Bayesian Inferential Approaches and Bootstrap for the Reliability and Hazard Rate Functions under Progressive First-Failure Censoring for Coronavirus Data from Asymmetric Model

Abstract

This paper deals with the estimation of the parameters for asymmetric distribution and some lifetime indices such as reliability and hazard rate functions based on progressive first-failure censoring. Maximum likelihood, bootstrap and Bayesian approaches of the distribution parameters and reliability characteristics are investigated. Furthermore, the approximate confidence intervals and highest posterior density credible intervals of the parameters are constructed based on the asymptotic distribution of the maximum likelihood estimators and Markov chain Monte Carlo technique, respectively. In addition, the delta method is implemented to obtain the variances of the reliability and hazard functions. Moreover, we apply two methods of bootstrap to construct the confidence intervals. The Bayes inference based on the squared error and LINEX loss functions is obtained. Extensive simulation studies are conducted to evaluate the behavior of the proposed methods. Finally, a real data set of the COVID-19 mortality rate is analyzed to illustrate the estimation methods developed here.

1. Introduction

In life testing and reliability analysis, some items may fail (called a censored sample) or be removed before failure. Hence, censoring occurs when the exact lifetimes for only some of the items in the test are known. One of the major reasons for the removal of an experimental item is saving the working experimental items for future use, saving the cost and time associated with testing.

In practice, there are usually two random variables: the time and the number of failures of items. This strategy of censoring schemes shows how the examiner imagines the experiment based on a predetermined time. A random number of items is accounted for the first Type-I of a censoring scheme, which means the exact time of stopping an experiment may be assumed, while there is a predetermined number of failure items and a random time in the type-II censoring scheme. In these two types of censoring schemes, items cannot be removed from an experiment until the final stage, or the number of items fail. These schemes allow the detection of some items that are defective after running the experiment. The proposed approach that has more flexibility than type-II censoring in allowing items to be withdrawn from the test at different observed failure times is the progressive type-II censoring.

Balakrishnan and Sandhu [1] generated a progressive type-II censoring, which is useful to achieve the desired aims of the used censoring schemes. The general idea of this type is to remove some items from the test at the observed failure times within the duration of the test until the test is terminated. Schematically: The independent items n are placed on a life test and the observed failure times m $(m\le n)$, which is called a progressive sample as $X_{1:m:n}<X_{2:m:n}<\cdots <X_{m:m:n}$, is prefixed. In addition, the scheme $\mathbf{R}=(R_1,R_2,\dots ,R_m)$ is a prefixed censoring plan. This is achieved as follows: At the first failure occurrence $X_1,$ the surviving items $R_1$ are randomly withdrawn from the test. Then, at the second failure occurrence $X_2$, the surviving items $R_2$ are randomly withdrawn from the test and so on until the occurrence of the m-th failure and therefore the remaining surviving items $R_m=n-m-R_1-R_2-\cdots -R_{m-1}$ are withdrawn from the test and the test is terminated. Several authors have discussed inference under a progressive type-II censoring scheme with applications; see, for example, Fu et al. [2], Chen et al. [3], Xu et al. [4] and Luo et al. [5]. Recently, El-Sagheer [6] discussed the estimation problem for the Weibull–Fréchet model based on the progressive type-II censoring schemes with applications on gastric cancer survival times.

Wu et al. [7] described a life test in which the experimenter can decide to divide the items under test into several groups and then run all the items simultaneously until the occurrence of the first failure in each group. Such a censoring scheme is called first-failure censoring. However, using this censoring scheme does not able the experimenter to remove experimental groups from the test until the first failure is observed. For this reason, Wu and Kuş [8] introduced life testing, which combines first-failure censoring with progressive type-II censoring, namely as a progressive first-failure censoring (PFFC) scheme. Several authors have discussed inference under a PFFC scheme for different lifetime distributions; see, for example, Haj et al. [9], Abushal [10], Soliman et al. [11,12], Mahmoud et al. [13], Ahmed [14], Xie and Gui [15] and Shi and Shi [16]. This censoring scheme has advantages in terms of reducing test time in which more items are used but only m of $n\times k$ items are failures. Note that some censoring rules can be accommodated: the complete sample case for $k=1,m=n,R_i=0,$ and $i=1,2,3,\dots ,m$; the first-failure censoring sample for $k\ne 1,m=n,R_i=0,$ and $i=1,2,3,\dots ,m$; the progressive type-II censoring sample for $k=1$; and the usual type-II censored sample case by taking $k=1,R_m=n-m,$ and $R_i=0$ for $i=1,2,3,\dots ,m-1$.

Afify et al. [17] introduced the four parameters Weibull–Fréchet distribution, WFr$(\alpha ,\beta ,\delta ,\lambda ),$ which is a generalization of both Weibull and Fréchet distributions. It is a significant continuous lifetime distribution and plays a key role in reliability problems. The probability density function (PDF) of a random variable $X\sim WFr(\alpha ,\beta ,\delta ,\lambda )$ is given by

$$f\left(x\right)=\delta \lambda \beta {\alpha }^{\beta }{x}^{-\left(\beta +1\right)}{e}^{-\lambda {\left(\frac{\alpha }{x}\right)}^{\beta }}\left[{\left(1-e^{-{\left(\frac{\alpha }{x}\right)}^{\beta }}\right)}^{-\left(\lambda +1\right)}\right]e^{-\delta \left\{{\left(e^{{\left(\frac{\alpha }{x}\right)}^{\beta }}-1\right)}^{-\lambda }\right\}};x>0,$$

(1)

and cumulative distribution function (CDF) is given by

$$F\left(x\right)=1-e^{-\delta \left\{{\left(e^{{\left(\frac{\alpha }{x}\right)}^{\beta }}-1\right)}^{-\lambda }\right\}};x>0,$$

(2)

where $\alpha$ is a scale parameter and $\beta ,$$\delta ,$$\lambda$ are shape parameters. The reliability and hazard rate functions of X are given by the following expressions

$$r\left(x\right)=e^{-\delta \left\{{\left(e^{{\left(\frac{\alpha }{x}\right)}^{\beta }}-1\right)}^{-\lambda }\right\}};x>0,$$

(3)

and

$$h\left(x\right)=\delta \lambda \beta {\alpha }^{\beta }{x}^{-\left(\beta +1\right)}{e}^{-\lambda {\left(\frac{\alpha }{x}\right)}^{\beta }}\left[{\left(1-e^{-{\left(\frac{\alpha }{x}\right)}^{\beta }}\right)}^{-\left(\lambda +1\right)}\right];x>0.$$

(4)

This distribution is a very exible model that approaches different distributions when its parameters are changed. It contains the following special models:

• WFr$(\alpha ,1,\delta ,\lambda )$ follows the Weibull inverse exponential model.
• WFr$\left(\alpha ,\beta ,\delta ,1\right)$ is the exponential Frechet model.
• WFr$\left(\alpha ,2,\delta ,\lambda \right)$ refers to the Weibull inverse Rayleigh model.
• WFr$\left(\alpha ,2,\delta ,1\right)$ reduces to the exponential inverse Rayleigh model.
• WFr$\left(\alpha ,1,\delta ,1\right)$ follows the exponential inverse exponential model.

The main aim of this article is to focus on the designing problem of a progressive first-failure censoring life test with a Weibull–Fréchet failure time distribution. Four methods—maximum likelihood, Bootstrap-p, Bootstrap-t and Bayes—are used to estimate four unknown parameters as well as reliability and hazard rate functions of the WFrD based on a PFFC scheme. The approximate confidence intervals (ACIs) for $\alpha ,$$\beta ,$$\delta$ and $\lambda$ are constructed based on the asymptotic normality of the MLEs. In addition, the delta method is implemented to construct the variances of the reliability $r\left(t\right)$ and hazard rate $h\left(t\right)$ functions. In the Bayesian framework, it is difficult to obtain the Bayes estimators in explicit forms. Thus, Markov chain Monte Carlo (MCMC) techniques are applied to compute the Bayes estimators. To this end, the Metropolis–Hastings (M-H) algorithm within the Gibbs sampler is implemented. The Bayes estimates are obtained under both symmetric (SE) and asymmetric (LINEX) loss functions. Moreover, the corresponding credible intervals (CRIs) are constructed under the MCMC technique. Finally, a simulation study is utilized to assay the performance of the proposed methods.

The rest of the paper is arranged as follows. Section 2 deals with the maximum likelihood estimate and asymptotic confidence intervals. Two parametric bootstrap methods are presented in Section 3. Bayesian estimates using the MCMC technique are provided in Section 4. In Section 5, a simulation study is conducted to compare the performance of these estimation methods. A real data set of COVID-19 is presented to illustrate the application of the proposed inference procedures in Section 6. Finally, a brief conclusion is furnished in Section 7.

