Analysis Simple Step Stress Model under Competing Extension Weibull Failure Distribution Based on Progressive Type-II Censoring

Analysis Simple Step Stress Model under Competing Extension Weibull Failure Distribution Based on Progressive Type-II Censoring

Abdallah M. Abdelfattah El-Sayed A. El-Sherpieny Osama F. Khalil* 

Faculty of Graduate Studies for Statistical Research (FGSSR), Cairo University, Giza 12613, Egypt

Corresponding Author Email: 
ousama_fathy2020@yahoo.com
Page: 
93-108
|
DOI: 
https://doi.org/10.18280/mmep.100111
Received: 
29 November 2022
|
Revised: 
19 January 2023
|
Accepted: 
3 February 2023
|
Available online: 
28 February 2023
| Citation

© 2023 IIETA. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).

OPEN ACCESS

Abstract: 

Accelerated life testing (ALT), a procedure utilized in reliability analysis, allows testing units to be subject to increasingly elevated grades of stress during an experiment. Step-stress tests are a subclass of accelerated tests in which the stress levels rise consecutively at prearranged cycles, consequently, the researcher might find out results more swiftly than in ordinary working settings about the parameter of the lifetime distribution. Moreover, there are frequently multiple fatal causes for a test element's failure, for instance, technical or electric. These causes are recognized as "competing risks". The purpose of the analysis is to assess simple step stress accelerated life testing (SS-ALT) with competing Risks originating from the extension of Weibull distribution by applying a progressive Type-II censoring scheme. In this case, under the assumption of a cumulative exposure model, the authors successfully obtained the Bayes estimates (BEs) and maximum likelihood estimators (MLEs) of the undetermined average parameters of the various causes. For Bayesian computations, the squared error loss functions are considered. Additionally, the estimators' asymptotic variance-covariance matrix was created. Additionally, credible intervals and asymptotic confidence intervals (CIs) are provided. For a large sample size, the CIs of the unidentified parameters are developed. A numerical study is also involved to exhibit the accuracy and variability of various estimators for several sample sizes. An example is being used to exemplify the inference method that's also considered here. This study concludes that the mean lengths of credible intervals and asymptotic confidence intervals get shorter as the number of failures rises. The credible interval technique is suggested, nevertheless.

Keywords: 

competing risks, progressive censored, maximum likelihood estimation, Bayesian estimation, simple step-stress accelerated life tests

1. Introduction

Due to continual improvement in manufacturing design, it is more difficult to obtain information about the lifetime of products or materials with high reliability at the time of testing under normal conditions. This makes lifetime testing under these conditions very costive and takes a long time. To get information about the lifetime distribution of these materials, a sample of these materials is subjected to more severe operation conditions than normal ones. These conditions are called stresses which may be in the form of temperature, voltage, pressure, vibration, cycling rate, load, etc. This kind of testing is called an accelerated life test (ALT), where products are put under stresses higher than usual to yield more failure data in a short time. Furthermore, it has a wide spread use in (i) Materials, which include metals, plastics, rubber and elastics, concrete and cement, ceramics, and building materials. (ii) Products, which include semiconductors, microelectronics, capacitors, electrical devices, and mechanical components. (iii) Degradation mechanism, life fatigue, creep, and cracking. For more details about the above uses of ALT, see studies [1-3].

There are mainly three ALT methods. The first method is called the constant stress accelerated life test (CS-ALT); the second one is referred to as the step stress accelerated life test (SS-ALT) and the third is the progressive stress accelerated life test (PS-ALT). The first method is used when the stress remains unchanged so that if the stress is weak, the test has to last for a long time. But, the other two methods can reduce the testing time and save a lot of manpower, material sources, and money.

The major assumption of ALT is that the observable behavior under accelerated conditions can be related to the behavior under normal use conditions through a mathematical model called an acceleration model. Thus, life tests conducted under accelerated conditions can be used to make inferences about the behavior of a device in normal use conditions. So, it is necessary to consider the relationship between one or more than one parameter of the failure distribution and the accelerated conditions. The main difficulty of ALT lies in using the failure data obtained at higher conditions to predict the reliability, mean life or other quantities under normal use conditions. The acceleration model is then used to extrapolate the reliability performance to the normal use conditions. Where these different types of methods met a lot of researchers and for more details, you can look at the researches [4-15].

Regarding time and expense reductions, censoring is commonly utilized in life tests. There are various patterns of censoring. Type-I and type-II censoring, whose the first is censored at a particular time and the last is censored at a stable number, are the two most often used censoring techniques. Furthermore, at distinct stages of the research, it can be vital to eradicate a number of test units for a diversity of reasons. Progressive censoring will be concluded from this. Utilizing the resources available is exceedingly efficient and effective with progressive censoring Type-II. Consistent with the research [16]. It enables be summarized as putting n identical units through a life test at zero time. The surviving items th, Rh are randomly omitted when the time of hth(h=1,2,…, m-1) failure occurs. While waiting for the m th failure tm to be noticed, the test is terminated, and the remaining Rm=n-$m-\sum_{h=1}^{m-1} R_h$ elements are all omitted, where m(m<n) and Rh are pre-fixed. Conventional Type-II censoring is a special case when $R_1=R_2=\cdots=R_{m-1}=0$ and Rm=n-m. Many authors have investigated competing risk data under this censoring strategy from other distributions; for more data [17-19], and others.

Based upon a reliability analysis, more than one fatal risk factor frequently contributes to a product's failure, as the inner structure and outer environmental are mutually complex. For instance, shaft or bearing failures may be linked to a bearing assembly failure. In actuality, the failure causes could be dependent or independent. Research [20] indicates that there is a difficulty with the underlying model's identifiability, even though a dependent risk temple would be further actual. The hypothesis of s-independent risks [21, 22], cannot be tested without the knowledge of the covariates. consequently, the failure causes are naturally expected to be independent of the objectives of exploring a competing risk model. One way to conceptualize a multi-component series system is as a prototype of an independent competing risk. Many investigators have explored competing risk models under the assumption that competing failure causes are independent. The analysis of ALT when more than one cause of failure is expressible was stated by Klein and Basu [23, 24]. The SS-ALT and CS-ALT in part have drawn the consideration of many researchers in realistic uses by employing life tests of several types with data from competing risks through different censored schemes and more clarifications can be seen in researches [25-41].

The Weibull model is superb at creating real phenomena with monotonous failure rates. However, the Weibull model should not be used for data with non-monotonous failure rates. The bathtub-shaped failure rate is one of the more practical non-monotonous failure rate functions, and it is used in a variety of literary contexts. For instance, in reliability engineering, it is observed that the lifecycle of an electronic constituent has a failure rate function with a bathtub shape, and also bio-analysis for the human death rate.

Chen [42] who examined a lifetime distribution with two-parameter with neither rising nor bathtub-shaped failure rates, inspected both possibilities. Its cumulative distribution function (CDF) is

$F(t)=1-\exp \left(\vartheta\left(1-e^{t^\eta}\right)\right) ; t>0, \quad \vartheta, \eta>0$       (1)

where, $\eta>0$ is the shape parameter and $\vartheta>0$ is the scale parameter. The equivalent survival function is

$S(t)=\exp \left(\vartheta\left(1-e^{t^\eta}\right)\right) ; \quad t>0$      (2)

The probability density function (pdf) is

$f(t)=\vartheta \eta t^{\eta-1} \exp \left(\vartheta\left(1-e^{t^\eta}\right)\right) ; t>0, \vartheta, \eta>0$     (3)

This article could be designed as exhibited further down. The extension Weibull distribution is presented as a lifetime model in Section 2, along with a description of the model. The ML method is employed in Section 3 to represent point estimates of the parameters for the expansion Weibull distribution under simple SS-ALT using progressively censored competing risks data. The asymptotic variance and covariance matrix are explored in section 4. In Section 5, the Bayesian approach for estimating the unknown parameters is derived. The simulation studies and the illustrative example used to illustrate the theoretical findings are explained in Section 6. Finally, section 7 deals with the findings and conclusions.

2. Description of the Model

Suppose that $S_0$ points to the normal level of stress. Assume $n$ identical units starting on SS-ALT beneath the premier level of stress $S_1\left(S_1>S_0\right)$. As well as information on which risk factor triggered each failure, the number of times it occurred successively is recorded. At a pre-specific time $\tau \in(0, \infty)$, there is a rise in the level of stress from $S_1$ up to $S_2$ and the life test maintains until the $m t h$ ( $m$ is pre-determined) failure is noticed. At the $h^{\text {th }}(1 \leq h \leq m-1)$ failure time, $R_h$ of the surviving elements are omitted, where $R_h$ is pre-specific and $R_m=n-m-\sum_{h=1}^{m-1} R_h$.

1. Only a single of two independent causes compete for failures with lifetimes T1 and T2 could cause a product to fail. Thus, the failure time for products T=min(T1, T2).

2. The lifetime of the cth failure cause Tlc follows an extension Weibull distribution with parameter scale ϑlc, the recognized shape parameter ηlc and l, c=1,2, under the level of stress Sl.

3. Under several levels of stress, the failure mechanisms are the same, i.e., η1c =η2c =η.

4. Following Nelson's cumulative exposure model (CEM) [3], the cumulative distribution exposure function of random variable T for simple SS-ALTs with one stress level change appears as the following:​

$F(t)=\left\{\begin{array}{lr}F_1(t) & 0<t<\tau \\ F_2\left(t-\tau-u_1\right) & t \geq \tau\end{array}\right.$

Consequently, the pdf and CDF of the lifetime $T_{l c}$ could be calculated according to the formula:

$\begin{aligned} & F_c(t) =\left\{\begin{array}{l}1-e^{\left[\vartheta_{1 c}\left(1-e^{t {\eta_c}}\right)\right] ;} \quad 0<t<\tau \\ 1-e^{\left[\vartheta_{2 c}\left(1-e^{t \eta_c-\tau \eta_c}\right)+\vartheta_{1 c}\left(1-e^{\tau \eta_c}\right)\right]} \quad; \quad t \geq \tau\end{array}\right.\end{aligned}$       (4)

and

$\begin{aligned} & f_c(t) = \begin{cases}\eta_c \vartheta_{1 c} t^{\eta_c-1} e^{\left[t^{\eta_c}+\vartheta_{1 c}\left(1-e^{\eta_c}\right)\right]} \quad; & 0<t<\tau \\ \eta_c \vartheta_{2 c} t^{\eta_c-1} e^{\left[\left(t^{\eta_c}-\tau^{\eta_c}\right)+\vartheta_{2 c}\left(1-e^{t^{\eta_{c-}}-\tau^{\eta_c}}\right)+\vartheta_{1 c}\left(1-e^{\left.\tau^{\eta_c}\right)}\right]\right.} \quad; t \geq \tau\end{cases} \\ & \end{aligned}$    (5)

5. The log-linear accelerated function (AF) of the cth cause of failure:

$\log \vartheta_{l c}=a_c+b_c \varphi\left(s_l\right)$                  (6)

whereas, ac, bc>0 are unidentified parameters, The provided decline function of the level of stress s is known as φ(s). In this research, the Arrhenius model is applied, so φ(s)=s-1.

