.991, 1.064, 1.088, 1.091, 1.174, 1.270, 1.275, 1.355, 1.397, 1.477, 1.578, 1.649, 1.702, 1.893, 1.932, 2.001, 2.161, 2.292, 2.326, 2.337, 2.628, 2.785, 2.811, 2.886, 2.993, 3.122, 3.248, 3.715, 3.790, 3.857, 3.912, 4.100, 4.106, 4.116, 4.315, 4.510, 4.580, 5.267, 5.299, 5.583, 6.065 and 9.701.

The K-S goodness-of-fit test is performed to check the validity of the proposed fitted model. The p values are given, respectively, as 0.9560 and 0.9866. The p-value given in each case showed that the proposed model fits the data very well.

Table 7. Bayes estimates and Ses of the parameters for the real data under the SE and LINEX loss functions.

Application	n	Estimators	SE		$\mathbf{LINEX} \left(\mathit{\nu }=0.1\right)$
			Estimates	Se	Estimates	Se
I	30	$\tilde{\mathit{\alpha }}$	1.5770	0.0060	1.4985	0.0025
		$\tilde{\mathit{\gamma }}$	0.4698	0.0033	0.5270	0.0031
		$\tilde{\mathit{\theta }}$	0.4610	0.0041	0.4920	0.0019
		$\tilde{\mathit{c}}$	0.6286	0.0020	0.5816	0.0012
		$\tilde{\mathit{d}}$	3.9958	0.0022	4.0038	0.0014
II	60	$\tilde{\mathit{\alpha }}$	1.9659	0.0020	1.9806	0.0016
		$\tilde{\mathit{\gamma }}$	0.9399	0.0042	0.9271	0.0015
		$\tilde{\mathit{\theta }}$	0.5288	0.0027	0.4968	0.0013
		$\tilde{\mathit{c}}$	0.1699	0.0022	0.2107	0.0019
		$\tilde{\mathit{d}}$	3.9783	0.0018	3.9890	0.0012

and

Table 8. Bayes estimates and Ses of the rf and hrf for the real data under the SE and LINEX loss functions.

Application	n	Estimators	SE		$\mathbf{LINEX} \left(\mathit{\nu }=0.1\right)$
			Estimates	Se	Estimates	Se
I	30	$\tilde{\mathit{R}}\left({\mathit{x}}_0\right)$	1.0640	0.0034	1.0787	0.0023
		$\tilde{\mathit{h}}\left({\mathit{x}}_0\right)$	0.3661	0.0025	0.3603	0.0013
II	60	$\tilde{\mathit{R}}\left({\mathit{x}}_0\right)$	0.9189	0.0030	0.9106	0.0012
		$\tilde{\mathit{h}}\left({\mathit{x}}_0\right)$	0.4182	0.0016	0.3992	0.0013

show the Bayes estimates of the parameters, rf, hrf and their standard errors (Ses) under the SE and LINEX loss functions. Also, Bayes point predictors and 95% CIs for the future observation from the DTZ-W ($\underset{\_}{\psi }$) distribution under two-sample prediction for the two datasets are displayed in

Table 9. Two-sample Bayesian prediction for two applications and 95% confidence intervals for the future observation from the DTZ-W distribution.

Application	s	SE				$\mathbf{LINEX} \left(\mathit{\nu }=0.1\right)$
		${\tilde{\mathit{y}}}_{\left(\mathit{s}\right)}$	UL	LL	Length	${\tilde{\mathit{y}}}_{\left(\mathit{s}\right)}$	UL	LL	Length
I	1	1.0013	1.0027	0.9999	0.0028	0.9992	1.0002	0.9980	0.0022
	10	3.0024	3.0046	2.9997	0.0049	2.9991	3.0007	2.9981	0.0026
	15	6.9962	6.9996	6.9946	0.0050	6.9985	7.0008	6.9971	0.0037
	25	9.9982	10.0003	9.9948	0.0055	9.9979	10.0003	9.9958	0.0045
II	1	0.9976	0.9997	0.9966	0.0030	0.9988	0.9999	0.9978	0.0021
	8	4.0016	4.0029	3.9998	0.0031	4.0017	4.0029	4.0003	0.0026
	20	5.9984	6.0001	5.9968	0.0033	6.0004	6.0023	5.9991	0.0032
	45	9.9975	9.9993	9.9952	0.0041	9.9987	10.0003	9.9966	0.0037

.

Ref. [39] utilized two real-world applications to demonstrate the superior performance of the DTZ-W distribution when compared against several alternative distributions. These alternatives included the Zubair–Weibull, doubly truncated exponentiated inverse Weibull, Truncated Weibull Power Lomax, Truncated Log-Logistic-Weibull, and Truncated Exponential Marshall Olkin-Weibull distributions.

The comparative analysis employed established goodness-of-fit criteria: the K-S test and its corresponding p-value, the negative two log-likelihood function, the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), and the Corrected Akaike Information Criterion (CAIC). Consistent with statistical best practices, the distribution exhibiting the lowest values for −2lnL, AIC, BIC and CAIC, coupled with the largest p-value for the K-S test, was deemed the best fit for the given data.

The results obtained from both applications, including ML estimates of parameters, Ses, K-S statistics, p-values, −2lnL statistics and AIC, BIC and CAIC values, consistently indicated that the DTZ-W distribution provided a superior fit to the observed data when compared with the other distributions considered.

4.3. Discussions

▪

According to Table 1 and Table 2, the RABs and REs of the Bayes averages for the parameters α, θ, γ, c and d perform better when the sample size increases. When the sample size increases, the lengths of the CIs decrease.

▪

It is observed from Table 3 and Table 4 that the RABs and REs of the Bayes averages for the rf and hrf perform better when the sample size increases. The lengths of the CIs decrease when the sample size increases.

▪

It is noticed from Table 5, Table 6 and Table 9 that the lengths of the BP increase when s increases. The BP for the future observation is located between the lower limit and upper limit.

▪

The lengths of the intervals of the BP under the LINEX loss function are less than the lengths of the intervals of the BP under the SE loss function.

5. Conclusions

In this paper, Bayesian estimation and prediction (point and interval) for the DTZ-W distribution are introduced. An informative prior is used by applying the joint bivariate prior that was used by [34]) and gamma prior to estimate the unknown parameters, rf and hrf under the SE loss function as a symmetric loss function and the LINEX loss function as an asymmetric loss function. The performance of the Bayes estimates was examined through some measurements of accuracy. The Bayes predictors (point and interval) for a future observation from the DTZ-W distribution are derived. From the numerical results, it is noticed that the Bayes estimates perform better when the sample size increases. The Bayes estimates of the parameters, rf and hrf of the DTZ-W distribution under the LINEX have smaller Ses than under the SE loss functions. Also, two real datasets are applied to show the flexibility and applicability of the distribution in practice.