2. Maximum Likelihood Inference

Suppose that $x_i=x_{i:m:n:k}^R$, $i=1,2,\dots ,m$, are PFFC order statistics from the WFrD with progressive censored scheme $R=\left(R_1,R_2,\dots ,R_m\right)$. According to Wu and Kuş [8], the log-likelihood function $\ell =logL\left(\alpha ,\beta ,\delta ,\lambda ;\underline{x}\right)$ without normalized constant can be written as

$$\begin{array}{ccc}\ell \hfill & \propto \hfill & mlogk+mlog\delta +mlog\lambda +mlog\beta +m\beta log\alpha -\left(\beta +1\right){\displaystyle \sum _{i=1}^m}log{x}_i-\lambda {\displaystyle \sum _{i=1}^m}{\left(\frac{\alpha }{x_i}\right)}^{\beta }\hfill \\ \hfill & \hfill & -\left(\lambda +1\right){\displaystyle \sum _{i=1}^m}log\left(1-e^{-{\left(\frac{\alpha }{x_i}\right)}^{\beta }}\right)-\delta {\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\lambda }.\hfill \end{array}$$

(5)

Calculating the first-order partial derivative of (5) with respect to $\alpha ,$$\beta ,$$\delta$ and $\lambda$, respectively, and equating each result to zero, we obtain the likelihood equations

$$\begin{array}{c}l\frac{m\beta }{\alpha }-\lambda {\displaystyle \sum _{i=1}^m}\frac{\beta }{x_i}{\left(\frac{\alpha }{x_i}\right)}^{\left(\beta -1\right)}-\left(\lambda +1\right){\displaystyle \sum _{i=1}^m}\frac{\frac{\beta }{x_i}{\left(\frac{\alpha }{x_i}\right)}^{\left(\beta -1\right)}{e}^{-{\left(\frac{\alpha }{x_i}\right)}^{\beta }}}{\left(1-e^{-{\left(\frac{\alpha }{x_i}\right)}^{\beta }}\right)}\hfill \\ +\delta \lambda {\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\left(\lambda +1\right)}\frac{\beta }{x_i}{\left(\frac{\alpha }{x_i}\right)}^{\left(\beta -1\right)}{e}^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}=0,\hfill \end{array}$$

(6)

$$\begin{array}{c}l\frac{m}{\beta }+mlog\alpha -{\displaystyle \sum _{i=1}^m}log{x}_i-\lambda {\displaystyle \sum _{i=1}^m}{\left(\frac{\alpha }{x_i}\right)}^{\beta }log\left(\frac{\alpha }{x_i}\right)-\left(\lambda +1\right){\displaystyle \sum _{i=1}^m}\frac{{\left(\frac{\alpha }{x_i}\right)}^{\beta }{e}^{-{\left(\frac{\alpha }{x_i}\right)}^{\beta }}log\left(\frac{\alpha }{x_i}\right)}{\left(1-e^{-{\left(\frac{\alpha }{x_i}\right)}^{\beta }}\right)}\hfill \\ +\delta \lambda {\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\left(\lambda +1\right)}{\left(\frac{\alpha }{x_i}\right)}^{\beta }{e}^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}log\left(\frac{\alpha }{x_i}\right)=0,\hfill \end{array}$$

(7)

$$\frac{m}{\delta }-{\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\lambda }=0,$$

(8)

and

$$\begin{array}{c}\frac{m}{\lambda }-{\displaystyle \sum _{i=1}^m}{\left(\frac{\alpha }{x_i}\right)}^{\beta }-{\displaystyle \sum _{i=1}^m}log\left(1-e^{-{\left(\frac{\alpha }{x_i}\right)}^{\beta }}\right)\hfill \\ -\delta {\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\lambda }log\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)=0.\hfill \end{array}$$

(9)

From (8), we obtain MLE of $\delta$ as

$$\widehat{\delta }=m{\left[{\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\widehat{\alpha }}{x_i}\right)}^{\widehat{\beta }}}-1\right)}^{-\widehat{\lambda }}\right]}^{-1}.$$

(10)

As such, (6), (7) and (9) do not have closed form solutions. Newton–Raphson iteration method is widely used to obtain the desired MLEs in such situations. For more details, see EL-Sagheer [18]. The algorithm is described as follows:

• Begin with a guess initial values $\left({\alpha }_0,{\beta }_0,{\delta }_0,{\lambda }_0\right)$ and set $k=0$.
• Calculate ${\left(\frac{\partial \ell }{\partial \alpha },\frac{\partial \ell }{\partial \beta },\frac{\partial \ell }{\partial \delta },\frac{\partial \ell }{\partial \lambda }\right)}_{\left({\alpha }_k,{\beta }_k,{\delta }_k,{\lambda }_k\right)}$ and the observed Fisher information matrix $I^{-1}\left(\alpha ,\beta ,\delta ,\lambda \right),$ given in Section 2.1.
• Update $\left(\alpha ,\beta ,\delta ,\lambda \right)$ as

  $$\begin{array}{ccc}\left({\alpha }_{k+1},{\beta }_{k+1},{\delta }_{k+1},{\lambda }_{k+1}\right)\hfill & =& \left({\alpha }_k,{\beta }_k,{\delta }_k,{\lambda }_k\right)+{\left(\frac{\partial \ell }{\partial \alpha },\frac{\partial \ell }{\partial \beta },\frac{\partial \ell }{\partial \delta },\frac{\partial \ell }{\partial \lambda }\right)}_{\left({\alpha }_k,{\beta }_k,{\delta }_k,{\lambda }_k\right)}\hfill \\ & & \times I^{-1}\left(\alpha ,\beta ,\delta ,\lambda \right).\hfill \end{array}$$
• Set $k=k+1$ and then go back to Step 1.
• Continue the iterative steps until $\left|\left({\alpha }_{k+1},{\beta }_{k+1},{\delta }_{k+1},{\lambda }_{k+1}\right)-\left({\alpha }_k,{\beta }_k,{\delta }_k,{\lambda }_k\right)\right|$ is smaller than a threshold value. The final estimates of $\alpha ,$$\beta ,$$\delta$ and $\lambda$ are the MLE of the parameters, denoted as $\widehat{\alpha },$$\widehat{\beta },$$\widehat{\delta }$ and $\widehat{\lambda }$.

Thus, the MLEs of $r\left(t\right)$ and $h\left(t\right)$ for given mission time t can be obtained according to the invariant property of the MLEs as

$$\widehat{r}\left(t\right)=e^{-\widehat{\delta }\left\{{\left(e^{{\left(\frac{\widehat{\alpha }}{x}\right)}^{\widehat{\beta }}}-1\right)}^{-\widehat{\lambda }}\right\}},$$

(11)

and

$$\widehat{h}\left(t\right)=\widehat{\delta }\widehat{\lambda }\widehat{\beta }{\widehat{\alpha }}^{\widehat{\beta }}{x}^{-\left(\widehat{\beta }+1\right)}{e}^{-\widehat{\lambda }{\left(\frac{\widehat{\alpha }}{x}\right)}^{\widehat{\beta }}}\left[{\left(1-e^{-{\left(\frac{\widehat{\alpha }}{x}\right)}^{\widehat{\beta }}}\right)}^{-\left(\widehat{\lambda }+1\right)}\right].$$

(12)

2.1. Asymptotic Confidence Intervals

In this subsection, we propose the asymptotic confidence intervals (ACIs) for unknown parameters $\Omega =({\Omega }_1,{\Omega }_2,{\Omega }_3,{\Omega }_4)=(\alpha ,\beta ,\delta ,\lambda )$. This requires to compute the Fisher information matrix (FIM), which is defined by the negative expectation of the second-order partial derivative of the log-likelihood function

$$I\left(\Omega \right)=E{\left[-{\displaystyle \frac{{\partial }^2\ell }{\partial {\Omega }_i\partial {\Omega }_j}}\right]}_{i,j=1,2,3,4}={\left[I_{ij}\right]}_{4\times 4}.$$

Because MLE has asymptotic normality property under certain regularity conditions, the estimator $\widehat{\Omega }=(\widehat{\alpha },\widehat{\beta },\widehat{\delta },\widehat{\lambda })$ has asymptotic distribution $\widehat{\Omega }-\Omega \to N(0,I^{-1}\left(\Omega \right)),$ and the inverse matrix of $I\left(\Omega \right)$ is

$$I^{-1}\left(\Omega \right)=I^{-1}\left(\widehat{\Omega }\right)=\left(\begin{array}{cccc}var\left(\widehat{\alpha }\right)& cov(\widehat{\alpha },\widehat{\beta })& cov(\widehat{\alpha },\widehat{\delta })& cov(\widehat{\alpha },\widehat{\lambda })\\ cov(\widehat{\beta },\widehat{\alpha })& var\left(\widehat{\beta }\right)& cov(\widehat{\beta },\widehat{\delta })& cov(\widehat{\beta },\widehat{\lambda })\\ cov(\widehat{\delta },\widehat{\alpha })& cov(\widehat{\delta },\widehat{\beta })& var\left(\widehat{\delta }\right)& cov(\widehat{\delta },\widehat{\lambda })\\ cov(\widehat{\lambda },\widehat{\alpha })& cov(\widehat{\lambda },\widehat{\beta })& cov(\widehat{\lambda },\widehat{\delta })& var\left(\widehat{\lambda }\right)\end{array}\right).$$

Thus, the $100(1-\gamma )\%$ ACIs of $\Omega$ is

$$\left({\widehat{\Omega }}_i-z_{\frac{\gamma }{2}}\sqrt{var\left({\widehat{\Omega }}_i\right)},{\widehat{\Omega }}_i+z_{\frac{\gamma }{2}}\sqrt{var\left({\widehat{\Omega }}_i\right)}\right),i=1,2,3,4,$$

where $var\left({\widehat{\Omega }}_i\right)=I_{ii}^{-1},i=1,2,3,4.$$z_{\frac{\gamma }{2}}$ is the percentile of the standard normal distribution with right-tail probability $\frac{\gamma }{2}$. In order to construct the ACIs of $r\left(t\right)$ and $h\left(t\right),$ we need to find the variances of them. Thus, Greene [19] implemented the delta method for this purpose as follows:

Assume that $G=(\frac{\partial \Lambda }{\partial \alpha },\frac{\partial \Lambda }{\partial \beta },\frac{\partial \Lambda }{\partial \delta },\frac{\partial \Lambda }{\partial \lambda })$ is first-order partial derivative of $\Lambda =\left(r\right(t),h(t\left)\right)$. Then, the asymptotic variances of $\widehat{\Lambda }=(\widehat{r}\left(t\right),\widehat{h}\left(t\right))$ can be written as

$$var\left(\widehat{\Lambda }\right)\simeq {\left(G^T{I}^{-1}\left(\widehat{\Omega }\right)G\right)}_{\downarrow (\Omega =\widehat{\Omega })},$$

where $G^T$ is the transpose matrix of G. Thus, the $(1-\gamma )100\%$ ACIs for $\widehat{\Lambda }=(\widehat{r}\left(t\right),\widehat{h}\left(t\right))$ can be given by

$$\left(\widehat{\Lambda }-z_{\frac{\gamma }{2}}\sqrt{var\left(\widehat{\Lambda }\right)},\widehat{\Lambda }+z_{\frac{\gamma }{2}}\sqrt{var\left(\widehat{\Lambda }\right)}\right).$$

Furthermore, to prevent the negative lower bound of the asymptotic confidence intervals, the use of the normal approximation for the log-transformed MLE is suggested by Meeker and Escobar [20]. Thus, two-sided $(1-\gamma )100\%$ normal ACIs for $\widehat{\theta }=\widehat{\alpha },\widehat{\beta },\widehat{\delta },\widehat{\lambda },\widehat{r}\left(t\right)$ or $\widehat{h}\left(t\right)$ are given by

$$\left(\widehat{\theta }.exp\left\{-\frac{z_{\frac{\gamma }{2}}\sqrt{var\left(\widehat{\theta }\right)}}{\widehat{\theta }}\right\},\widehat{\theta }.exp\left\{\frac{z_{\frac{\gamma }{2}}\sqrt{var\left(\widehat{\theta }\right)}}{\widehat{\theta }}\right\}\right).$$

3. Parametric Bootstrap Methods

It is known that the normal approximation is inappropriate when the sample is small. In this section, we propose a resampling technique, bootstrap procedure, to obtain more widely used confidence intervals. For this, two parametric bootstrap procedures are discussed to construct the bootstrap confidence intervals of $\theta =\alpha$, $\beta$, $\delta$, $\lambda ,$$r\left(t\right)$ or $h\left(t\right)$. The percentile bootstrap (boot-p) and bootstrap-t (boot-t) confidence intervals using the same steps, which were mentioned in DiCiccio and Efron [21] and Hall [22], respectively, are discussed. Several authors have studied these two types of bootstrap, for example, Reiser et al. [23] and Besseris [24].