For c=1,2. Since we will observe only the smaller of T1 and T2, let T =min(T1, T2) refer to the overall failure time of a test unit. Then, its CDF and PDF are easily achieved to be:

$F_T(t)=1-\left(1-G_1(t)\right)\left(1-G_2(t)\right)$

$F_T(t)$$=\left\{\begin{array}{cc}1-e^{\left[\vartheta_{11}\left(1-e^{t^{\eta_1}}\right)+\vartheta_{12}\left(1-e^{t^{\eta_2}}\right)\right]} \quad; & 0<t<\tau \\ 1-e^{\left[\vartheta_{21}\left(1-e^{t^{\eta_1}- \tau^{\eta_1}}\right)+\vartheta_{11}\left(1-e ^{\tau^{\eta_1}}\right)+\vartheta_{22}\left(1-e^{t^{\eta_2-} \tau^{\eta_2}}\right)+\vartheta_{12}\left(1-e^{\tau ^{\eta_2}}\right)\right]} \quad; t \geq \tau\end{array}\right.$     (7)

$f_T(t)$$=\left\{\begin{array}{c}{\left[\eta_1 \vartheta_{11} t^{\eta_1-1} e^{\left(t ^{\eta_1}\right)}+\eta_2 \vartheta_{12} t^{\eta_2-1} e^{\left(t^{\eta_2}\right)}\right]} \\e^{\left[\vartheta_{11}\left(1-e^{t^{\eta_1}}\right)+\vartheta_{12}\left(1-e^{t^{\eta_2}}\right)\right]}\quad ; \quad 0<t<\tau \\ {\left[\eta_1 \vartheta_{21} t^{\eta_1-1} e^{\left(t^{\eta_1-} \tau^{{\eta_1}}\right)}+\eta_2 \vartheta_{22} t^{\eta_2-1} e^{\left(t^{\eta_2-}\tau^{{\eta_2}}\right)}\right]} \\ e^{\left[\vartheta_{21}\left(1-e^{t^{\eta_1-} \tau^{\eta_1}}\right)+\vartheta_{11}\left(1-e^{\tau^{\eta_1}}\right)+\vartheta_{22}\left(1-e^{t^{\eta_{2-}} \tau^{\eta_2}}\right)+\vartheta_{12}\left(1-e^{\tau^{\eta_2}}\right)\right]}\quad ; \quad t \geq \tau\end{array}\right.$    (8)

Let $\xi$ be the indicator of the failure cause, then we derive the joint PDF of $(T, \xi)$ as:

$f_{T, \xi}(t)=g_c(t)\left(1-G_{c^{\prime}}(t)\right)$

$f_T(t)=\left\{\begin{array}{c}{\left[\eta_1 \vartheta_{11} t^{\eta_1-1} e^{\left(t \eta_1\right)}+\eta_2 \vartheta_{12} t^{\eta_2-1} e^{\left(t^{\eta_2}\right)}\right]} \\ e^{\left[\vartheta_{11}\left(1-e^{t^{\eta_1}}\right)+\vartheta_{12}\left(1-e^{t^{\eta_2}}\right)\right]}\quad ; \quad 0<t<\tau \\ {\left[\eta_1 \vartheta_{21} t^{\eta_1-1} e^{\left(t^{\eta_1}-\tau^{\eta_1}\right)}+\eta_2 \vartheta_{22} t^{\eta_2-1} e^{\left(t^{ \eta_2}-\tau^{\eta_2}\right)}\right]} \\ e^{\left[\vartheta_{21}\left(1-e^{t^{\eta_1}-\tau^{\eta_1}}\right)+\vartheta_{11}\left(1-e^{\tau^{\eta_1}}\right)+\vartheta_{22}\left(1-e^{t^{\eta_2}-\tau^{\eta_2}}\right)+\vartheta_{12}\left(1-e^{\tau^{\eta_2}}\right)\right]} \quad; t \geq \tau\end{array}\right.$     (9)

for c, c'=1,2 and cc'.

3. Maximum Likelihood Estimation

Recognizing that there are N1 failures prior to changing the stress time τ. When we point out η1c and η2c that signifies the sum of failures associated with a failure caused c under the level of stress s1 and s2, in turn, so N1=n11+n12 is the sum of failures under the level of stress s1 and m- N1= N2 n21+n22 is the sum of failures under the level of stress s2. Given that the resultant cause of failure takes place contained by each failure time, let ξ=(ξ1,ξ2,…,ξm )characterize the observed failure cause indicator series relating to the recorded failure time t=(t1, t2,…, tm). The likelihood function is then constructed using the progressively censoring scheme R1, R2 …, Rm and presupposition 4 as shown by Balakrishnan and Aggarwala [16].

$L\left(\frac{\psi_c}{t}\right) \propto \prod_{h=1}^{N_1} f_1\left(t_h, \xi_h\right)\left[1-F\left(t_h\right)\right]^{R_h}$

$\prod_{h=N_1+1}^m f_2\left(t_h, \xi_h\right)\left[1-F\left(t_h\right)\right]^{R_h}$              (10)

Then

$L\left(\psi_c / t\right) \propto U_1 U_2 \exp \left[\sum_{h=1}^{N_1} t_h^{\eta_c}+\sum_{h=N_1+1}^m\left(t_h^{\eta_c}-\tau^{\eta_c}\right)+\sum_{c=1}^2 \vartheta_{1 c}\left(U_{1 c}+U_{2 c}\right)+\sum_{c=1}^2 \vartheta_{2 c} U_{3 c}\right]$     (11)

where,

$U_1=\prod_{i, c=1}^2\left[\eta_c^{n_{i c}} \vartheta_{i c}^{n_{i c}}\right], U_2=\prod_{h=1}^m t_h^{\eta_c-1}$,

$U_{1 c}=\sum_{h=1}^{N_1}\left(1+R_h\right)\left(1-e^{t_h^{\eta_c}}\right)$,

$E_{2 c}=\sum_{h=N_1+1}^m\left(1+R_h\right)\left\{e^{\left(t_h^{\eta_c}-\tau^{\eta_c}\right)}\left[t_h^{\eta_c} \log t_h\left(\log t_h+\log t_h t_h^{\eta_c}-\tau^{\eta_c} \log \tau\right)-\log \tau \tau^{\eta_c}\left(\log \tau+t_h^{\eta_c} \log t_h-\tau^{\eta_c} \log \tau\right)\right]\right\}$. 

Utilizing the likelihood function (11), henceforth the MLE of ψc=(ϑ1c, ϑ2c) and c=1,2. Thus, to estimate ϑlc, we may fairly presume that ϑlc≥1(l, c=1,2), i.e., At least one failure must be observed under each failure caused by each level of stress. The MLEs of the parameters for Eq. (11) are got by maximizing the logarithm of the likelihood function stated as:

$\begin{aligned} \log L & \propto \sum_{i, c=1}^2 n_{i c}\left(\log \eta_c+\log \vartheta_{i c}\right) \\ & +\left(\eta_c-1\right) \sum_{h=1}^m \log \left(t_h\right)+\sum_{h=1}^{N_1} t_h^{\eta_c} \\ & +\sum_{h=N_1+1}^m\left(t_h^{\eta_c}-\tau^{\eta_c}\right)+\sum_{c=1}^2 \vartheta_{1 c}\left(U_{1 c}+U_{2 c}\right) \\ & +\sum_{c=1}^2 \vartheta_{2 c} U_{3 c}\end{aligned}$        (12)

The log-likelihood function's first partial derivative with respect to the parameters $\psi_c=\left(\eta_c, \vartheta_{1 c}, \vartheta_{2 c}\right)$ and $c=1,2$ respectively as follows:

$\begin{aligned} & \frac{\partial \log L}{\partial \eta_c}=\hat{\eta}_c^{-1} \sum_i^2 n_{i c}+Q^*+\sum_{h=N_1+1}^m Q_{h c}-\hat{\vartheta}_{1 c}\left[Q_{1 c}+Q_{2 c}\right]-\hat{\vartheta}_{2 c} Q_{3 c}=0\end{aligned}$    (13)

$\frac{\partial \log L}{\partial \vartheta_{1 c}}=n_{1 c} \hat{\vartheta}_{1 c}^{-1}+U_{1 c}+U_{2 c}=0$     (14)

and

$\frac{\partial \log L}{\partial \vartheta_{2 c}}=n_{2 c} \hat{\vartheta}_{2 c}^{-1}+U_{3 c}=0$      (15)

where, $Q^*=\sum_{h=1}^m \log \left(t_h\right)+\sum_{h=1}^{N_1} t_h^{\hat{\eta}_c} \log t_h$,

$Q_{h c}=\left(t_h^{\hat{\eta}_c} \log t_h-\tau^{\hat{\eta}_c} \log \tau\right)$,

$Q_{1 c}=\sum_{h=1}^{N_1}\left(1+R_h\right)\left(e^{t_h^{\hat{\eta}_c}} t_h^{\eta_c} \log \left(t_h\right)\right)$,

$Q_{2 c}=\sum_{h=N_1+1}^m\left(1+R_h\right)\left(e^{\tau^{\hat{\eta}_c}} \tau^{\hat{\eta}_c} \log \tau\right)$ and

$Q_{3 c}=\sum_{h=N_1+1}^m\left(1+R_h\right)\left(e^{\left(t_h^{\widehat{\eta}_c}-\tau^{\widehat{\eta}_c}\right)}\left(t_h^{\widehat{\eta}_c} \log t_h-\tau^{\widehat{\eta}_c} \log \tau\right)\right)$.

As of (14) and (15), the MLEs of $\vartheta_{1 c}$ and $\vartheta_{2 c}$ are easily acquired as:

$\hat{\vartheta}_{1 c}=-n_{1 c}\left[U_{1 c}+U_{2 c}\right]^{-1}$    (16)

$\hat{\vartheta}_{2 c}=-n_{2 c}\left[U_{3 c}\right]^{-1}$     (17)

The nonlinear Eq. (13) have difficult closed-form solutions. Therefore, a numerical approach ought to be utilized to resolve these concurrent equations to finding $\hat{\eta}_c ; c=1,2$.

4. Asymptotic Variance and Covariance’s of Estimation

The asymptotic Fisher information matrix of the parameter's MLE could be approximated by numerically inverting the asymptotic Fisher information matrix Ic. It is made up of the negative second and mixed partial derivatives of the likelihood function's actual logarithm as defined by the MLE. It can be computed utilizing the following matrix:

$I_c=\left[\begin{array}{lll}I_{c 11} & I_{c 12} & I_{c 13} \\ I_{c 21} & I_{c 22} & I_{c 23} \\ I_{c 31} & I_{c 32} & I_{c 33}\end{array}\right]$$=\left[\begin{array}{ccc}\frac{-\partial^2 \log L}{\partial \eta_c^2} & \frac{-\partial^2 \log L}{\partial \eta_c \partial \vartheta_{1 c}} & \frac{-\partial^2 \log L}{\partial \eta_c \partial \vartheta_{2 c}} \\ \frac{-\partial^2 \log L}{\partial \vartheta_{1 c} \partial \eta_c} & \frac{-\partial^2 \log L}{\partial \vartheta_{1 c}^2} & \frac{-\partial^2 \log L}{\partial \vartheta_{1 c} \partial \vartheta_{2 c}} \\ \frac{-\partial^2 \log L}{\partial \vartheta_{2 c} \partial \eta_c} & \frac{-\partial^2 \log L}{\partial \vartheta_{1 c} \partial \vartheta_{2 c}} & \frac{-\partial^2 \log L}{\partial \vartheta_{2 c}^2}\end{array}\right]$

The components of the noticed Fisher information matrix Ic11, Ic22, Ic33, Ic12, and Ic13 are obtained as the following:

$I_{c 11}=\frac{-\partial^2 \log L}{\partial \eta_c^2}=\eta_c^{-2} \sum_i^2 n_{i c}-E^*-E^{* *}+\vartheta_{1 c} E_{1 c}+\vartheta_{2 c} E_{2 c}$,