3.1. Percentile Bootstrap

• From the original data $\underline{x}\equiv x_{1:m:n:k}^{\mathbf{R}},$ $x_{2:m:n:k}^{\mathbf{R}},\dots ,$$x_{m:m:n:k}^{\mathbf{R}}$, compute the MLEs of the unknown parameters $\alpha ,$$\beta ,$$\delta ,$ and $\lambda$ by solving the nonlinear Equations (6)–(10), denoted the estimates as $\widehat{\theta }=(\widehat{\alpha },\widehat{\beta },\widehat{\delta },\widehat{\lambda },r\left(t\right),h\left(t\right))$.
• Use $\widehat{\alpha },$$\widehat{\beta },$$\widehat{\delta }$ and $\widehat{\lambda }$ to generate PFFC sample ${\underline{x}}^{*}$ with the same values of $R_i,$$m;(i=1,2,\dots ,m)$ and compute the bootstrap estimates ${\widehat{\theta }}^{*}$.
• Repeat Step 2 N times; then, we have ${\widehat{\theta }}_1^{*},$${\widehat{\theta }}_2^{*},\dots ,{\widehat{\theta }}_N^{*}.$
• Arrange all components in ascending order; the bootstrap estimates are ${\widehat{\theta }}_{\left(1\right)}^{*},$${\widehat{\theta }}_{\left(2\right)}^{*},\dots ,{\widehat{\theta }}_{\left(N\right)}^{*}$.
• Let $\phi \left(x\right)=P\left({\widehat{\theta }}^{*}\le x\right)$ be the CDF of ${\widehat{\theta }}^{*}$. Define ${\widehat{\theta }}_{boot-p}\left(x\right)={\phi }^{-1}\left(x\right)$ for given x. Then, two-side $100\left(1-\gamma \right)\%$ percentile confidence intervals of $\theta$ given by

  $$\left[{\widehat{\theta }}_{boot-p}\left(\frac{\gamma }{2}\right),{\widehat{\theta }}_{boot-p}\left(1-\frac{\gamma }{2}\right)\right].$$

3.2. Bootstrap-t

1,2.

The same as the percentile bootstrap.

3.

Compute the t-statistics $T_{\theta }^{*}=\frac{{\widehat{\theta }}^{*}-\widehat{\theta }}{\sqrt{var\left({\widehat{\theta }}^{*}\right)}},$ where $var\left({\widehat{\theta }}^{*}\right)$ obtained using the FIM for $\alpha$, $\beta$, $\delta$, $\lambda ,$ and delta method for $r\left(t\right)$ and $h\left(t\right)$.

4.

Repeat Steps 2, 3, 4 N times; then, we have $T_{\theta }^{*\left(1\right)},T_{\theta }^{*\left(2\right)},\dots ,T_{\theta }^{*\left(N\right)}$.

5.

Let $\psi \left(x\right)=P\left(T_{\theta }^{*}\le x\right)$ be the CDF of $T_{\theta }^{*}$. Define ${\widehat{\theta }}_{boot-t}\left(x\right)=\widehat{\theta }+{\psi }^{-1}\left(x\right)\sqrt{var\left({\widehat{\theta }}^{*}\right)}$ for given x. Then, two-side $100\left(1-\gamma \right)\%$ bootstrap-t confidence intervals of $\theta =\alpha$, $\beta$, $\delta$, $\lambda ,$$r\left(t\right)$ or $h\left(t\right)$ given by

$$\left[{\widehat{\theta }}_{boot-t}\left(\frac{\gamma }{2}\right),{\widehat{\theta }}_{boot-t}\left(1-\frac{\gamma }{2}\right)\right].$$

4. Bayes Inference

Bayes estimation is quite different from MLE and bootstrap methods because it takes into consideration both the information from observed sample data and the prior information, in addition to symmetric and asymmetric loss functions. It can characterize the problems more rationally and reasonably. In this section, Bayesian inference procedure using MCMC technique are proposed to estimate the parameters $\alpha ,$$\beta ,$$\delta$ and $\lambda$ as well as $r\left(t\right)$ or $h\left(t\right)$ under both SE and LINEX loss functions. Moreover, the corresponding CRIs are constructed under MCMC technique. It is known that the family of gamma distributions is flexible enough to cover a large variety of prior beliefs of the experimenter. Therefore, we consider that the unknown parameters $\alpha ,$$\beta ,$$\delta$ and $\lambda$ are stochastically independently distributed with conjugate gamma prior as follows

$$\left\{\begin{array}{c}{\pi }_1\left(\alpha \right)=\frac{b_1^{a_1}}{\Gamma \left(a_1\right)}{\alpha }^{a_1-1}{e}^{-b_1\alpha },\alpha >0,{\pi }_2\left(\beta \right)=\frac{b_2^{a_2}}{\Gamma \left(a_1\right)}{\beta }^{a_2-1}{e}^{-b_2\beta },\beta >0,\\ \\ {\pi }_3\left(\lambda \right)=\frac{b_3^{a_3}}{\Gamma \left(a_3\right)}{\delta }^{a_3-1}{e}^{-b_3\delta },\delta >0,{\pi }_4\left(\lambda \right)=\frac{b_4^{a_4}}{\Gamma \left(a_3\right)}{\lambda }^{a_4-1}{e}^{-b_4\lambda },\lambda >0,\end{array}\right.,$$

where $a_i$ and $b_i,$$i=1,2,3,4$ are the hyper-parameters which reflect the prior knowledge about the unknown parameters, and they are assumed to be known and non-negative. The joint prior of unknown parameters $\alpha ,$$\beta ,$$\delta$ and $\lambda$ is

$$\pi (\alpha ,\beta ,\delta ,\lambda )\propto {\alpha }^{a_1-1}{\beta }^{a_2-1}{\delta }^{a_3-1}{\lambda }^{a_4-1}{e}^{-\left(b_1\alpha +b_2\beta +b_3\delta +b_4\lambda \right)}.$$

(13)

Then, the joint posterior density of the unknown parameters $\Omega =(\alpha ,\beta ,\delta ,\lambda )$ is

$${\pi }^{*}(\Omega |\underline{x})=\frac{\pi (\Omega )L(\Omega |\underline{x})}{{\int }_{\Omega }\pi (\Omega )L(\Omega |\underline{x})d\alpha d\beta d\delta d\lambda }.$$

(14)

Under SE loss function $\phi \left(\widehat{\eta },\eta \right)={\left(\widehat{\eta }-\eta \right)}^2,$ the Bayes estimation of any function $W(\Omega )$ is

$$W^{*}(\Omega )=\frac{{\int }_{\Omega }W(\Omega )\pi (\Omega )L(\Omega |\underline{x})d\alpha d\beta d\delta d\lambda }{{\int }_{\Omega }L(\Omega |\underline{x})\pi (\Omega )d\alpha d\beta d\delta d\lambda }.$$

(15)

Under asymmetric LINEX loss function $\psi \left(\widehat{\eta },\eta \right)=e^{\epsilon \left(\widehat{\eta }-\eta \right)}-\epsilon \left(\widehat{\eta }-\eta \right)-1,$ we have

$$\widehat{\zeta }=-\frac{1}{\epsilon }logE\left(e^{-\epsilon \zeta }\right),\epsilon \ne 0,$$

(16)

where $\zeta =\alpha$, $\beta$, $\delta$, $\lambda ,$$r\left(t\right)$ or $h\left(t\right)$. For more accurate and important details, see EL-Sagheer et al. [25]. It is clear that the integrals in (15) and (16) cannot be computed in closed forms. Therefore, the Metropolis–Hastings (MH) within Gibbs sampling is a useful MCMC technique to estimate complex Bayes problems. Thus, the conditional posterior densities of $\alpha ,$$\beta ,$$\delta$ and $\lambda$ can be written as

$$\begin{array}{ccc}{\pi }_1^{*}\left(\alpha \right|\beta ,\delta ,\lambda ,\underline{x})\hfill & \propto \hfill & {\alpha }^{m\beta +a_1-1}\left[{\displaystyle \prod _{i=1}^m}{\left(1-e^{-{\left(\frac{\alpha }{x_i}\right)}^{\beta }}\right)}^{-\left(\lambda +1\right)}\right]\hfill \\ \hfill & \hfill & \times e^{\left\{-b_1\alpha -\lambda {\displaystyle \sum _{i=1}^m}{\left(\frac{\alpha }{x_i}\right)}^{\beta }-\delta {\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\lambda }\right\}},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}{\pi }_2^{*}\left(\beta \right|\alpha ,\delta ,\lambda ,\underline{x})\hfill & \propto \hfill & {\beta }^{m+a_2-1}{\alpha }^{m\beta }\left[{\displaystyle \prod _{i=1}^m}{x}_i^{-\beta }{\left(1-e^{-{\left(\frac{\alpha }{x_i}\right)}^{\beta }}\right)}^{-\left(\lambda +1\right)}\right]\hfill \\ \hfill & \hfill & \times e^{\left\{-b_2\beta -\lambda {\displaystyle \sum _{i=1}^m}{\left(\frac{\alpha }{x_i}\right)}^{\beta }-\delta {\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\lambda }\right\}},\hfill \end{array}$$

(18)

$${\pi }_3^{*}\left(\delta \right|\alpha ,\beta ,\lambda ,\underline{x})\propto {\delta }^{m+a_3-1}\times e^{-\delta \left\{b_3+{\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\lambda }\right\}},$$

(19)

and

$$\begin{array}{ccc}{\pi }_4^{*}\left(\lambda \right|\alpha ,\beta ,\delta ,\underline{x})\hfill & \propto \hfill & {\lambda }^{m+a_4-1}\left[{\displaystyle \prod _{i=1}^m}{\left(1-e^{-{\left(\frac{\alpha }{x_i}\right)}^{\beta }}\right)}^{-\lambda }\right]\hfill \\ \hfill & \hfill & \times e^{\left\{-\lambda \left(b_4+{\displaystyle \sum _{i=1}^m}{\left(\frac{\alpha }{x_i}\right)}^{\beta }\right)-\delta {\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\lambda }\right\}}.\hfill \end{array}$$

(20)

It is noticeable that the conditional posterior density of $\delta$ given in (19) is Gamma density with shape parameter $(m+a_3)$ and scale parameter $b_3+{\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\lambda }$. Thus, by implementing any Gamma-generating routine, samples of $\delta$ can be simply generated. In addition, the conditional posterior densities of $\alpha ,$$\beta$ and $\lambda$ cannot be reduced analytically to well-known distributions. Thus, M-H algorithm within Gibbs sampling is implemented to conduct the MCMC methodology; see Tierney [26]. This procedure is illustrated in the following algorithm steps:

• Use the MLEs as the initial value, denoted by ${\widehat{\alpha }}^{\left(0\right)},{\widehat{\beta }}^{\left(0\right)},{\widehat{\delta }}^{\left(0\right)}$ and ${\widehat{\lambda }}^{\left(0\right)}$.
• Set $j=1.$
• Generate ${\delta }^{\left(j\right)}$ from Gamma $\left(m+a_3,b_3+{\displaystyle \sum _{i=1}^m}k\left(R_i+1\right){\left(e^{{\left(\frac{\alpha }{x_i}\right)}^{\beta }}-1\right)}^{-\lambda }\right)$.
• Using M-H algorithm, generate ${\alpha }^{\left(j\right)},$${\beta }^{\left(j\right)}$ and ${\lambda }^{\left(j\right)}$ from ${\pi }_1^{*}\left({\alpha }^{\left(j-1\right)}\right|{\beta }^{\left(j-1\right)},{\delta }^{\left(j\right)},{\lambda }^{\left(j-1\right)},\underline{x}),$${\pi }_2^{*}\left({\beta }^{\left(j-1\right)}\right|{\alpha }^{\left(j-1\right)},{\delta }^{\left(j\right)},{\lambda }^{\left(j-1\right)},\underline{x})$ and ${\pi }_4^{*}\left({\lambda }^{\left(j-1\right)}\right|{\alpha }^{\left(j-1\right)},{\beta }^{\left(j-1\right)},{\delta }^{\left(j\right)},\underline{x})$, respectively, with the normal distributions $N\left({\alpha }^{\left(j-1\right)},var\left(\widehat{\alpha }\right)\right)$, $N\left({\beta }^{\left(j-1\right)},var\left(\widehat{\beta }\right)\right)$ and $N\left({\lambda }^{\left(j-1\right)},var\left(\widehat{\lambda }\right)\right)$.

  (a)

  Generate ${\alpha }^{*}$ from $N\left({\alpha }^{\left(j-1\right)},var\left(\widehat{\alpha }\right)\right),$${\beta }^{*}$ from $N\left({\beta }^{\left(j-1\right)},var\left(\widehat{\beta }\right)\right)$ and ${\lambda }^{*}$ from $N\left({\lambda }^{\left(j-1\right)},var\left(\widehat{\lambda }\right)\right)$.

  (b)

  Evaluate the acceptance probabilities

  $$\begin{array}{ccc}{Q}_{\alpha }\hfill & =& min\left[1,\frac{{\pi }_2^{*}\left({\alpha }^{*}\right|{\beta }^{\left(j-1\right)},{\delta }^{\left(j\right)},{\lambda }^{(j-1)},\underline{x})}{{\pi }_2^{*}\left({\alpha }^{\left(j-1\right)}\right|{\beta }^{\left(j-1\right)},{\delta }^{\left(j\right)},{\lambda }^{(j-1)},\underline{x})}\right],\hfill \\ Q_{\beta }\hfill & =& min\left[1,\frac{{\pi }_2^{*}\left({\beta }^{*}\right|{\alpha }^{\left(j\right)},{\delta }^{\left(j\right)},{\lambda }^{(j-1)},\underline{x})}{{\pi }_2^{*}\left({\beta }^{\left(j-1\right)}\right|{\alpha }^{\left(j\right)},{\delta }^{\left(j\right)},{\lambda }^{(j-1)},\underline{x})}\right],\hfill \\ Q_{\lambda }\hfill & =& min\left[1,\frac{{\pi }_4^{*}\left({\lambda }^{*}\right|{\alpha }^{\left(j\right)},{\beta }^{\left(j\right)},{\delta }^{\left(j\right)},\underline{x})}{{\pi }_4^{*}\left({\lambda }^{\left(j-1\right)}\right|{\alpha }^{\left(j\right)},{\beta }^{\left(j\right)},{\delta }^{\left(j\right)},\underline{x})}\right].\hfill \end{array}$$

  (c)

  Generate a ${\rho }_1,$${\rho }_2$ and ${\rho }_3$ from a Uniform $\left(0,1\right)$ distribution.

  (d)

  If ${\rho }_1<Q_{\alpha }$, accept the proposal and set ${\alpha }^{*}={\alpha }^{\left(j\right)}$, else set ${\alpha }^{\left(j\right)}={\alpha }^{\left(j-1\right)}$.

  (e)

  If ${\rho }_2<Q_{\beta }$, accept the proposal and set ${\beta }^{*}={\beta }^{\left(j\right)}$, else set ${\beta }^{\left(j\right)}={\beta }^{\left(j-1\right)}$.

  (f)

  If ${\rho }_3<Q_{\lambda }$, accept the proposal and set ${\lambda }^{*}={\lambda }^{\left(j\right)}$, else set ${\lambda }^{\left(j\right)}={\lambda }^{\left(j-1\right)}$.

• Compute $r\left(t\right)$ and $h\left(t\right)$ as

  $$r^{\left(j\right)}\left(t\right)=e^{-{\delta }^{\left(j\right)}\left\{{\left(e^{{\left(\frac{{\alpha }^{\left(j\right)}}{x}\right)}^{{\beta }^{\left(j\right)}}}-1\right)}^{-{\lambda }^{\left(j\right)}}\right\}},$$

  $$h^{\left(j\right)}\left(t\right)={\delta }^{\left(j\right)}{\lambda }^{\left(j\right)}{\beta }^{\left(j\right)}{\left({\alpha }^{\left(j\right)}\right)}^{{\beta }^{\left(j\right)}}{x}^{-\left({\beta }^{\left(j\right)}+1\right)}{e}^{-{\lambda }^{\left(j\right)}{\left(\frac{{\alpha }^{\left(j\right)}}{x}\right)}^{{\beta }^{\left(j\right)}}}\left[{\left(1-e^{-{\left(\frac{{\alpha }^{\left(j\right)}}{x}\right)}^{{\beta }^{\left(j\right)}}}\right)}^{-\left({\lambda }^{\left(j\right)}+1\right)}\right].$$
• Set $j=j+1$.
• Reiterate Steps (3)–(6) M times. The first $M_0$ simulated varieties are ignored to remove the affection of the selection of initial value and to guarantee the convergence. Then, the selected sample is ${\alpha }^{\left(j\right)},{\beta }^{\left(j\right)},{\delta }^{\left(j\right)}, {\lambda }^{\left(j\right)},r^{{}^{\left(j\right)}}\left(t\right)$ and $h^{{}^{\left(j\right)}}\left(t\right),$$j=M_0+1,\dots ,M,$ for sufficiently large M, forms an approximate posterior sample which can be used to develop the Bayesian inference.
• Under SE and LINEX loss functions, the approximate Bayes estimate of $\zeta$ (where $\zeta =$$\alpha ,$$\beta ,$$\delta ,$$\lambda ,$$r\left(t\right)$ and $h\left(t\right)$) can be obtained by

  $${\widehat{\zeta }}_{SE}=\frac{1}{M-M_0}{\displaystyle \sum _{j=M_0+1}^M}{\zeta }^{\left(j\right)},{\widehat{\zeta }}_{LE}=\frac{-1}{\epsilon }log\left(\frac{1}{M-M_0}{\displaystyle \sum _{j=M_0+1}^M}{e}^{-\epsilon {\zeta }^{\left(j\right)}}\right),\epsilon \ne 0,$$

  where $M_0$ is the burn-in period and ${\zeta }^{\left(j\right)}=$${\alpha }^{\left(j\right)},$${\beta }^{\left(j\right)},$${\delta }^{\left(j\right)},$${\lambda }^{\left(j\right)},$$r^{\left(j\right)}\left(t\right)$ and $h^{\left(j\right)}\left(t\right)$.
• To compute the CRIs of $\zeta$, order $\left\{{\zeta }^{M_0+1},{\zeta }^{M_0+2},\dots ,{\zeta }^M\right\}$ as $\left\{{\zeta }^{\left[1\right]},{\zeta }^{\left[2\right]},\dots ,{\zeta }^{\left[M-M_0\right]}\right\}$. Then, the $\left(1-\gamma \right)100\%$ CRIs of $\zeta$ can be given by

  $$\left[{\zeta }_{\left(M-M_0\right)\left(\frac{\gamma }{2}\right)},{\zeta }_{\left(M-M_0\right)\left(1-\frac{\gamma }{2}\right)}\right].$$