$I_{c 22}=\frac{-\partial^2 \log L}{\partial \vartheta_{1 c}^2}=n_{1 c} \vartheta_{1 c}^{-2}, I_{c 33}=\frac{-\partial^2 \log L}{\partial \vartheta_{2 c}^2}=n_{2 c} \vartheta_{2 c}^{-2}$,

$I_{c 12}=\frac{-\partial^2 \log L}{\partial \eta_c \partial \vartheta_{1 c}}=I_{c 21}=\frac{-\partial^2 \log L}{\partial \vartheta_{1 c} \partial \eta_c}=Q_{1 c}+Q_{2 c}$

$I_{c 13}=\frac{-\partial^2 \log L}{\partial \eta_c \partial \vartheta_{2 c}}=I_{c 31}=\frac{-\partial^2 \log L}{\partial \vartheta_{2 c} \partial \eta_c}=-Q_{3 c}$ and

$I_{c 23}=\frac{-\partial^2 \log L}{\partial \vartheta_{1 c} \partial \vartheta_{2 c}}=I_{c 32}=\frac{-\partial^2 \log L}{\partial \vartheta_{2 c} \partial \vartheta_{1 c}}=0$

where, $E^*=\sum_{h=1}^{N_1} t_h^{\eta_c}\left(\log t_h\right)^2$,

$E^{* *}=\sum_{h=N_1+1}^m\left(t_h^{\eta_c}\left(\log t_h\right)^2-\tau^{\eta_c}(\log \tau)^2\right)$,

$E_{1 c}=\sum_{h=1}^{N_1}\left(1+R_h\right)\left[e^{t_h^{\eta_c}} t_h^{\eta_c}\left(\log t_h\right)^2\left(1+t_h^{\eta_c}\right)\right]$$+\sum_{h=N_1+1}^m\left(1+R_h\right)\left[e^{\tau^{\eta_c}} \tau^{\eta_c}(\log \tau)^2\left(1+\tau^{\eta_c}\right)\right]$,

$\begin{aligned} & E_{2 c}=\sum_{h=N_1+1}^m\left(1+R_h\right)\left\{e^{\left(t_h^{\eta_c c}-\tau^{\eta_c}\right)}\left[t_h^{\eta_c} \log t_h\left(\log t_h+\right.\right.\right. \left.\log t_h t_h^{\eta_c}-\tau^{\eta_c} \log \tau\right)-\log \tau \tau^{\eta_c}\left(\log \tau+t_h^{\eta_c} \log t_h-\right. \left.\left.\left.\tau^{\eta_c} \log \tau\right)\right]\right\}\end{aligned}.$

Substitute the MLEs $\hat{\vartheta}_{1 c}$ for $\vartheta_{1 c}, \hat{\vartheta}_{2 c}$ for $\vartheta_{2 c}$ and $c=1,2$. It is feasible to acquire the noticed Fisher information matrix $\hat{I}_c$. when this matrix is inverted and symbolized by $\hat{V}_c=\hat{I}_c^{-1}$.

One might obtain the convergent CIs of the parameters on the asymptotic distribution of the MLEs of the units of the vector of unidentified parameters $\psi_c=\left(\vartheta_{1 c}, \vartheta_{2 c}\right), c=1,2$. The asymptotic distribution of the MLEs $\left(\hat{\psi}_c-\psi_c / \sqrt{\operatorname{var}\left(\hat{\psi}_c\right)}\right), c=1,2$ is known to be approximated by a standard normal distribution, whereas $\operatorname{var}\left(\hat{\psi}_c\right)$ is assessed as the asymptotic variance, then, the estimated $100(1-\gamma) \%$ two-sided CI for $\psi_c=\left(\vartheta_{1 c}, \vartheta_{2 c}\right), c=1,2$ are achieved, therefore;

$p\left[\widehat{\psi}_c-z_{\gamma / 2} \sqrt{\operatorname{var}\left(\hat{\psi}_c\right)} \leq \psi_c \leq \hat{\psi}_c+z_{\gamma / 2} \sqrt{\operatorname{var}\left(\hat{\psi}_c\right)}\right] \cong \alpha$

whereas, $z_{\gamma / 2}$ is the $100(1-\gamma) \%$ standard normal percentile.

5. Bayesian Estimation

In this section, established on data of competing risks, the Bayesian estimation employing square error loss functions is obtained based on a simple step-stress model with type II progressive censoring. One could propose utilizing independently distributed gamma priors with known parameters ϑ1c, ϑ2c and η1c where c=1,2 of the extension Weibull distribution (EWD) as:

$\pi_1\left(\eta_c\right)=\eta_c^{-1} ; 0<\eta_c<1$

and

$\begin{gathered}\pi_2\left(\vartheta_{l c}\right)=\left[\Gamma\left(a_{l c}\right)\right]^{-1} b_{l c}^{a_{l c}} \vartheta_{l c}^{a_{l c}^{-1}} \exp \left(-b_{l c} \vartheta_{l c}\right) ; \\ \vartheta_{l c}, a_{l c}, b_{l c}>0, \quad l, c=1,2\end{gathered}$

where, the hyper-parameters $a_{l c}, b_{l c}$ and $l, c=1,2$ are elected to mirror prior knowledge of the unknown parameters and the parametric space of $\vartheta_{1 c}$ and $\vartheta_{2 c}$ should be $K_{\vartheta_{l c}}=\left\{\vartheta_{l c}, \vartheta_{1 c} \leq \vartheta_{2 c}\right\}$, $c=1,2$.

Hence, the jointly prior densities of $\psi_c=\left(\eta_c, \vartheta_{1 c}, \vartheta_{2 c}\right), c=1,2$ can then be written as:

$\begin{gathered}\pi\left(\psi_c\right) \propto \eta_c^{-1} \vartheta_{1 c}^{a_{1 c}-1} \vartheta_{2 c}^{a_{2 c}-1} e^{\left(-b_{1 c} \vartheta_{1 c}-b_{2 c} \vartheta_{2 c}\right)} I_{\left(\vartheta_{1 c} \leq \vartheta_{2 c}\right)} ; c \quad=1,2\end{gathered}$       (18)

For the noticed data t obtained from a life test experiment's type II progressive censoring with four independent the extension Weibull distribution ϑlc and l, c=1,2 and from Eq. (11) of the likelihood function and Eq. (18) of prior distribution the equivalent posterior density of ψc=(ηc, ϑ1c,ϑ2c), c=1,2 and is given by:

$\pi\left(\psi_c \mid \underline{t}\right) \propto L\left(\psi_c \mid \underline{t}\right) . \pi\left(\eta_1, \eta_2, \vartheta_{11}, \vartheta_{12}, \vartheta_{21}, \vartheta_{22}\right)$

The posterior density function is given by:

$\pi\left(\psi_c \mid \underline{t}\right) \propto$$\left[\eta_c^{\sum_{l=1}^2 n_{l c}-1}\right]\left[\prod_{l=1}^2 \vartheta_{l c}^{n_{l c}+a_{l c}-1}\right]\left(\prod_{h=1}^m t_h^{\eta_c-1}\right)$

$\exp \left[\sum_{h=1}^{N_1} t_h^{\eta_c}+\sum_{h=N_1+1}^m\left(t_h^{\eta_c}-\tau^{\eta_c}\right)+\vartheta_{1 c}\left(U_{1 c}+U_{2 c}-b_{1 c}\right)+\vartheta_{2 c}\left(U_{3 c}-b_{2 c}\right)\right] I_{\left(\vartheta_{1 c} \leq \vartheta_{2 c}\right)} ; c=1,2$      (19)

By integrating $\pi\left(\psi_c \mid \underline{t}\right)$ with regard to $\vartheta_1$ and $\vartheta_{2 c}$, the marginal posterior density function of $\eta_c$ is displayed to be proportional to:

$m\left(\eta_c \mid t\right) \propto \eta_c^{\sum_{l=1}^2 n_{l c}-1}\left(\prod_{h=1}^m t_h^{\eta_c-1}\right) \Gamma\left(n_{1 c}+a_{1 c}\right)$

$\Gamma\left(n_{2 c}+a_{2 c}\right) \exp \left[\sum_{h=1}^{N_1} t_h^{\eta_c}+\sum_{h=N_1+1}^m\left(t_h^{\eta_c}-\tau^{\eta_c}\right)\right]$

$\left[\left(U_{1 c}+U_{2 c}-b_{1 c}\right)^{n_{1 c}+a_{1 c}}\left(U_{3 c}-b_{1 c}\right)^{n_{2 c}+a_{2 c}}\right]^{-1}$

To achieve the most suitable estimator using Bayes' technique, one must select a loss function that corresponds with all of the potential estimators. In this section the estimates are obtained for two various kinds of loss functions, explicitly, squared error loss function (SELF) as an illustrative of the first type, and LINEX loss function (LLF) as an exemplar of another type. The SELF is improper when there is an exaggeration or underestimation. In this situation, LLF could be utilized as an alternate option for an estimate the parameters. Additionally, it is helpful when exaggeration and underestimation are both serious issues. The loss functions that are currently researched could be shown as follows, assuming that $\omega_c$ is an estimator for the unknown parameter $\psi_c$.

  • SELF: From "$\left(\omega_c-\psi_c\right)^2$" and Bayes estimate "$E_{\psi_c}\left(\psi_c \mid \underline{t}\right)$".
  • LLF: "$\exp \left[u\left(\omega_c-\psi_c\right)\right]-u\left(\omega_c-\psi_c\right)-1, u \neq 0$" and Bayes estimate "$-\frac{1}{u} \ln \left[E_{\psi_c}\left(\exp \left(\left(-u \psi_c\right) \mid \underline{t}\right)\right)\right]$".

Referring to Eq. (19), one could observe the difficulty of computing integrals, followed by the incapability to acquire the closed form for the joint posterior that allows us to calculate Bayes estimations of the unknown parameters ψc=(ηc, ϑ1c,ϑ2c), c=1,2. As a result, we will use the MCMC approach to obtain these estimates, which allows us to generate simulated samples from the parameter posterior distributions. These generated samples will be used to calculate the interval and point estimation of unknown parameters. The mechanism of this approach is based on the computation of conditional posterior functions wherein a conditional distribution of $\left(\vartheta_{1 c} \mid \eta_c\right)$ and $\left(\vartheta_{2 c} \mid \eta_c\right)$ is gamma distribution with respective PDF:

$\begin{aligned} & \pi_1^*\left(\vartheta_{1 c} \mid \eta_c\right)=D_{1 c} \vartheta_{1 c}^{n_{1 c}+a_{1 c}-1} \exp \left[-\vartheta_{1 c}\left(b_{1 c}-U_{1 c}-U_{2 c}\right)\right] \\ & \simeq \operatorname{Gamma}\left(n_{1 c}+a_{1 c}, b_{1 c}-U_{1 c}-U_{2 c}\right) \\ & \pi_2^*\left(\vartheta_{2 c} \mid \eta_c\right)=D_{2 c} \vartheta_{1 c}^{n_{2 c}+a_{2 c}-1} \exp \left[-\vartheta_{2 c}\left(b_{2 c}-U_{3 c}\right)\right] \\ & \simeq \operatorname{Gamma}\left(n_{2 c}+a_{2 c}, b_{2 c}-U_{3 c}\right)\end{aligned}$

where, $D_{1 c}=\left(b_{1 c}-U_{1 c}-U_{2 c}\right)^{n_{1 c}+a_{1 c} / \Gamma\left(n_{1 c}+a_{1 c}\right)}$ and $D_{1 c}=\left(b_{2 c}-U_{3 c}\right)^{n_{2 c}+a_{2 c}} / \Gamma\left(n_{2 c}+a_{2 c}\right)$.