5. Simulation Study

In this section, to compare the performance of the proposed Bayes estimators with the MLEs and bootstrap, we perform a Monte Carlo simulation study using different combinations of n, m, k and the different censoring scheme $\mathbf{R}$ (different $R_i$ values). Using the algorithm introduced by Balakrishnan and Sandhu [1], with distribution function $1-{\left(1-F\left(x\right)\right)}^k$, we generate PFFC samples from WFrD with the parameters $\left(\alpha ,\beta ,\delta ,\lambda \right)=\left(0.5,1,1.5,5\right)$, respectively. Then, the true values of $r\left(t\right)$ and $h\left(t\right)$ at time $t=0.4$ are evaluated to be $0.984462$ and $0.342945$. The performance of estimators are evaluated in terms of mean square error (MSE) which is computed as $MSE=\frac{1}{N}{\sum }_{i=1}^N{({\widehat{\zeta }}_j^{\left(i\right)}-{\zeta }_j)}^2$, where N = 10,000, $j=1,2,\dots ,6,$${\zeta }_1=\alpha ,$${\zeta }_2=\beta ,$${\zeta }_3=\delta ,$${\zeta }_4=\lambda ,$${\zeta }_5=r\left(t\right)$ and ${\zeta }_6=h\left(t\right)$ for the point estimates and also average lengths (ALs) and coverage probability (CPs) for interval estimates (asymptotic, bootstrap, HPD). Bayes estimates and the CRIs are computed based on 12,000 MCMC samples and discard the first 2000 values as “burn-in”. In addition, we assume the informative Gamma priors for $\alpha ,$$\beta ,$$\delta$ and $\lambda$, that is, when the hyper-parameters are $a_i=1$ and $b_i=2,i=1,2,3,4$. Further, $95\%$ CRIs were computed for each simulated sample. In our study, we consider two different group sizes $k=2,5$ and the following censoring schemes:

CS I: 

$R_1=n-m,$$R_i=0$ for $i\ne 1.$

CS II: 

$R_{(m+1)/2}=n-m,$$R_i=0$ for $i\ne (m+1)/2$ if m odd; $R_{m/2}=n-m,$$R_i=0$ for $i\ne m/2$ if m even.

CS III: 

$R_m=n-m,$$R_i=0$ for $i\ne m.$

All the computations were performed by MATHEMATICA ver. 12. The results of the MSE of the estimates are reported in

Table 1. MSE of estimates for the parameter $\alpha$.

(k, n, m)	SC	MLE	Bootstrap		Bayesian
			Boot-p	Boot-t	SE	LINEX
						$\mathbf{\epsilon }=-\mathbf{0}.\mathbf{5}$	$\mathbf{\epsilon }=\mathbf{0}.\mathbf{5}$
$(2,30,15)$	I	0.00413	0.00384	0.00352	0.00331	0.00313	0.00289
	II	0.00492	0.00493	0.00474	0.00452	0.00425	0.00372
	III	0.00526	0.00530	0.00513	0.00482	0.00467	0.00443
$(2,30,20)$	I	0.00384	0.00371	0.00332	0.00324	0.00274	0.00248
	II	0.00435	0.00429	0.00391	0.00378	0.00346	0.00326
	III	0.00467	0.00462	0.00449	0.00399	0.00378	0.00352
$(2,50,35)$	I	0.00238	0.00236	0.00219	0.00208	0.00201	0.00189
	II	0.00257	0.00249	0.00239	0.00232	0.00228	0.00215
	III	0.00295	0.00288	0.00273	0.00265	0.00259	0.00239
$\left(5,30,15\right)$	I	0.00522	0.00523	0.00458	0.00425	0.00412	0.00399
	II	0.00593	0.00584	0.00529	0.00497	0.00469	0.00448
	III	0.00619	0.00618	0.00586	0.00553	0.00513	0.00494
$\left(5,30,20\right)$	I	0.00459	0.00461	0.00385	0.00367	0.00351	0.00336
	II	0.00472	0.00471	0.00452	0.00436	0.00404	0.00384
	III	0.00515	0.00508	0.00477	0.00454	0.00441	0.00419
$\left(5,50,35\right)$	I	0.00312	0.00299	0.00287	0.00266	0.00259	0.00228
	II	0.00375	0.00368	0.00347	0.00334	0.00325	0.00249
	III	0.00428	0.00421	0.00384	0.00371	0.00364	0.00291

,

Table 2. MSE of estimates for the parameter $\beta$.

(k, n, m)	SC	MLE	Bootstrap		Bayesian
			Boot-p	Boot-t	SE	LINEX
						$\mathbf{\epsilon }=-\mathbf{0}.\mathbf{5}$	$\mathbf{\epsilon }=\mathbf{0}.\mathbf{5}$
$(2,30,15)$	I	0.00799	0.00785	0.00771	0.00753	0.00748	0.00713
	II	0.00825	0.00823	0.00794	0.00775	0.00762	0.00746
	III	0.00886	0.00881	0.00854	0.00834	0.00817	0.00792
$(2,30,20)$	I	0.00722	0.00718	0.00699	0.00675	0.00664	0.00625
	II	0.00763	0.00755	0.00712	0.00699	0.00687	0.00652
	III	0.00796	0.00785	0.00764	0.00733	0.00729	0.00706
$(2,50,35)$	I	0.00561	0.00557	0.00513	0.00498	0.00486	0.00445
	II	0.00615	0.00608	0.00579	0.00564	0.00558	0.00498
	III	0.00669	0.00661	0.00617	0.00599	0.00584	0.00537
$\left(5,30,15\right)$	I	0.00854	0.00847	0.00812	0.00798	0.00782	0.00746
	II	0.00884	0.00878	0.00835	0.00824	0.00823	0.00785
	III	0.00922	0.00918	0.00896	0.00881	0.00867	0.00842
$\left(5,30,20\right)$	I	0.00813	0.00811	0.00785	0.00778	0.00761	0.00713
	II	0.00842	0.00842	0.00814	0.00805	0.00793	0.00762
	III	0.00884	0.00879	0.00856	0.00847	0.00839	0.00786
$\left(5,50,35\right)$	I	0.00751	0.00749	0.00724	0.00705	0.00692	0.00665
	II	0.00792	0.00783	0.00764	0.00751	0.00726	0.00689
	III	0.00833	0.00828	0.00814	0.00783	0.00775	0.00744

,

Table 3. MSE of estimates for the parameter $\delta$.

(k, n, m)	SC	MLE	Bootstrap		Bayesian
			Boot-p	Boot-t	SE	LINEX
						$\mathbf{\epsilon }=-\mathbf{0}.\mathbf{5}$	$\mathbf{\epsilon }=\mathbf{0}.\mathbf{5}$
$(2,30,15)$	I	0.00954	0.00947	0.00912	0.00911	0.00883	0.00856
	II	0.00982	0.00978	0.00954	0.00946	0.00932	0.00897
	III	0.01102	0.01087	0.00986	0.00978	0.00969	0.00942
$(2,30,20)$	I	0.00912	0.00908	0.00883	0.00879	0.00854	0.00817
	II	0.00955	0.00951	0.00942	0.00935	0.00894	0.00876
	III	0.00984	0.00982	0.00961	0.00954	0.00925	0.00913
$(2,50,35)$	I	0.00823	0.00817	0.00806	0.00783	0.00775	0.00737
	II	0.00865	0.00855	0.00839	0.00810	0.00808	0.00777
	III	0.00899	0.00892	0.00875	0.00863	0.00841	0.00813
$\left(5,30,15\right)$	I	0.00994	0.00982	0.00954	0.00949	0.00945	0.00916
	II	0.01243	0.01239	0.00994	0.00975	0.00964	0.00943
	III	0.01354	0.01351	0.01162	0.00993	0.00984	0.00971
$\left(5,30,20\right)$	I	0.00947	0.00945	0.00925	0.00918	0.00914	0.00873
	II	0.00984	0.00985	0.00961	0.00954	0.00943	0.00915
	III	0.01195	0.01179	0.01101	0.00994	0.00982	0.00960
$\left(5,50,35\right)$	I	0.00911	0.00907	0.00883	0.00879	0.00869	0.00826
	II	0.00945	0.00941	0.00926	0.00911	0.00903	0.00874
	III	0.00989	0.00987	0.00962	0.00955	0.00947	0.00914

,

Table 4. MSE of estimates for the parameter $\lambda$.

(k, n, m)	SC	MLE	Bootstrap		Bayesian
			Boot-p	Boot-t	SE	LINEX
						$\mathbf{\epsilon }=-\mathbf{0}.\mathbf{5}$	$\mathbf{\epsilon }=\mathbf{0}.\mathbf{5}$
$(2,30,15)$	I	0.04128	0.04064	0.03944	0.03784	0.03694	0.03351
	II	0.04456	0.04441	0.04291	0.04123	0.04098	0.03756
	III	0.04922	0.04913	0.04621	0.04523	0.04428	0.04154
$(2,30,20)$	I	0.03788	0.03781	0.03599	0.03387	0.03317	0.02999
	II	0.04245	0.04234	0.04045	0.03911	0.03894	0.03541
	III	0.04623	0.04512	0.04459	0.04230	0.04155	0.03864
$(2,50,35)$	I	0.02541	0.02511	0.02317	0.02274	0.02199	0.01884
	II	0.02845	0.02756	0.02547	0.02489	0.02397	0.02103
	III	0.03254	0.03184	0.02987	0.02754	0.02699	0.02480
$\left(5,30,15\right)$	I	0.05233	0.05178	0.04865	0.04796	0.04785	0.04211
	II	0.05536	0.05524	0.05261	0.05012	0.05001	0.04699
	III	0.05866	0.05736	0.05527	0.05324	0.05311	0.04986
$\left(5,30,20\right)$	I	0.04864	0.04799	0.04562	0.04467	0.04413	0.03950
	II	0.05155	0.05132	0.04835	0.04793	0.04681	0.04356
	III	0.05687	0.05646	0.05345	0.05189	0.05099	0.04786
$\left(5,50,35\right)$	I	0.02945	0.02874	0.02614	0.02416	0.02354	0.02147
	II	0.03255	0.03187	0.03002	0.02994	0.02758	0.02540
	III	0.03761	0.03697	0.03354	0.03300	0.03214	0.02893

,

Table 5. MSE of estimates for the parameter $r\left(t\right)$.

(k, n, m)	SC	MLE	Bootstrap		Bayesian
			Boot-p	Boot-t	SE	LINEX
						$\mathbf{\epsilon }=-\mathbf{0}.\mathbf{5}$	$\mathbf{\epsilon }=\mathbf{0}.\mathbf{5}$
$(2,30,15)$	I	0.00224	0.00221	0.00210	0.00199	0.00195	0.00189
	II	0.00245	0.00241	0.00233	0.00211	0.00205	0.00195
	III	0.00271	0.00275	0.00267	0.00255	0.00239	0.00210
$(2,30,20)$	I	0.00201	0.00199	0.00193	0.00191	0.00189	0.00171
	II	0.00235	0.00232	0.00205	0.00203	0.00198	0.00183
	III	0.00253	0.00254	0.00235	0.00224	0.00218	0.00197
$(2,50,35)$	I	0.00159	0.00149	0.00135	0.00131	0.00129	0.00119
	II	0.00176	0.00171	0.00152	0.00149	0.00138	0.00127
	III	0.00202	0.00198	0.00176	0.00169	0.00155	0.00139
$\left(5,30,15\right)$	I	0.00331	0.00332	0.00314	0.00298	0.00298	0.00271
	II	0.00361	0.00361	0.00354	0.00314	0.00314	0.00305
	III	0.00401	0.00411	0.00382	0.00367	0.00365	0.00342
$\left(5,30,20\right)$	I	0.00255	0.00249	0.00235	0.00227	0.00227	0.00199
	II	0.00269	0.00268	0.00254	0.00248	0.00247	0.00212
	III	0.00293	0.00287	0.00278	0.00265	0.00264	0.00240
$\left(5,50,35\right)$	I	0.00198	0.00191	0.00185	0.00185	0.00181	0.00164
	II	0.00221	0.00213	0.00204	0.00190	0.00190	0.00171
	III	0.00251	0.00249	0.00238	0.00238	0.00237	0.00219

and

Table 6. MSE of estimates for the parameter $h\left(t\right)$.

(k, n, m)	SC	MLE	Bootstrap		Bayesian
			Boot-p	Boot-t	SE	LINEX
						$\mathbf{\epsilon }=-\mathbf{0}.\mathbf{5}$	$\mathbf{\epsilon }=\mathbf{0}.\mathbf{5}$
$(2,30,15)$	I	0.00112	0.00110	0.00105	0.00100	0.00099	0.00078
	II	0.00129	0.00125	0.00115	0.00109	0.00106	0.00091
	III	0.00139	0.00138	0.00125	0.00116	0.00111	0.0099
$(2,30,20)$	I	0.00101	0.00101	0.00097	0.00091	0.00091	0.00071
	II	0.00120	0.00122	0.00110	0.00101	0.00101	0.00082
	III	0.00133	0.00131	0.00127	0.00125	0.00115	0.00093
$(2,50,35)$	I	0.00089	0.00088	0.00083	0.00081	0.00082	0.00067
	II	0.00095	0.00094	0.00089	0.00089	0.00090	0.00078
	III	0.00101	0.0099	0.0096	0.00095	0.00095	0.00086
$\left(5,30,15\right)$	I	0.00125	0.00124	0.00121	0.00121	0.00120	0.00191
	II	0.00146	0.00144	0.00136	0.00132	0.00132	0.00119
	III	0.00175	0.00168	0.00154	0.00142	0.00141	0.00128
$\left(5,30,20\right)$	I	0.00115	0.00114	0.00106	0.00106	0.00105	0.00092
	II	0.00123	0.00123	0.00118	0.00117	0.00117	0.00112
	III	0.00145	0.00144	0.00138	0.00137	0.00135	0.00127
$\left(5,50,35\right)$	I	0.00101	0.00101	0.00091	0.00089	0.00089	0.00075
	II	0.00117	0.00116	0.00099	0.00097	0.00095	0.00086
	III	0.00135	0.00133	0.00110	0.00109	0.00108	0.00093

, while the results of the ALs and CPs of the estimates are shown in

Table 7. ALs (first row) and CPs (second row) of $95\%$ ACIs for $\alpha$ and $\beta$.

		$\mathit{\alpha }$				$\mathit{\beta }$
(k, n, m)	SC	MLE	Bootstrap		Bayes	MLE	Bootstrap		Bayes
		ACI	Boot-p	Boot-t	CRI	ACI	Boot-p	Boot-t	CRI
$(2,30,15)$	I	1.1245	1.0899	1.0745	0.9987	2.5263	2.1891	1.9466	1.6469
		0.9254	0.9341	0.9294	0.9466	0.9241	0.9233	0.9456	0.9541
	II	1.2954	1.1123	1.1118	1.0894	2.6451	2.3654	2.1457	1.9546
		0.9355	0.9455	0.9399	0.9471	0.9345	0.9256	0.9388	0.9491
	III	1.3781	1.2457	1.2018	1.1101	2.7491	2.5331	2.2478	2.1564
		0.9289	0.9511	0.9342	0.9522	0.9287	0.9293	0.9238	0.9487
$(2,30,20)$	I	1.1146	1.0647	1.0532	0.9145	2.4136	1.9845	1.7941	1.5342
		0.9426	0.9452	0.9366	0.9688	0.9299	0.9374	0.9461	0.9588
	II	1.2346	1.0914	1.0698	1.0245	2.5461	2.2145	1.9963	1.7645
		0.9234	0.9345	0.9284	0.9544	0.9324	0.9287	0.9356	0.9455
	III	1.2935	1.1141	1.1126	1.0963	2.6314	2.4665	2.1328	1.9746
		0.9194	0.9234	0.9199	0.9763	0.9297	0.9366	0.9388	0.9541
$(2,50,35)$	I	0.9984	0.9832	0.8843	0.7536	2.2146	1.7478	1.5399	1.3457
		0.9148	0.9248	0.9362	0.9612	0.9354	0.9247	0.9455	0.9593
	II	1.0874	1.0228	0.9874	0.8345	2.3498	1.8999	1.6794	1.5617
		0.9541	0.9312	0.9433	0.9542	0.9411	0.9399	0.9541	0.9641
	III	1.2012	1.1099	1.1096	0.9144	2.5666	2.2456	1.8974	1.7224
		0.9258	0.9235	0.9347	0.9366	0.9320	0.9388	0.9421	0.9510
$\left(5,30,15\right)$	I	1.3442	1.2457	1.1157	1.1099	2.8941	2.7462	2.3471	1.9946
		0.9422	0.9461	0.9541	0.9655	0.9294	0.9323	0.9410	0.9500
	II	1.4457	1.3564	1.2434	1.1187	2.9361	2.9002	2.5941	2.2340
		0.9199	0.9278	0.9471	0.9547	0.9342	0.9326	0.9478	0.9544
	III	1.4963	1.4100	1.3210	1.2914	2.9833	2.9541	2.6574	2.4562
		0.9197	0.9236	0.9473	0.9631	0.9411	0.9514	0.9399	0.9481
$\left(5,30,20\right)$	I	1.2354	1.1135	1.1086	1.0894	2.6124	2.5841	2.4189	2.139
		0.9411	0.9345	0.9154	0.9594	0.9399	0.9432	0.9477	0.9521
	II	1.3547	1.2936	1.11365	1.1094	2.7942	2.7155	2.6084	2.3667
		0.9344	0.9247	0.9536	0.9496	0.9412	0.9324	0.9487	0.9527
	III	1.4612	1.3745	1.2364	1.1102	2.8354	2.7941	2.6263	2.4110
		0.9418	0.9510	0.9355	0.9543	0.9147	0.9214	0.9248	0.9392
$\left(5,50,35\right)$	I	1.2031	1.1978	1.1311	0.9965	2.3451	2.22147	2.0145	1.9746
		0.9399	0.9344	0.9255	0.9468	0.9398	0.9322	0.9475	0.9532
	II	1.3124	1.2347	1.2004	1.1139	2.4652	2.3654	2.1657	2.0197
		0.9438	0.9512	0.9478	0.9587	0.9412	0.9539	0.9511	0.9746
	III	1.3924	1.2963	1.2594	1.2077	2.5631	2.4719	2.2948	2.1345
		0.9214	0.9398	0.9455	0.9641	0.9281	0.9278	0.9356	0.9523

,

Table 8. ALs (first row) and CPs (second row) of $95\%$ ACIs for $\delta$ and $\lambda$.