Gilks and Wild [43] proposed a simple and applies solution to this problem. We can sample from a full conditional distribution that ought to be log-concave utilizing technique. As a result, we must establish whether the condition is satisfied. Given ϑ1c, ϑ2c and c=1,2 the logarithm of the density of conditional posterior function of ηc is:

$\begin{aligned} & \log \pi\left(\eta_c \mid \vartheta_{1 c}, \vartheta_{2 c}\right) \propto\left(\sum_{l=1}^2 n_{l c}-1\right) \log \eta_c \\ &+\left(\eta_c-1\right) \sum_{h=1}^m \log t_h+\sum_{h=1}^{N_1} t_h^{\eta_c} \\ &+\sum_{h=N_1+1}^m\left(t_h^{\eta_c}-\tau^{\eta_c}\right) \\ &+\vartheta_{1 c}\left(U_{1 c}+U_{2 c}-b_{1 c}\right) \\ &+\vartheta_{2 c}\left(U_{3 c}-b_{2 c}\right)\end{aligned}$      (20)

Compute the second derivative of (20) as:

$\begin{aligned} & \partial^2 \log \pi\left(\eta_c \mid \vartheta_{1 c}, \vartheta_{2 c}\right) / \partial \eta_c^2 \\ &=-\eta_c^{-2}\left(\sum_{l=1}^2 n_{l c}-1\right)+E^*+E^{* *} \\ &-\vartheta_{1 c} E_{1 c}-\vartheta_{2 c} E_{2 c}\end{aligned}$

It is clear that $\pi_1^*\left(\vartheta_{1 c} \mid \eta_c\right)$ and $\pi_2^*\left(\vartheta_{2 c} \mid \eta_c\right)$ are gamma distributions. Therefore, employing a gamma generator, samples of ϑ1c and ϑ2c can be generated. Furthermore, π(ηc|ϑ1c, ϑ2c) cannot be directly shrunk for drawing samples utilizing standard techniques. For this type of situation, acquiring Bayes’ estimate for ηc, we can employ one of the very well algorithms in the MCMC approach, the Metropolis-Hastings (MH) algorithm model, which was published in the study by Metropolis et al. [44]. To employ this algorithm, we must first presume a suggestion function for a sample from it. Throughout this technique, we can choose either to employ a non-symmetric or a symmetric suggestion distribution to reduce the rejection rate as much as potential. Because the marginal distribution of ηc is unknown, the normal distribution is designated as a symmetric proposal distribution. The Metropolis-Hastings steps are employed by the Gibbs sampler to update ηc, whereas ϑ1c and ϑ2c are updated directly from their full conditionals. The following is a hybrid algorithm that utilizes Gibbs sampling steps to update the parameters ϑ1c and ϑ2c with MH steps to update ηc and create the associated HPD credible intervals. Typically, we select the MLEs $\hat{\vartheta}_{1 c}, \hat{\vartheta}_{2 c}$ and c=1,2 as primary values.

Step 1: Given ϑ1c, ϑ2c, according to the sampling algorithm suggested by Gilks and Wild [43] of the adaptive rejection, generate ηc.

Step 2: Based upon ηc from step 1, generate ϑ1c from Gamma(n1c+a1c, b1c-U1c-U2c) and ϑ2c from Gamma(n2c+a2c,b2c-U3c).

Step 3: Corroborate that ϑ1cϑ2c, if not, return to step 2. If this is the case, repeat 1,000 times for steps 1 to 3.

Signify the kth Gibbs sampler by $\left(\eta_c^{(k)}, \vartheta_{1 c}^{(k)}, \vartheta_{2 c}^{(k)}, k=\right.$ $1,2, \ldots, 1000)$. Before stationary, iterations ( $N$ times in total) are discarded, therefore, the Bayesian estimates of $\eta_c, \vartheta_{1 c}, \vartheta_{2 c}$ and $c=1,2$ are:

$\begin{aligned} & \eta_{c B S}=\sum_{k=N+1}^{1000} \eta_c^{(k)} /(1000-N), \text { and } \\ & \vartheta_{l c B S}=\sum_{k=N+1}^{1000} \vartheta_{l c}^{(k)} /(1000-N), \text { for } l, c=1,2\end{aligned}$.

Step 4: Calculate all conceivable 100(1-γ)% credible intervals of the form $\left[\eta_{c(p)}, \eta_{c(p+100(1-\gamma))}\right]$ and $\left[\vartheta_{l c(p)}, \vartheta_{l c(p+100(1-\gamma))}\right]$ where $l, c=1,2 ; p=1,2, \ldots, 1000-100(1-\gamma)$ respectively, by sorting the Gibbs sampler $\eta_c^{(k)}, \vartheta_{1 c}^{(k)}, \vartheta_{2 c}^{(k)}, k=1,2, \ldots, 1000$ in ascending order.

Step 5: Calculate the lengths of each credible interval, then select the lowest interval to serve as the HPD credible interval of ηc, ϑ1c, ϑ2c and c=1,2.

6. Simulation Study and Illustrative Example

The goal of this section is to compare the performance of the different estimation methods introduced in the preceding sections. A real given dataset is used for illustration purposes; furthermore, a simulation study is used to evaluate the behavior of the suggested methods and to test the statistical performances of the estimators given a Progressive Censoring Type-II scheme under step-stress for extension Weibull distribution in the presence of competing risks. The R statistical software program was used to accomplish the calculations. Computing MLEs and HPD intervals in the program R is done by employing the bbmle and HDInterval packages.

6.1 Simulation study

In this sub-section, to analyze the accuracy of estimation methods, including MLE and Bayesian estimation, a Monte Carlo simulation study is employed, under progressive Type-II under step-stress for extension Weibull distribution in presence of competing causes of risks. For the MLEs, 1,000 observations are generated from the extension Weibull distribution based on the following assumptions:

  1. Parameters are given by: ϑ11=0.5, ϑ12=0.75, ϑ21=1.25, ϑ22=1.5, η1=1.75, η2=2.
  2. Sample sizes of n=40, n=80, and n=100.
  3. The number of stages of progressive censoring: m=30, 40, 60, 80.
  4. Removed items Ri are assumed at different sample sizes n and number of stages m as shown in Table 1.

Table 1. Numerous patterns for removing items from life tests at different stages

n

m

Censoring Schemes

S1

S2

S3

S4

S5

60

30

(30, 0*29)

(15, 0*28, 15)

(0*14,15,15, 0*14)

(1*30, 0*0)

(0*29, 30)

40

(20, 0*39)

(10, 0*38, 10)

(0*19,10,10, 0*19)

(1*20, 0*20)

(0*39, 10)

80

40

(40, 0*39)

(20, 0*38, 20)

(0*19,20,20, 0*19)

(1*40, 0*0)

(0*39, 40)

60

(20, 0*59)

(10, 0*58, 10)

(0*29,10,10, 0*29)

(1*20, 0*40)

(0*59, 20)

100

60

(40, 0*59)

(20, 0*58, 20)

(0*29,20,20, 0*29)

(1*40, 0*20)

(0*59, 40)

80

(20, 0*79)

(10, 0*78, 10)

(0*39,10,10, 0*39)

(1*20, 0*60)

(0*79, 20)

Here, $\left(5^{* 3}, 0\right)$, for instance, implies that the censorship scheme utilized is $(5,5,5,0)$.

Suppose that the levels of accelerated temperature S1=270℉, S2=320℉ and the utilized temperature is S0=210℉. The lifetime of the two failure causes follows an expansion of Weibull distribution with recognized shape parameter η1, η2, respectively. The amount of the parameter was selected to be ϑ11, ϑ12, ϑ21, ϑ22. To clarify a specific scenario underneath each cause of failure the increase of stress level in our case of the study with the extension Weibull model will be achieved by increasing the rate of the scale parameter ϑc, which will be reflected in shrinking the main time to failure.

Before continuing, first, the progressively censored Type-II is created through competing risks data utilizing the extension Weibull cumulative exposure model (CEM) for constant m, (R1, R2…, Rm, n), as shown below:

Step 1: Based on the algorithm proposed by Balakrishnan and Sandu [45], generating two samples that are progressively censored W1, W2…, Wm and U1,U2,…Um from Uniform distribution (0,1).

Step 2: Calculating $t_{11 h}$ and $t_{12 h}$ using $U_h=1-\exp \left[\vartheta_{11}(1-\right.$ $\left.\left.e^{t_{11 h}^{\eta_1}}\right)\right]$ and $W_h=1-\exp \left[\vartheta_{12}\left(1-e^{t_{12 h}^{\eta_2}}\right)\right]$, the minimum of $\left(t_{11 h,}, t_{12 h}\right)$ is registered as $t_h^*$, while the corresponding minimum index comes out of this condition $\left(t_{11 h}<t_{12 h}\right.$, set $\xi_h^*=1$; else set $\xi_h^*=2$ ) for $1 \leq h \leq N_1$.

Step 3: Let's assume that various values of stress changing time $\tau$ as follows: $\tau=\left(\operatorname{mean}\left(t_h^*\right)+\operatorname{median}\left(t_h^*\right)\right) / 2$.

Step 4: Find $N_1$ such that $t_{N_1}^*<\tau<t_{N_1+1}^*$. Hence, put $t_h=$$t_h^*$ and $\xi_h=\xi_h^*$ for $1 \leq h \leq N_1$.

Step 5: Generating $t_{21 h}$ and $t_{22 h}$ utilizing:

$U_h=1-\exp \left[\vartheta_{21}\left(1-e^{t_{21 h}^{\eta_1}-\tau^{\eta_1}}\right)+\vartheta_{11}\left(1-e^{\tau^{\eta_1}}\right)\right]$ and

$W_h=1-\exp \left[\vartheta_{22}\left(1-e^{t_{22 h}^{\eta_2}-\tau^{\eta_2}}\right)+\vartheta_{12}\left(1-e^{\tau^{\eta_2}}\right)\right]$, for $N_1+$ $1 \leq h \leq m$. The minimum values of $\left(t_{21 h}, t_{22 h}\right)$ assigned as $t_h^*$.

Step 6: Setting the value of $t_h=t_h^*$ and $\xi_h=\xi_h^*$ for $N_1+1 \leq$ $h \leq m$. Then $t_1, t_2, \ldots, t_m$ are the required order observation and $\xi=\left[\xi_1, \xi_2, \ldots, \xi_m\right]$ the vector of the indices.

MLEs and related 95% asymptotic confidence intervals (ACIs) are produced based on the generated data. On deriving MLEs, be aware that the initial assume values are regarded as true parameter values.

We compute Bayesian estimates using informative priors for the Bayesian estimation method using (18) as the value for all hyperparameters. Such values of informative priors are plugged-in to evaluate the required estimates. Through implementing the MH algorithm, the MLEs are used as initial guess values, as well as the corresponding variance-covariance matrix $I_c$ of $(\hat{\psi})$, where $\psi=\left(\vartheta_{11}, \vartheta_{12}, \vartheta_{21}, \vartheta_{22}, \eta_1, \eta_2\right)$. In the end, 1500 burn-in samples were deleted from the total of 6,000 generated samples by the posterior density and produced Bayes estimates under two loss functions, namely: squared error loss (SEL) function LINEX at $\tau=-1.5,1.5$ Also, HPD interval estimates have been computed according to the technique of Chen and Shao [46].