		$\mathit{\delta }$				$\mathit{\lambda }$
(k, n, m)	SC	MLE	Bootstrap		Bayes	MLE	Bootstrap		Bayes
		ACI	Boot-p	Boot-t	CRI	ACI	Boot-p	Boot-t	CRI
$(2,30,15)$	I	3.2478	3.2147	3.1148	2.2462	8.6443	7.5468	7.2631	6.1946
		0.9248	0.9387	0.9347	0.9544	0.9298	0.9248	0.9389	0.9536
	II	3.5247	3.3456	3.2467	2.3233	8.8561	7.7314	7.5615	6.3210
		0.9354	0.9288	0.9412	0.9611	0.9451	0.9381	0.9256	0.9498
	III	3.7356	3.6564	3.4326	2.4854	8.9232	7.8765	7.6694	6.5328
		0.9199	0.9366	0.9255	0.9498	0.9193	0.9288	0.9481	0.9611
$(2,30,20)$	I	3.0124	2.9991	2.8974	1.9998	8.5326	7.2657	6.8364	5.7698
		0.9288	0.9149	0.9258	0.9399	0.9288	0.9266	0.9387	0.9642
	II	3.2321	3.2110	3.1342	2.2110	8.6345	7.4458	7.0326	5.9461
		0.9487	0.9511	0.9347	0.9555	0.9475	0.9354	0.9492	0.9782
	III	3.4298	3.3457	3.2645	2.3214	8.7456	7.5472	7.2884	6.1632
		0.9411	0.9482	0.9541	0.9614	0.9265	0.9199	0.9325	0.9611
$(2,50,35)$	I	2.6312	2.4871	2.4357	1.7245	8.1457	6.8635	6.1473	5.2654
		0.9388	0.9287	0.9571	0.9632	0.9473	0.9355	0.9456	0.952
	II	2.7564	2.5634	2.5166	1.9846	8.2465	6.9984	6.3145	5.4231
		0.9277	0.9237	0.9345	0.9544	0.9498	0.9514	0.9256	0.9558
	III	2.8553	2.7651	2.7033	2.1144	8.3658	7.1984	6.5327	5.5321
		0.9398	0.9412	0.9425	0.9499	0.9354	0.9355	0.9281	0.9521
(k, n, m)	SC	MLE	Bootstrap		Bayes	MLE	Bootstrap		Bayes
		ACI	Boot-p	Boot-t	CRI	ACI	Boot-p	Boot-t	CRI
$\left(5,30,15\right)$	I	3.6178	3.5148	3.3149	2.6460	8.9443	7.8469	7.7632	6.5942
		0.9397	0.9398	0.9475	0.9577	0.9500	0.9344	0.9387	0.9499
	II	3.7123	3.6478	3.5124	2.7341	9.0456	8.0113	7.9635	6.7251
		0.9278	0.9385	0.9422	0.9524	0.9326	0.9487	0.9286	0.9621
	III	3.8254	3.7694	3.6324	2.8211	9.1236	8.2654	8.0321	6.8654
		0.9541	0.9497	0.9514	0.9578	0.9156	0.9236	0.9244	0.9762
$\left(5,30,20\right)$	I	3.3245	3.2568	3.1456	2.2754	8.6624	7.3548	6.9365	5.9699
		0.9199	0.9287	0.9345	0.9487	0.9345	0.9288	0.9356	0.9566
	II	3.4358	3.3457	3.2147	2.3645	8.7635	7.4553	7.1247	6.2354
		0.9315	0.9288	0.9487	0.9514	0.9243	0.9265	0.9199	0.9499
	III	3.5684	3.4935	3.3978	2.4391	8.8365	7.5647	7.3546	6.4321
		0.9484	0.9347	0.3981	0.9577	0.9255	0.9366	0.9478	0.9516
$\left(5,50,35\right)$	I	3.1457	3.1147	2.8224	1.9974	8.3458	7.0636	6.4472	5.5651
		0.9500	0.9487	0.9390	0.9535	0.9366	0.9264	0.9287	0.9578
	II	3.2586	3.2114	3.0051	2.2457	8.4652	7.1879	6.5541	5.6234
		0.9294	0.9411	0.9388	0.9547	0.9392	0.9481	0.9356	0.9614
	III	3.3451	3.3347	3.1963	2.3551	8.5326	7.3365	6.7654	5.7653
		0.9379	0.9299	0.9378	0.9499	0.9421	0.9387	0.9456	0.9526

and

Table 9. ALs (first row) and CPs (second row) of $95\%$ ACIs for $r\left(t\right)$ and $h\left(t\right)$.

		r(t)				h(t)
(k, n, m)	SC	MLE	Bootstrap		Bayes	MLE	Bootstrap		Bayes
		ACI	Boot-p	Boot-t	CRI	ACI	Boot-p	Boot-t	CRI
$(2,30,15)$	I	0.6532	0.6333	0.6154	0.5642	0.3565	0.3354	0.3125	0.2547
		0.9399	0.9421	0.9485	0.9651	0.9410	0.9389	0.9348	0.9587
	II	0.6698	0.6475	0.6235	0.5723	0.3654	0.3462	0.3247	0.2645
		0.9189	0.9245	0.9366	0.9546	0.9243	0.9246	0.9423	0.9641
	III	0.6754	0.6587	0.6378	0.5836	0.3765	0.3564	0.3312	0.2765
		0.9245	0.9289	0.9355	0.9487	0.9123	0.9188	0.9265	0.9499
$(2,30,20)$	I	0.6324	0.6054	0.5732	0.5236	0.3148	0.2987	0.2634	0.2198
		0.9412	0.9493	0.9568	0.9712	0.92478	0.9251	0.9238	0.9514
	II	0.6436	0.6135	0.5846	0.5366	0.3254	0.3158	0.2794	0.2365
		0.9256	0.9245	0.9356	0.9614	0.9245	0.9345	0.9356	0.9651
	III	0.6536	0.6245	0.5964	0.5476	0.3365	0.3298	0.2897	0.2476
		0.9136	0.9254	0.9289	0.9566	0.9298	0.9356	0.9324	0.9558
$(2,50,35)$	I	0.5864	0.5668	0.5463	0.4977	0.2736	0.2699	0.2298	0.1899
		0.9456	0.9345	0.9362	0.9584	0.9314	0.9216	0.9245	0.9499
	II	0.5932	0.5736	0.5559	0.5146	0.2836	0.2796	0.2345	0.1997
		0.9246	0.9258	0.9356	0.9583	0.9256	0.9361	0.9384	0.9544
	III	0.6123	0.5864	0.5678	0.5346	0.2936	0.2874	0.2455	0.2195
		0.9156	0.9184	0.9365	0.9511	0.9199	0.9236	0.9326	0.9562
$\left(5,30,15\right)$	I	0.6835	0.6736	0.6455	0.5947	0.3964	0.3857	0.3526	0.2945
		0.9365	0.9387	0.9314	0.9614	0.9265	0.9366	0.9214	0.9745
	II	0.6998	0.6835	0.6545	0.6014	0.4110	0.3984	0.3676	0.3009
		0.9322	0.9423	0.9411	0.9498	0.9347	0.9348	0.9365	0.9633
	III	0.7136	0.6936	0.6634	0.6201	0.4265	0.4105	0.3765	0.3187
		0.9187	0.9265	0.9199	0.9547	0.9321	0.9354	0.9399	0.9584
$\left(5,30,20\right)$	I	0.6655	0.6355	0.6134	0.5633	0.3623	0.3564	0.3254	0.2551
		0.9256	0.9264	0.9345	0.9582	0.9265	0.9324	0.9478	0.9614
	II	0.6774	0.6468	0.6247	0.5763	0.3765	0.3658	0.3304	0.2669
		0.9288	0.9235	0.9451	0.9547	0.9351	0.9415	0.9387	0.9526
	III	0.6836	0.6523	0.6348	0.5863	0.3864	0.3784	0.3466	0.2863
		0.9284	0.9123	0.9354	0.9645	0.9247	0.9481	0.9369	0.9485
$\left(5,50,35\right)$	I	0.6168	0.5967	0.5763	0.5275	0.3265	0.3198	0.2796	0.2187
		0.9471	0.9395	0.9455	0.9610	0.9512	0.9498	0.9399	0.9587
	II	0.6258	0.6145	0.5889	0.5302	0.3359	0.3257	0.2900	0.2341
		0.9345	0.9289	0.9356	0.9584	0.9245	0.9355	0.9387	0.9612
	III	0.6321	0.6235	0.5994	0.5467	0.3462	0.3364	0.3166	0.2499
		0.9263	0.9265	0.9371	0.9498	0.9187	0.9325	0.9289	0.9499

.

From the results in Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, we observe the following:

• As expected, from Table 1, Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, as sample sizes $\left(n,m\right)$ increase, the MSEs and ALs decrease.
• The Bayes estimates have the smallest MSEs and ALs for the unknown quantities $\zeta =\alpha$, $\beta$, $\delta$, $\lambda ,$$r\left(t\right)$ and $h\left(t\right)$. Hence, Bayes estimates performed better than the MLEs and bootstrap methods.
• The Bootstrap methods performed better than the ML method of $\zeta$ in terms of the MSEs and ALs. Moreover, boot-t performs better than boot-p in terms of the MSEs and ALs.
• The Bayes estimate under LINEX with $\epsilon =0.5$ provides better estimates for $\zeta$ because of having the smallest MSEs.
• The Bayes estimates of $\zeta$ under LINEX for the choice $\epsilon =0.5$ performed better than their estimates for the choice $\epsilon =-0.5$ in the sense of having smaller MSEs.
• From all of the tables, we observe that as the group size k increases, the MSEs and ALs associated with $\zeta$ increase.
• For fixed sample sizes and observed failures, the first scheme I is the best scheme in the sense of having smaller MSEs and ALs.
• The MLE, bootstrap and Bayesian methods have very close estimates and their ACIs have quite high CPs (around $0.95$). Moreover, the Bayesian CRIs have the highest CPs.

In general, we can conclude that the best estimation method is the Bayes method using the asymmetric loss function, especially if the prior information about the problem under study is available. If the prior information about the problem under study is not available, it is preferable to use bootstrap methods that depend primarily on the maximum likelihood estimates.

6. Applications of COVID-19 Data

To clarify the inference methods discussed in the previous sections, we present an application of real-life data for the coronavirus. We consider two sets of real-life data, which were reported in Almongy et al. [27]. The first set represents COVID-19 mortality rate data belonging to Mexico over 108 days, recorded from 4 March to 20 July 2020. The second set represents COVID-19 data belonging to the Netherlands over 30 days, recorded from 31 March to 30 April 2020. The initial density shape is explored using the nonparametric kernel density (KD) estimation approach in Figure 1 and Figure 2, and the density is asymmetric and unimodal (bimodal) functions for data sets I and II, respectively, are noted. The normality condition and hazard rate shapes are tested via the quantile–quantile (QQ) and TTT plots in Figure 1 and Figure 2. The extremes are spotted from the box and violin plots in Figure 1 and Figure 2, and it is showed that some extreme observations were found.

For the purpose of the goodness-of-fit test, the K-S distance between the empirical and the fitted distribution functions have been computed. It is $0.05698$ and the associated p-value is $0.8744$ for the first set, while it is $0.09230$ and the associated p-value is $0.93963$ for the second set. Hence, the p-value for K-S has the highest value for the first and second sets. This leads us to conclude that the WFr model is the best fit for the two real data sets. Empirical plots are shown in Figure 3, which make it clear that the WFr model fits the data very well. Figure 4 and Figure 5 show that the ML estimators are unique.

It was found that the log-likelihood function is unimodal-shaped for each parameter.