All the average estimates for methods are reported in Tables 3 to 8 for different combinations of sample size n and number of stages m. Further, the first column donates the average estimates (Avg.) and in the second column, related means square errors (MSEs). For confidence intervals, we have asymptotic confidence intervals for MLEs and HPD for Bayesian estimates based on MCMC which are reported in Tables 9 to 14 for different combinations of sample size n and number of stages m. Further, the first column represents the lower bound confidence interval, the second column represents the upper bound of CI, the third column represents the average interval lengths (AILs), and in the last column, related coverage probabilities (CPs) in percentage (%).

According to the tabulated values, one can indicate that:

  1. As n raises and m fixed, Avg. estimates for MLE and BE using MCMC tend to gradually converge for the initial parameter values.
  2. When the sample size increases, the MSE and average of MLE and BE of the considered parameters decrease.
  3. The MLE and Bayesian estimates for Scheme 5 have good statistical properties than other Schemes.
  4. As small as the sample size, Bayesian estimates can be offered.
  5. As n fixed and m increases, AIL for MLE and BE using MCMC tend to decrease.
  6. MSE for BE under LINEX with (τ=1.5) is smaller than other BE methods for all parameters expect.
  7. For fixed sample size and $m$ small, the AIL for BE is shorter than the AIL for MLE using MCMC, but in most cases, for $m$ increasing and fixed sample size, the AIL for MLE is shorter than the AIL for BE.
  8. For fixed m and n increasing, the AIL for MLE and BE using MCMC tend to decrease.
  9. For the same sample size, the average estimate for MLE tends to the initial values of the parameter with m small.
  10. For fixed $n$ and $m$ small, the MSE of MLE decreases.
  11. All most cases, the point estimates of the scale parameters for BEs are better than for MLEs and the opposite is true for the shape parameter.

Also, a comparison of Bayesian estimation under MCMC for all different combinations of samples size $n$ and number of stages $m$ in the case of Scheme $3\left(S_3\right)$ can be shown graphically the graphs of MCMC estimates for $\vartheta_{11}, \vartheta_{12}, \vartheta_{21}, \vartheta_{22}$ using the $\mathrm{MH}$ algorithm and $\eta_1$ and $\eta_2$ using Gibbs sampling algorithm in cases of informative priors are the plotting of estimates, histogram of estimates, and cumulative mean of estimates. These graphs can be shown in Figure 1 to Figure 3. Also, one can conclude the convergence of estimates under different sample sizes $n$ and number of stages $m$.

6.2 Illustrative example

In this section, we simulate progressive Type-II censored samples under step stress for extension Weibull distribution in presence of a competing failure model. The dataset is generated with the following selections of parameters: ϑ11=0.75, ϑ12=0.80, ϑ21=0.65, ϑ22=0.70, η1=2.0, and η2=2.10 and sample size n=50, number of stages m=30 with censoring scheme:

$R_i=(2,0,2,0,2,0,2,0,2,0,0,0,0,0,0,0,0,0,0,0,0,2,0,2,0,2,0,2,0,2)$.

Also, the assumed stress level is τ=0.5. The progressively censored Type-II data are given in the following Table 2.

From this dataset, the estimates (Est.) and standard error (St.Er) of MLEs and BEs using MCMC of the parameters are derived. Also, ACIs and HPD intervals are computed. The results are presented in Table 15 for Est. and St.Ers and in Table 16 for Cis (lower, upper, interval length (IL)).

The graphs of MCMC estimates for ϑ11, ϑ12, ϑ21, ϑ22, η1 and η2 using the MH algorithm can be shown in Figures 4 to 6 which indicates the normality of posterior samples generated using the MH algorithm and utilizing MCMC.

Table 2. Data of the progressively censored Type-II for the illustrative example

The first level of stress

(0.1742090, 2) (0.2031259, 1) (0.2315669, 1) (0.2759920, 1) (0.2877801, 1) (0.3045739, 1) (0.3457952, 1) (0.3508297, 1) (0.3648599, 1) (0.4635089, 1)

The second level of stress

(0.7515350, 2) (0.7987907, 2) (0.8096729, 2) (0.8502840, 2) (0.8565298, 1) (0.8593854, 1) (0.8725126, 1) (0.8737639, 1) (0.8747619, 1) (0.8775376, 1) (0.9101309 1) (0.9281317, 1) (0.9294583, 1) (0.9319161, 1) (0.9823121, 1) (1.0200595, 1) (1.0678518, 1) (1.0796584, 1) (1.0945277, 1) (1.1951210, 2)

The first element represents failure time and the second element represents the failure cause.

Table 3. Avg. estimated values and MSEs of the ML and BE using MCMC for different schemes of progressive Type-II censoring step-stress for extension Weibull distribution at n=60 and m=30

Sch.

Parm.

MLE

BE MCMC: SEL

BE MCMC: LINEX

τ=-1.5

τ=1.5

Avg.

MSE

Avg.

MSE

Avg.

MSE

Avg.

MSE

S1

ϑ11

0.3629

0.0207

0.3357

0.0288

0.3292

0.0309

0.3426

0.0267

ϑ12

0.3730

0.1448

0.3467

0.1652

0.3387

0.1716

0.3552

0.1586

ϑ21

1.2385

0.0275

1.1466

0.0350

1.0284

0.0660

1.3410

0.0784

ϑ22

1.1036

0.1861

1.0193

0.2560

0.9262

0.3476

1.1575

0.1585

η1

0.9362

0.6697

0.9749

0.6085

0.9481

0.6500

1.0040

0.5649

η2

1.0171

0.9772

1.0603

0.8947

1.0326

0.9465

1.0903

0.8401

S2

ϑ11

0.4409

0.0062

0.4033

0.0120

0.3948

0.0135

0.4124

0.0105

ϑ12

0.4525

0.0917

0.4147

0.1154

0.4055

0.1215

0.4245

0.1091

ϑ21

2.0345

0.6859

1.9390

0.5395

1.7121

0.2578

2.3654

1.4249

ϑ22

1.9388

0.2649

1.8426

0.1824

1.6442

0.0663

2.1837

0.6085

η1

1.0923

0.4454

1.1542

0.3686

1.1069

0.4252

1.2082

0.3099

η2

1.1413

0.7528

1.2050

0.6485

1.1586

0.7224

1.2581

0.5698

S3

ϑ11

0.3311

0.0294

0.3128

0.0359

0.3082

0.0376

0.3176

0.0342

ϑ12

0.5035

0.0650

0.4777

0.0785

0.4659

0.0847

0.4905

0.0721

ϑ21

1.1781

0.2045

1.1521

0.1849

1.0398

0.1700

1.3411

0.3655

ϑ22

0.9829

0.4289

0.9456

0.4540

0.8726

0.5033

1.0529

0.4327

η1

1.0842

0.4547

1.0801

0.4591

1.0442

0.5076

1.1188

0.4101

η2

1.2664

0.5535

1.2766

0.5375

1.2398

0.5908

1.3157

0.4838

S4

ϑ11

0.4002

0.0118

0.3784

0.0234

0.3683

0.0218

0.3997

0.2040

ϑ12

0.4715

0.0811

0.4965

0.7273

0.4371

0.1385

0.5964

2.2996

ϑ21

2.0791

0.7487

2.0115

0.6452

1.7817

0.3232

2.4575

1.7950

ϑ22

1.8916

0.2060

1.8397

0.2331

1.6519

0.0646

2.1967

2.6084

η1

1.0718

0.4701

1.1064

0.4259

1.0654

0.4793

1.1550

0.3716

η2

1.1806

0.6831

1.2170

0.6304

1.1747

0.6973

1.2753

0.7254

S5

ϑ11

0.5518

0.0072

0.5089

0.0043

0.4951

0.0038

0.5236

0.0052

ϑ12

0.5518

0.0438

0.5081

0.0628

0.4944

0.0692

0.5229

0.0563

ϑ21

2.8508

2.6734

2.7456

2.3323

2.4121

1.4165

2.4604

0.2327

ϑ22

2.8509

1.9355

2.7525

1.6630

2.4167

0.9036

2.4886

0.3228

η1

1.0986

0.4385

1.1635

0.3584

1.1076

0.4247

1.2288

0.2897

η2

1.0986

0.8267

1.1604

0.7188

1.1047

0.8133

1.2252

0.6175

Table 4. Avg. estimated values and MSEs of the ML and BE using MCMC for different schemes of progressive Type-II censoring step-stress for extension Weibull distribution at n=60 and m=40

Sch.

Parm.

MLE

BE MCMC: SEL

BE MCMC: LINEX

τ=-1.5

τ=1.5

Avg.

MSE

Avg.

MSE

Avg.

MSE

Avg.

MSE

S1

ϑ11

0.3892

0.0136

0.3651

0.0195

0.3590

0.0212

0.3715

0.0179

ϑ12

0.3969

0.1263

0.3738

0.1431

0.3669

0.1484

0.3812

0.1377

ϑ21

1.3560

0.0344

1.2689

0.0231

1.1497

0.0262

1.4521

0.0844

ϑ22

1.2656

0.0783

1.1816

0.1236

1.0795

0.1931

1.3329

0.0666

η1

0.8742

0.7717

0.9108

0.7093

0.8898

0.7446

0.9334

0.6723

η2

0.9193

1.1736

0.9578

1.0927

0.9364

1.1373

0.9807

1.0460

S2

ϑ11

0.4091

0.0100

0.3791

0.0163

0.3732

0.0177

0.3852

0.0150

ϑ12

0.4134

0.1152

0.3837

0.1361

0.3775

0.1406

0.3902

0.1314

ϑ21

1.7902

0.3373

1.6945

0.2392

1.5147

0.0997

1.9987

0.6505

ϑ22

1.7244

0.0961

1.6321

0.0595

1.4676

0.0311

1.8979

0.2409

η1

0.9821

0.5966

1.0388

0.5136

1.0071

0.5588

1.0736

0.4663

η2

1.0120

0.9841

1.0688

0.8759

1.0373

0.9346

1.1034

0.8137

S3

ϑ11

0.3521

0.0226

0.3353

0.0279

0.3313

0.0293

0.3396

0.0266

ϑ12

0.4592

0.0881

0.4372

0.1030

0.4293

0.1076

0.4455

0.0983

ϑ21

1.3993

0.2220

1.3528

0.1898

1.2249

0.1284

1.5608

0.4178

ϑ22

1.3259

0.2300

1.2686

0.2314

1.1561

0.2466

1.4429

0.3035

η1

0.9734

0.6100

0.9924

0.5803

0.9668

0.6193

1.0196

0.5403

η2

1.0236

0.9614

1.0507

0.9093

1.0240

0.9600

1.0790

0.8570

S4

ϑ11

0.3484

0.0240

0.5273

1.3523

0.3939

0.2632

0.5502

1.2756

ϑ12

0.4201

0.1108

0.7327

3.2772

0.5134

0.5491

0.8500

1.7366

ϑ21

1.2498

0.0225

1.8732

1.2202

1.1313

1.1070

2.5090

1.0604

ϑ22

1.1521

0.1397

1.6107

0.6030

1.0096

0.5902

2.1884

0.1731

η1

0.9481

0.6478

1.7780

2.5824

1.1314

2.8737

2.0254

2.1397

η2

1.0082

0.9889

1.3636

2.9432

1.0327

2.0952

2.0950

2.08196

S5

ϑ11

0.4570

0.0041

0.4257

0.0078

0.4185

0.0088

0.4333

0.0068

ϑ12

0.4570

0.0881

0.4256

0.1075

0.4185

0.1120

0.4331

0.1028

ϑ21

2.2047

0.9737

2.0990

0.7770

1.8605

0.4113

2.5440

1.8374

ϑ22

2.2047

0.5589

2.0947

0.4124

1.8562

0.1671

2.5273

1.2076

η1

1.0184

0.5441

1.0824

0.4553

1.0438

0.5072

1.1255

0.4013

η2

1.0184

0.9725

1.0845

0.8482

1.0457

0.9195

1.1279

0.7722

Table 5. Avg. estimated values and MSEs of the ML and BE using MCMC for different schemes of progressive Type-II censoring step-stress for extension Weibull distribution at n=80 and m=40

Sch.