The first set consisting of 108 survival times for 108 days. The data are randomly divided into 27 groups with $k=4$ items within each group. The groups can be divided as follows: {1.041, 2.988, 4.344, 8.826}, {4.424, 5.242, 9.284, 10.855}, {6.105, 7.630, 7.903, 14.604}, {1.402, 6.327, 9.391, 14.962}, {5.143, 7.840, 9.550, 9.935}, {1.800, 6.015, 7.267, 13.220}, {3.286, 3.537, 6.370, 6.968}, {1.815, 4.949, 5.406, 6.182}, {1.867, 2.601, 2.926, 3.218}, {3.228, 7.854, 8.325, 8.551}, {1.923, 3.233, 3.257, 3.632}, {6.697, 6.560, 7.151, 8.108}, {2.077, 4.292, 4.557, 5.442}, {2.326, 2.500, 5.317, 6.535}, {3.298, 4.648, 4.697, 5.459}, {2.438, 6.122, 10.035, 10.685}, {3.440, 5.854, 6.656, 8.813}, {2.352, 5.985, 7.260, 10.043}, {3.215, 4.730, 11.665, 16.498}, {4.661, 4.909, 6.140, 6.625}, {3.499, 3.751, 6.814, 7.486}, {3.395, 6.412, 8.696, 12.042}, {2.070, 3.336, 10.158, 12.878}, {2.506, 2.838, 3.029, 5.392}, {2.065, 3.327, 3.359, 4.089}, {1.205, 3.027, 3.778, 10.383}, {2.058, 3.219, 3.922, 4.120}.

A PFFC sample, say $x^{\left(1\right)},$ is obtained as follows: from the original data, the items are divided into $n=27$ (number of groups) based on $k=4$ (number of items within each group) with $R=(2,0,2,0,2,0,2,0,2,0,2,0,2)$ (13 first-failure times are observed $(m=13)$ and 14 groups are removed). The result in the obtained PFFC sample is $x^{\left(1\right)}=$ (1.041, 1.402, 1.800, 1.815, 1.867, 1.923, 2.058, 2.065, 2.070, 2.077, 2.326, 2.352, 2.438).

The second set consists of 30 survival times for 30 days. The data are randomly divided into 15 groups with $k=2$ items within each group. The groups can be divided as follows: {1.273, 6.027}, {10.656, 12.274}, {1.974, 4.960}, {5.555, 7.584}, {3.883, 4.462}, {4.235, 5.307}, {7.968, 13.211}, {3.611, 3.647}, {6.940, 7.498}, {5.928, 7.099}, {2.254, 5.431}, {10.289, 10.832}, {4.097, 5.048}, {1.416, 2.857}, {3.461, 14.918}.

Suppose that a PFFC scheme is given by $R=(2,1,1,1,1,1,1)$; then, a PFFC sample of size 7 out of 15 groups of data is obtained as $x^{\left(2\right)}=$ (1.273, 1.974, 2.254, 3.461, 3.611, 4.097, 4.235).

Under the previous data $x^{\left(1\right)}$ and $x^{\left(2\right)}$, the MLEs and bootstrap estimates of $\alpha ,$$\beta ,$$\delta ,$$\lambda ,$$r\left(t\right)$ and $h\left(t\right)$ are determined to be as in

Table 10. Point estimates of $\alpha ,$$\beta ,$$\delta ,$$\lambda ,$$r\left(t\right)$ and $h\left(t\right)$.

Data	(k, n, m)	Parameter	MLE	Bootstrap		Bayesian
				Boot-p	Boot-t	SE	LINEX
							$\mathbf{\epsilon }=-\mathbf{0}.\mathbf{5}$	$\mathbf{\epsilon }=\mathbf{0}.\mathbf{5}$
$x^{\left(1\right)}$	$(4,27,13)$	$\alpha$	2.8565	2.6784	2.5899	2.7112	2.7047	2.4999
		$\beta$	1.5481	1.4977	1.4723	1.5189	1.4566	1.4119
		$\delta$	0.9851	0.9963	0.9664	0.9547	0.9687	0.9489
		$\lambda$	0.3532	0.3751	0.3465	0.3398	0.3456	0.3297
		$r\left(0.4\right)$	0.9997	0.9767	0.9567	0.9767	0.9744	0.9653
		$h\left(0.4\right)$	0.0041	0.0045	0.0039	0.0042	0.0043	0.0038
$x^{\left(2\right)}$	$(2,15,7)$	$\alpha$	1.5034	1.5149	1.4957	1.4768	1.4722	1.4701
		$\beta$	5.0484	5.1556	4.9974	4.9643	4.9107	4.8777
		$\delta$	0.0504	0.0506	0.0501	0.0497	0.0499	0.04875
		$\lambda$	0.5779	0.6144	0.5997	0.5668	0.5757	0.5638
		$r\left(0.4\right)$	0.9987	0.9887	0.9767	0.9856	0.9777	0.9686
		$h\left(0.4\right)$	0.0031	0.0027	0.0025	0.0024	0.0023	0.0021

. Moreover,

Table 11. The $95\%$ ACIs and CRIs of $\alpha ,$$\beta ,$$\delta ,$$\lambda ,$$r\left(t\right)$ and $h\left(t\right)$ based on $x^{\left(1\right)}$.

Method	$\mathit{\alpha }$		$\mathit{\beta }$		$\mathit{\delta }$
	Interval	Length	Interval	Length	Interval	Length
$ACI$	[1.2485,4.4645]	3.2160	[0.0523,8.1486]	8.0963	[0.1205,3.0907]	2.9702
$boot-p$	[1.3234,4.7217]	3.3983	[0.2357,7.6541]	7.4184	[0.2456,3.2641]	3.0185
$boot-t$	[0.9984,4.1568]	3.1584	[0.0584,6.7542]	6.6958	[0.0874,2.8456]	2.7582
$CRI$	[1.0749,4.2559]	3.1810	[0.1425,5.3622]	5.2197	[0.1005,2.5312]	2.4307
	$\mathbf{\lambda }$		$\mathit{r}\left(\mathit{t}\right)$		$\mathit{h}\left(\mathit{t}\right)$
	Interval	Length	Interval	Length	Interval	Length
$ACI$	[0.0485,0.6579]	0.6094	[0.9918,1.0076]	0.0158	[0.0008,1.0076]	1.0068
$boot-p$	[0.0524,0.6149]	0.5625	[0.9484,1.0099]	0.0615	[0.0009,0.9456]	0.9447
$boot-t$	[0.0147,0.6037]	0.5890	[0.9473,1.0087]	0.0614	[0.0007,0.9249]	0.9242
$CRI$	[0.0099,0.5831]	0.5732	[0.9402,1.0035]	0.0633	[0.0004,0.9189]	0.9185

and

Table 12. The $95\%$ ACIs and CRIs of $\alpha ,$$\beta ,$$\delta ,$$\lambda ,$$r\left(t\right)$ and $h\left(t\right)$ based on $x^{\left(2\right)}$.

Method	$\mathit{\alpha }$		$\mathit{\beta }$		$\mathit{\delta }$
	Interval	Length	Interval	Length	Interval	Length
$ACI$	[0.6567,2.3502]	1.6935	[1.2560,9.3527]	8.0967	[0.0017,0.2025]	0.2008
$boot-p$	[0.6241,2.2633]	1.6392	[1.5476,9.8475]	8.2999	[0.0014,0.2361]	0.2347
$boot-t$	[0.5147,2.1685]	1.6538	[1.8947,9.7566]	7.8619	[0.0011,0.2287]	0.2276
$CRI$	[0.4566,2.0472]	1.5906	[1.9965,8.1253]	6.1288	[0.0009,0.2154]	0.2145
	$\mathbf{\lambda }$		$\mathit{r}\left(\mathit{t}\right)$		$\mathit{h}\left(\mathit{t}\right)$
	Interval	Length	Interval	Length	Interval	Length
$ACI$	[0.0826,1.9817]	1.8991	[0.9517,1.0023]	0.0506	[0.0005,1.0047]	1.0042
$boot-p$	[0.1145,2.1475]	2.0330	[0.9454,1.0025]	0.0571	[0.0003,0.9599]	0.9596
$boot-t$	[0.0967,1.9987]	1.9020	[0.9321,0.9997]	0.0676	[0.0003,0.9467]	0.9464
$CRI$	[0.0845,1.9632]	1.8787	[0.9254,0.9873]	0.0619	[0.0002,0.9354]	0.9352

show the $95\%$ ACIs for $\alpha ,$$\beta ,$$\delta ,$$\lambda ,$$r\left(t\right)$ and $h\left(t\right)$. Now, to compute the Bayesian estimates, the prior distributions of the parameters need to be specified. Because we have no prior information, we assume the noninformative Gamma priors for $\alpha ,$$\beta ,$$\delta$ and $\lambda$, that is, when the hyper-parameters are $a_i=0.0001$ and $b_i=0.0001,$$i=1,2,3,4$. Under the MCMC technique, the posterior analysis was performed across the combined M-H algorithm within the Gibbs sampler. To conduct the MCMC algorithm, which was described in Section 4, the initial values for the parameters $\alpha ,$$\beta ,$$\delta$ and $\lambda$ were taken to be their MLEs. In addition, 12,000 MCMC samples were generated. To avoid the effect of the initial values (starting point), we expunge the first 2000 samples as “burn-in”. Table 10, Table 11 and Table 12 show the Bayesian estimates as well as the $95\%$ CRIs for $\alpha ,$$\beta ,$$\delta ,$$\lambda ,$$r\left(t\right)$ and $h\left(t\right)$.

From the results in Table 10, Table 11 and Table 12, we observe the following:

• The model under study fits the data very well, and this is quite clear from the figures.
• The estimation of the four methods is somewhat similar, with very slight differences, which gives a good impression to the reader.
• Approximate confidence intervals are good, and all point estimates fall within them. In addition, there are slight differences in the lengths of the intervals, as expected.

7. Conclusions

The main aim of this paper was to develop different methods to estimate the unknown quantities $\zeta =\alpha$, $\beta$, $\delta$, $\lambda ,$$r\left(t\right)$ and $h\left(t\right)$ of the WFrD based on a PFFC scheme, which was introduced by Wu and Kuş [8]. The MLEs as well as the ACIs using asymptotic distributions were obtained. Furthermore, to obtain the CIs of the reliability and hazard functions, we used the delta method. Two parametric bootstrap procedures (boot-p and boot-t) were discussed to obtain more widely used confidence intervals. It is clear that, after studying the Bayesian estimates, the posterior distribution equations of the unknown quantities are complicated and so hard to reduce analytically to well-known forms. For this reason, we applied the MCMC method to compute the Bayes estimators. The Bayes estimates were computed under both the SE and LINEX loss functions. For illustrative purposes, two real data sets of COVID-19 mortality rates were considered. To check and compare the performance of the proposed methods, a simulation study was implemented with different sample sizes $\left(n,m,k\right)$ and different CSs $(I,II,III)$.