Parm.

MLE

BE MCMC: SEL

BE MCMC: LINEX

τ=-1.5

τ=1.5

Avg.

MSE

Avg.

MSE

Avg.

MSE

Avg.

MSE

S1

ϑ11

0.3694

0.0186

0.3481

0.0246

0.3428

0.0262

0.3537

0.0230

ϑ12

0.3778

0.1406

0.3571

0.1565

0.3508

0.1614

0.3637

0.1514

ϑ21

1.2576

0.0202

1.1842

0.0234

1.0848

0.0416

1.3264

0.0369

ϑ22

1.1311

0.1575

1.0620

0.2113

0.9837

0.2815

1.1679

0.1386

η1

0.9228

0.6894

0.9538

0.6393

0.9338

0.6712

0.9752

0.6061

η2

0.9957

1.0162

1.0297

0.9494

1.0094

0.9887

1.0514

0.9084

S2

ϑ11

0.4429

0.0054

0.4142

0.0095

0.4074

0.0106

0.4213

0.0084

ϑ12

0.4527

0.0909

0.4235

0.1091

0.4163

0.1137

0.4310

0.1043

ϑ21

2.0347

0.6699

1.9604

0.5531

1.7732

0.3099

2.2637

1.1197

ϑ22

1.9470

0.2561

1.8665

0.1866

1.7026

0.0801

2.1173

0.4696

η1

1.0862

0.4507

1.1350

0.3888

1.0989

0.4333

1.1752

0.3427

η2

1.1303

0.7685

1.1828

0.6806

1.1474

0.7383

1.2220

0.6198

S3

ϑ11

0.3239

0.0317

0.3107

0.0365

0.3073

0.0378

0.3142

0.0352

ϑ12

0.4971

0.0666

0.4776

0.0768

0.4685

0.0817

0.4872

0.0718

ϑ21

1.1052

0.1151

1.0950

0.1123

1.0132

0.1231

1.2128

0.1386

ϑ22

0.9101

0.4242

0.8824

0.4530

0.8329

0.5022

0.9466

0.4057

η1

1.0876

0.4456

1.0820

0.4528

1.0541

0.4904

1.1114

0.4148

η2

1.2773

0.5315

1.2855

0.5196

1.2573

0.5601

1.3152

0.4788

S4

ϑ11

0.3934

0.0126

0.3711

0.0179

0.3660

0.0192

0.3764

0.0166

ϑ12

0.4623

0.0852

0.4348

0.1020

0.4274

0.1064

0.4433

0.0991

ϑ21

2.0810

0.7315

2.0416

0.6693

1.8547

0.3947

2.3692

1.8543

ϑ22

1.8883

0.1855

1.8397

0.1496

1.6970

0.0649

2.0566

0.3963

η1

1.0663

0.4748

1.0967

0.4460

1.0626

0.4794

1.1267

0.3970

η2

1.1763

0.6869

1.2087

0.6352

1.1776

0.6847

1.2419

0.5846

S5

ϑ11

0.5493

0.0059

0.5166

0.0037

0.5059

0.0032

0.5278

0.0044

ϑ12

0.5493

0.0438

0.5171

0.0577

0.5065

0.0625

0.5283

0.0528

ϑ21

2.8325

2.5929

2.7607

2.3644

2.4830

1.5786

3.2832

4.3588

ϑ22

2.8325

1.8641

2.7575

1.6595

2.4823

1.0216

3.2661

3.3091

η1

1.1032

0.4300

1.1529

0.3687

1.1087

0.4217

1.2027

0.3140

η2

1.1032

0.8159

1.1534

0.7286

1.1095

0.8031

1.2027

0.6496

Table 6. Avg. estimated values and MSEs of the ML and BE using MCMC for different schemes of progressive Type-II censoring step-stress for extension Weibull distribution at n=80 and m=60

Sch.

Parm.

MLE

BE MCMC: SEL

BE MCMC: LINEX

τ=-1.5

τ=1.5

Avg.

MSE

Avg.

MSE

Avg.

MSE

Avg.

MSE

S1

ϑ11

0.4043

0.0101

0.3880

0.0136

0.3834

0.0146

0.3927

0.0126

ϑ12

0.4097

0.1169

0.3931

0.1286

0.3880

0.1322

0.3983

0.1249

ϑ21

1.4104

0.0421

1.3467

0.0264

1.2507

0.0133

1.4762

0.0760

ϑ22

1.3470

0.0400

1.2843

0.0630

1.1973

0.1045

1.3998

0.0337

η1

0.8456

0.8209

0.8715

0.7751

0.8575

0.7997

0.8862

0.7498

η2

0.8747

1.2698

0.9017

1.2102

0.8874

1.2415

0.9166

1.1777

S2

ϑ11

0.4066

0.0097

0.3860

0.0140

0.3818

0.0149

0.3903

0.0131

ϑ12

0.4088

0.1175

0.3877

0.1324

0.3833

0.1355

0.3921

0.1292

ϑ21

1.7094

0.2353

1.6390

0.1764

1.5076

0.0852

1.8287

0.3794

ϑ22

1.6664

0.0522

1.5971

0.0340

1.4731

0.0193

1.7733

0.1158

η1

0.9361

0.6665

0.9760

0.6038

0.9561

0.6345

0.9971

0.5719

η2

0.9546

1.0972

0.9947

1.0155

0.9749

1.0554

1.0158

0.9739

S3

ϑ11

0.3629

0.0194

0.3493

0.0233

0.3462

0.0243

0.3525

0.0224

ϑ12

0.4429

0.0969

0.4265

0.1079

0.4213

0.1112

0.4319

0.1046

ϑ21

1.4474

0.2020

1.4022

0.1637

1.3010

0.1105

1.5459

0.2950

ϑ22

1.4307

0.1696

1.3822

0.1600

1.2827

0.1588

1.5221

0.2184

η1

0.9332

0.6721

0.9529

0.6398

0.9359

0.6669

0.9706

0.6122

η2

0.9459

1.1161

0.9692

1.0676

0.9518

1.1035

0.9875

1.0307

S4

ϑ11

0.3673

0.0183

0.3536

0.0222

0.3500

0.0232

0.3574

0.0211

ϑ12

0.4108

0.1161

0.3953

0.1269

0.3901

0.1306

0.4007

0.1232

ϑ21

1.3179

0.0166

1.2731

0.0131

1.1869

0.0139

1.3891

0.0378

ϑ22

1.2448

0.0764

1.1992

0.1027

1.1220

0.1525

1.3017

0.0568

η1

0.8945

0.7344

0.9150

0.7002

0.8996

0.7259

0.9311

0.6737

η2

0.9325

1.1424

0.9567

1.0919

0.9404

1.1260

0.9737

1.0568

S5

ϑ11

0.4308

0.0060

0.4087

0.0096

0.4041

0.0104

0.4134

0.0088

ϑ12

0.4308

0.1031

0.4092

0.1174

0.4046

0.1205

0.4139

0.1142

ϑ21

1.9786

0.5693

1.8992

0.4587

1.7365

0.2644

2.1466

0.8714

ϑ22

1.9786

0.2675

1.9056

0.2020

1.7407

0.0858

2.1572

0.4997

η1

0.9785

0.6007

1.0244

0.5325

1.0005

0.5674

1.0502

0.4964

η2

0.9785

1.0490

1.0227

0.9613

0.9986

1.0084

1.0484

0.9122

Table 7. Avg. estimated values and MSEs of the ML and BE using MCMC for different schemes of progressive Type-II censoring step-stress for extension Weibull distribution at n=100 and m=60

Sch.

Parm.

MLE

BE MCMC: SEL

BE MCMC: LINEX

τ=-1.5

τ=1.5

Avg.

MSE

Avg.

MSE

Avg.

MSE

Avg.

MSE

S1

ϑ11

0.3842

0.0144

0.3683

0.0184

0.3642

0.0194

0.3725

0.0173

ϑ12

0.3901

0.1308

0.3749

0.1420

0.3703

0.1454

0.3797

0.1385

ϑ21

1.3256

0.0223

1.2668

0.0166

1.1849

0.0172

1.3735

0.0381

ϑ22

1.2279

0.0915

1.1738

0.1226

1.1047

0.1695

1.2611

0.0785

η1

0.8865

0.7492

0.9111

0.7076

0.8975

0.7304

0.9253

0.6840

η2

0.9369

1.1350

0.9619

1.0826

0.9481

1.1112

0.9764

1.0530

S2

ϑ11

0.4191

0.0077

0.3992

0.0113

0.3949

0.0121

0.4035

0.0105

ϑ12

0.4236

0.1078

0.4032

0.1216

0.3987

0.1247

0.4078

0.1185

ϑ21

1.8622

0.4042

1.8011

0.3339

1.6663

0.1966

1.9919

0.5976

ϑ22

1.7941

0.1170

1.7310

0.0836

1.6098

0.0360

1.8981

0.2041

η1

1.0232

0.5334

1.0614

0.4800

1.0388

0.5112

1.0855

0.4478

η2

1.0548

0.8992

1.0947

0.8261

1.0724

0.8666

1.1186

0.7839

S3

ϑ11

0.3354

0.0276

0.3258

0.0308

0.3232

0.0317

0.3284

0.0300

ϑ12

0.4749

0.0775

0.4616

0.0851

0.4556

0.0885

0.4678

0.0816

ϑ21

1.1647

0.0847

1.1473

0.0827

1.0810

0.0866

1.2334

0.0970

ϑ22

1.0375

0.2858

1.0158

0.3005

0.9650

0.3400

1.0798

0.2647

η1

1.0323

0.5200

1.0374

0.5125

1.0188

0.5392

1.0568

0.4855

η2

1.1337

0.7566

1.1445

0.7378

1.1250

0.7713

1.1648

0.7039

S4

ϑ11

0.3392

0.0264

0.3867

0.5193

0.3410

0.0590

0.4260

0.6208

ϑ12

0.3975

0.1253

0.5161

1.4882

0.4062

0.1539

0.5695

1.5474

ϑ21

1.6016

0.1443

1.7484

4.4111

1.4686

0.5586

2.0205

5.0282

ϑ22

1.4456

0.0193

1.5201

1.6944

1.2953

0.0800

1.8940

7.3793

η1

0.9737

0.6056

1.1568

2.3779

1.0243

1.6474

1.5549

1.5995

η2

1.0666

0.8745

1.2143

2.8129

1.0548

0.9438

1.4546

0.7479

S5

ϑ11

0.4859

0.0019

0.4642

0.0030

0.4586

0.0033

0.4700

0.0026

ϑ12

0.4859

0.0715

0.4647

0.0831

0.4591

0.0863

0.4705

0.0799

ϑ21

2.3918

1.3499

2.3262

1.2039

2.1391

0.8254

2.6164

1.9514

ϑ22

2.3919

0.8416

2.3335

0.7397

2.1437

0.4488

2.6248

1.3447

η1

1.0562

0.4877

1.0986

0.4313

1.0705

0.4681

1.1290

0.3934

η2

1.0562

0.8971

1.0966

0.8229

1.0683

0.8743

1.1272

0.7694

Table 8. Avg. estimated values and MSEs of the ML and BE using MCMC for different schemes of progressive Type-II censoring step-stress for extension Weibull distribution at n=100 and m=80

Sch.

Parm.

MLE

BE MCMC: SEL

BE MCMC: LINEX

τ=-1.5

τ=1.5

Avg.

MSE

Avg.

MSE

Avg.

MSE

Avg.

MSE

S1

ϑ11

0.4098

0.0088

0.3971

0.0113

0.3934

0.0121

0.4008

0.0106

ϑ12

0.4134

0.1140

0.4008

0.1228

0.3968

0.1255

0.4048

0.1200

ϑ21

1.4411

0.0513

1.3911

0.0346

1.3103

0.0156

1.4939

0.0792

ϑ22

1.3924

0.0265

1.3427

0.0396

1.2682

0.0658

1.4365

0.0238

η1

0.8334

0.8426

0.8538

0.8059

0.8431

0.8250

0.8649

0.7862

η2

0.8548

1.3143

0.8754

1.2675

0.8647

1.2917

0.8866

1.2427

S2

ϑ11

0.4054

0.0097

0.3894

0.0130

0.3861

0.0137

0.3927

0.0123

ϑ12

0.4067

0.1187

0.3905

0.1301

0.3871

0.1325

0.3939

0.1276

ϑ21

1.6607

0.1870

1.6058

0.1449

1.5033

0.0790

1.7418

0.2679

ϑ22

1.6295

0.0352

1.5744

0.0255

1.4769

0.0166

1.7014

0.0683

η1

0.9111

0.7066

0.9409

0.6578

0.9267

0.6809

0.9558

0.6341

η2

0.9243

1.1603

0.9551

1.0954

0.9409

1.1251

0.9700

1.0647

S3

ϑ11

0.3702

0.0173

0.3590

0.0204

0.3565

0.0211

0.3616

0.0197

ϑ12

0.4316

0.1036

0.4175

0.1130

0.4137

0.1154

0.4214

0.1106

ϑ21

1.5007

0.1988

1.4619

0.1679

1.3759

0.1152

1.5740

0.2679

ϑ22

1.5065

0.1342

1.4603

0.1228

1.3737

0.1134

1.5733

0.1680

η1

0.9055

0.7164

0.9234

0.6863

0.9109

0.7069

0.9362

0.6653

η2

0.9053

1.2014

0.9268

1.1548

0.9142

1.1818

0.9398

1.1271

S4

ϑ11

0.3807

0.0148

0.3694

0.0177

0.3664

0.0185

0.3726

0.0169

ϑ12

0.4101

0.1163

0.3982

0.1246

0.3942

0.1274

0.4023

0.1217

ϑ21

1.3603

0.0232

1.3193

0.0166

1.2466

0.0097

1.4110

0.0416

ϑ22

1.3018

0.0498

1.2626

0.0674

1.1952

0.1020

1.3475

0.0378

η1

0.8706

0.7757

0.8888

0.7442

0.8774

0.7638

0.9005

0.7242

η2

0.8989

1.2150

0.9181

1.1733

0.9061

1.1993

0.9305

1.1466

S5

ϑ11

0.4195

0.0073

0.4029

0.0103

0.3995

0.0109

0.4064

0.0096

ϑ12

0.4195

0.1100

0.4029

0.1213

0.3995

0.1237

0.4064

0.1189

ϑ21

1.8658

0.4024

1.8097

0.3372

1.6862

0.2095

1.9791

0.5673

ϑ22

1.8658

0.1570

1.8071

0.1185

1.6837

0.0530

1.9755

0.2621

η1

0.9501

0.6431

0.9833

0.5915

0.9664

0.6174

1.0011

0.5648

η2

0.9501

1.1056

0.9842

1.0357

0.9673

1.0701

1.0020

1.0001

Table 9. Interval estimates, AILs, and CP(%) values of the ML and BE using MCMC for different schemes of progressive Type-II step-stress for extension Weibull distribution at n=60 and m=30

Sch.

Parm.

Asy-CI

HPD

Lower

Upper

AIL

CP (%)

Lower

Upper

AIL

CP (%)

S1

ϑ11

0.2768

0.4490

0.1723

97.2

0.2433

0.4134

0.1701

96.6

ϑ12

0.2710

0.4750

0.2040

97.6

0.2480

0.4445

0.1964

97.3

ϑ21

0.9142

1.5628

0.6485

95.3

0.8733

1.4830

0.6097

97.5

ϑ22

0.7699

1.4373

0.6673

96.2

0.6950

1.3074

0.6124

95.7

η1

0.7683

1.1040

0.3357

97.1

0.8192

1.1567

0.3375

97.8

η2

0.8104

1.2239

0.4135

96.6

0.8582

1.2615

0.4033

96.8

S2

ϑ11

0.3386

0.5433

0.2047

96.7

0.3040

0.5088

0.2048

97.2

ϑ12

0.3415

0.5636

0.2220

96.4

0.3051

0.5189

0.2138

96.5

ϑ21

1.5141

2.5550

1.0409

96.9

1.4686

2.4419

0.9733

97.3

ϑ22

1.4114

2.4662

1.0548

97.4

1.3600

2.3445

0.9845

97.2

η1

0.8703

1.3143

0.4440

97.1

0.9270

1.3741

0.4471

96.9

η2

0.8984

1.3842

0.4858

96.7

0.9442

1.4268

0.4826

96.0

S3

ϑ11

0.2734

0.3888

0.1154

95.9

0.2624

0.3730

0.1105

96.8

ϑ12

0.3762

0.6308

0.2546

97.2

0.3306

0.5897

0.2590

95.8

ϑ21

0.3027

2.0535

1.7508

90.0

0.7458

2.3516

1.6057

95.1

ϑ22

0.1950

1.7708

1.5758

90.0

0.6290

2.0645

1.4355

95.5

η1

0.8742

1.2942

0.4200

98.4

0.8644

1.2708

0.4065

96.8

η2

1.0235

1.5094

0.4859

98.3

0.9992

1.4775

0.4783

96.7

S4

ϑ11

0.3161

0.4842

0.1681

96.4

0.2886

0.4689

0.1804

96.7

ϑ12

0.3543

0.5887

0.2344

96.3

0.3250

0.5653

0.2403

96.6

ϑ21

1.5932

2.5649

0.9717

96.1

1.5706

2.4748

0.9042

97.0

ϑ22

1.4417

2.3415

0.8998

96.0

1.3673

2.2548

0.8875

95.9

η1

0.8741

1.2695

0.3954

96.9

0.8999

1.2975

0.3976

96.8

η2

0.9690

1.3922

0.4233

97.0

1.0096

1.4586

0.4490

97.7

S5

ϑ11

0.4199

0.6838

0.2639

95.7

0.3956

0.6484

0.2528

97.1

ϑ12

0.4199

0.6838

0.2639

95.7

0.3912

0.6392

0.2480

96.8

ϑ21

2.1985

3.5032

1.3046

95.8

2.1178

3.3580

1.2402

96.0

ϑ22

2.1985

3.5032

1.3046

95.8

2.1797

3.3652

1.1855

96.6

η1

0.8652

1.3319

0.4667

96.8

0.9401

1.4123

0.4722

97.8

η2

0.8652

1.3319

0.4667

96.8

0.9559

1.4233

0.4675

98.0

Table 10. Interval estimates, AILs, and CP(%) values of the ML and BE using MCMC for different schemes of progressive Type-II step-stress for extension Weibull distribution at n=60 and m=40

Sch.

Parm.

Asy-CI

HPD

Lower

Upper

AIL

CP (%)

Lower

Upper

AIL

CP (%)

S1

ϑ11

0.3182

0.4601

0.1419

97.3

0.2953

0.4378

0.1425

97.6

ϑ12

0.3186

0.4752

0.1566

97.4

0.3007

0.4580

0.1573

98.1

ϑ21

1.0577

1.6542

0.5965

97.1

0.9923

1.5594

0.5671

97.2

ϑ22

0.9662

1.5649

0.5987

97.5

0.9167

1.4872

0.5705

98.0

η1

0.7419

1.0064

0.2646

97.4

0.7672

1.0448

0.2776

96.4

η2

0.7703

1.0683

0.2980

97.0

0.8061

1.1148

0.3088

97.1

S2

ϑ11

0.3279

0.4903

0.1624

97.0

0.3006

0.4627

0.1622

96.9

ϑ12

0.3278

0.4990

0.1712

97.0

0.2998

0.4672

0.1674

97.0

ϑ21

1.3722

2.2083

0.8361

96.3

1.3296

2.1015

0.7720

97.4

ϑ22

1.3048

2.1440

0.8392

96.3

1.2701

2.0452

0.7751

97.2

η1

0.8181

1.1462

0.3282

97.5

0.8738

1.2168

0.3430

97.3

η2

0.8376

1.1864

0.3488

97.3

0.8982

1.2549

0.3567

97.9

S3

ϑ11

0.2990

0.4053

0.1063

96.7

0.2836

0.3921

0.1085

97.3

ϑ12

0.3427

0.5757

0.2330

99.1

0.3185

0.5382

0.2197

97.0

ϑ21

0.5230

2.2757

1.7527

96.1

0.8883

2.2028

1.3145

95.9

ϑ22

0.4495

2.2023

1.7528

96.4

0.7913

2.0283

1.2370

95.3

η1

0.8101

1.1368

0.3267

98.3

0.8380

1.1465

0.3086

98.0

η2

0.8487

1.1984

0.3498

98.3

0.8716

1.2199

0.3483

97.4

S4

ϑ11

0.2873

0.4095

0.1222

97.2

0.0017

0.8938

0.8921

95.1

ϑ12

0.3328

0.5075

0.1747

97.9

0.0048

1.5740

1.5692

95.1

ϑ21

0.9554

1.5443

0.5889

95.2

0.0032

5.1306

5.1274

95.1

ϑ22

0.8841

1.4202

0.5361

95.9

0.0049

3.8354

3.8305

95.1

η1

0.8143

1.0818

0.2675

97.7

0.0055

3.9074

3.9019

95.0

η2

0.8670

1.1494

0.2825

98.0

0.0081

2.0450

2.0370

95.2

S5

ϑ11

0.3642

0.5498

0.1855

96.7

0.3380

0.5225

0.1844

97.4

ϑ12

0.3642

0.5498

0.1855

96.7

0.3395

0.5225

0.1830

97.1

ϑ21

1.7150

2.6943

0.9793

96.8

1.6774

2.5784

0.9010

97.4

ϑ22

1.7150

2.6943

0.9793

96.8

1.6525

2.5848

0.9322

97.3

η1

0.8341

1.2026

0.3686

97.2

0.8860

1.2688

0.3828

96.7

η2

0.8341

1.2026

0.3686

97.2

0.9076

1.2952

0.3876

98.2

Table 11. Interval estimates, AILs, and CP (%) values of the ML and BE using MCMC for different schemes of progressive Type-II step-stress for extension Weibull distribution at n=80 and m=40

Sch.

Parm.

Asy-CI

HPD

Lower

Upper

AIL

CP (%)

Lower

Upper

AIL

CP (%)

S1

ϑ11

0.2917

0.4471

0.1553

97.0

0.2737

0.4246

0.1509

97.1

ϑ12

0.2880

0.4676

0.1796

96.6

0.2774

0.4561

0.1788

98.4

ϑ21

0.9792

1.5361

0.5569

97.1

0.9196

1.4466

0.5270

96.8

ϑ22

0.8439

1.4183

0.5744

97.4

0.8034

1.3408

0.5374

97.4

η1

0.7829

1.0627

0.2798

96.2

0.8153

1.0954

0.2801

96.8

η2

0.8252

1.1662

0.3410

96.8

0.8602

1.1981

0.3379

96.6

S2

ϑ11

0.3516

0.5342

0.1826

96.4

0.3295

0.5063

0.1768

97.1

ϑ12

0.3548

0.5505

0.1957

96.4

0.3289

0.5202

0.1913

96.3

ϑ21

1.5788

2.4906

0.9118

97.0

1.5448

2.3890

0.8442

97.1

ϑ22

1.4819

2.4122

0.9303

97.1

1.4224

2.2947

0.8723

96.8

η1

0.8897

1.2827

0.3930

97.1

0.9461

1.3340

0.3879

97.1

η2

0.9154

1.3451

0.4297

96.5

0.9773

1.4157

0.4383

97.3

S3

ϑ11

0.2738

0.3741

0.1003

96.1

0.2666

0.3631

0.0965

97.1

ϑ12

0.3960

0.5983

0.2023

97.1

0.3789

0.5921

0.2132

98.6

ϑ21

0.5037

1.7068

1.2031

95.5

0.8083

1.3053

0.4970

95.4

ϑ22

0.3688

1.4513

1.0825

95.5

0.6460

1.0450

0.3989

95.3

η1

0.9251

1.2501

0.3249

98.7

0.9141

1.2268

0.3127

97.2

η2

1.0891

1.4656

0.3765

98.8

1.1183

1.4931

0.3749

99.7

S4

ϑ11

0.3242

0.4626

0.1385

96.4

0.3091

0.4439

0.1347

97.2

ϑ12

0.3658

0.5588

0.1930

96.7

0.3503

0.5334

0.1831

97.2

ϑ21

1.6843

2.4777

0.7934

97.0

1.6679

2.4458

0.7778

97.2

ϑ22

1.5230

2.2536

0.7307

96.8

1.4982

2.2085

0.7103

97.2

η1

0.8980

1.2346

0.3366

96.9

0.9424

1.2898

0.3474

98.3

η2

0.9957

1.3570

0.3613

96.7

1.0290

1.3914

0.3624

97.2

S5

ϑ11

0.4330

0.6656

0.2325

95.4

0.4121

0.6347

0.2226

96.6

ϑ12

0.4330

0.6656

0.2326

95.4

0.4126

0.6383

0.2257

96.7

ϑ21

2.2489

3.4161

1.1671

96.7

2.1897

3.3144

1.1246

96.4

ϑ22

2.2489

3.4161

1.1671

96.7

2.2537

3.3410

1.0873

97.2

η1

0.8914

1.3150

0.4236

97.2

0.9392

1.3607

0.4214

96.9

η2

0.8914

1.3150

0.4236

97.2

0.9497

1.3780

0.4283

97.9

Table 12. Interval estimates, AILs, and CP (%) values of the ML and BE using MCMC for different schemes of progressive Type-II step-stress for extension Weibull distribution at n=80 and m=60

Sch.

Parm.

Asy-CI

HPD

Lower

Upper

AIL

CP (%)

Lower

Upper

AIL

CP (%)

S1

ϑ11

0.3431

0.4655

0.1224

97.3

0.3250

0.4516

0.1266

97.7

ϑ12

0.3445

0.4749

0.1304

97.6

0.3306

0.4614

0.1308

97.9

ϑ21

1.1599

1.6610

0.5012

97.1

1.1062

1.6044

0.4982

97.1

ϑ22

1.0941

1.6000

0.5059

97.5

1.0377

1.5204

0.4827

96.2

η1

0.7390

0.9523

0.2133

97.6

0.7713

0.9891

0.2179

98.0

η2

0.7584

0.9910

0.2325

97.2

0.7895

1.0305

0.2410

98.3

S2

ϑ11

0.3447

0.4685

0.1238

97.5

0.3273

0.4495

0.1222

97.9

ϑ12

0.3446

0.4731

0.1285

97.5

0.3209

0.4508

0.1298

97.1

ϑ21

1.4038

2.0149

0.6111

96.8

1.3480

1.9411

0.5931

97.1

ϑ22

1.3595

1.9733

0.6137

97.1

1.3127

1.9008

0.5881

97.5

η1

0.8119

1.0602

0.2483

97.7

0.8482

1.1180

0.2698

97.7

η2

0.8252

1.0841

0.2589

97.5

0.8622

1.1350

0.2728

97.5

S3

ϑ11

0.3161

0.4097

0.0936

96.1

0.3044

0.4004

0.0960

96.8

ϑ12

0.3430

0.5427

0.1997

98.6

0.3263

0.5162

0.1899

96.8

ϑ21

0.6557

2.2392

1.5835

94.5

0.9218

2.1655

1.2437

95.2

ϑ22

0.6347

2.2267

1.5920

94.5

0.9025

2.1642

1.2617

95.2

η1

0.7945

1.0720

0.2776

98.1

0.8174

1.0772

0.2598

97.5

η2

0.8067

1.0852

0.2785

98.3

0.8147

1.0960

0.2813

97.1

S4

ϑ11

0.3142

0.4205

0.1063

96.0

0.3004

0.4098

0.1095

96.9

ϑ12

0.3464

0.4752

0.1288

96.2

0.3355

0.4693

0.1338

98.7

ϑ21

1.1030

1.5328

0.4298

95.6

1.0799

1.5179

0.4380

97.5

ϑ22

1.0365

1.4532

0.4167

95.4

0.9915

1.4196

0.4282

96.7

η1

0.7946

0.9944

0.1998

97.8

0.8174

1.0304

0.2130

98.7

η2

0.8259

1.0392

0.2133

97.7

0.8389

1.0643

0.2254

97.1

S5

ϑ11

0.3625

0.4991

0.1366

97.2

0.3411

0.4786

0.1375

97.1

ϑ12

0.3625

0.4991

0.1366

97.2

0.3388

0.4762

0.1373

97.0

ϑ21

1.5941

2.3631

0.7690

96.4

1.5291

2.2705

0.7414

96.7

ϑ22

1.5941

2.3631

0.7690

96.4

1.5310

2.3108

0.7798

96.8

η1

0.8330

1.1239

0.2909

97.9

0.8688

1.1747

0.3059

97.6

η2

0.8330

1.1239

0.2909

97.9

0.8577

1.1688

0.3112

96.7

Table 13. Interval estimates, AILs, and CP (%) values of the ML and BE using MCMC for different schemes of progressive Type-II step-stress for extension Weibull distribution at n=100 and m=60

Sch.

Parm.

Asy-CI

HPD

Lower

Upper

AIL

CP (%)

Lower

Upper

AIL

CP (%)

S1

ϑ11

0.3212

0.4473

0.1261

98.0

0.3073

0.4318

0.1245

97.5

ϑ12

0.3206

0.4595

0.1389

97.6

0.3043

0.4406

0.1364

97.5

ϑ21

1.0732

1.5779

0.5047

96.7

1.0013

1.5056

0.5043

96.0

ϑ22

0.9685

1.4873

0.5188

97.7

0.9281

1.4168

0.4887

96.8

η1

0.7697

1.0032

0.2335

97.9

0.7887

1.0205

0.2319

96.5

η2

0.8008

1.0730

0.2721

97.3

0.8415

1.1042

0.2627

97.8

S2

ϑ11

0.3537

0.4845

0.1308

96.6

0.3412

0.4697

0.1285

97.9

ϑ12

0.3548

0.4923

0.1375

96.8

0.3343

0.4767

0.1424

97.2

ϑ21

1.5254

2.1989

0.6735

96.7

1.4651

2.1332

0.6681

96.1

ϑ22

1.4520

2.1363

0.6843

96.9

1.4264

2.0823

0.6559

98.0

η1

0.8830

1.1634

0.2804

97.7

0.9210

1.2102

0.2892

97.8

η2

0.9043

1.2054

0.3010

97.6

0.9588

1.2633

0.3046

98.2

S3

ϑ11

0.2916

0.3792

0.0875

96.3

0.2821

0.3685

0.0865

96.4

ϑ12

0.3917

0.5582

0.1665

97.7

0.3643

0.5430

0.1787

97.9

ϑ21

0.6190

1.7104

1.0913

93.8

0.8651

1.9844

1.1193

95.2

ϑ22

0.5120

1.5630

1.0510

93.8

0.7837

1.8617

1.0780

95.7

η1

0.8940

1.1707

0.2767

98.2

0.8973

1.1658

0.2684

97.6

η2

0.9810

1.2863

0.3053

98.5

0.9910

1.2898

0.2988

98.0

S4

ϑ11

0.2937

0.3846

0.0909

96.0

0.2615

0.4839

0.2223

96.6

ϑ12

0.3339

0.4612

0.1273

95.8

0.2868

0.6525

0.3656

95.7

ϑ21

1.3194

1.8837

0.5643

96.7

1.1930

1.9534

0.7604

96.7

ϑ22

1.1946

1.6966

0.5020

96.6

1.0525

1.8074

0.7549

97.2

η1

0.8670

1.0805

0.2134

///97.7

0.8111

1.1666

0.3555

95.7

η2

0.9539

1.1794

0.2255

97.6

0.8409

1.2991

0.4581

97.4

S5

ϑ11

0.4054

0.5663

0.1609

96.7

0.3909

0.5517

0.1607

98.1

ϑ12

0.4054

0.5663

0.1609

96.7

0.3912

0.5516

0.1604

97.9

ϑ21

1.9706

2.8131

0.8425

96.9

1.9024

2.7379

0.8356

96.8

ϑ22

1.9706

2.8131

0.8425

96.9

1.9181

2.7482

0.8301

97.0

η1

0.8997

1.2127

0.3131

97.6

0.9319

1.2515

0.3196

96.7

η2

0.8997

1.2127

0.3131

97.6

0.9262

1.2503

0.3242

96.6

Table 14. Interval estimates, AILs, and CP (%) values of the ML and BE using MCMC for different schemes of progressive Type-II step-stress for extension Weibull distribution at n=100 and m=80

Sch.

Parm.

Asy-CI

HPD

Lower

Upper

AIL

CP (%)

Lower

Upper

AIL

CP (%)

S1

ϑ11

0.3583

0.4613

0.1029

97.2

0.3446

0.4520

0.1074

97.8

ϑ12

0.3593

0.4676

0.1083

97.6

0.3427

0.4576

0.1149

97.5

ϑ21

1.2029

1.6792

0.4763

96.5

1.1373

1.6201

0.4828

96.3

ϑ22

1.1533

1.6315

0.4782

96.6

1.0956

1.5755

0.4799

96.6

η1

0.7368

0.9299

0.1931

97.2

0.7602

0.9601

0.1998

97.6

η2

0.7521

0.9574

0.2053

96.9

0.7730

0.9885

0.2155

97.4

S2

ϑ11

0.3520

0.4589

0.1069

97.3

0.3364

0.4447

0.1083

97.2

ϑ12

0.3518

0.4615

0.1097

97.1

0.3268

0.4396

0.1128

96.1

ϑ21

1.3953

1.9262

0.5309

96.4

1.3579

1.8758

0.5180

97.2

ϑ22

1.3632

1.8958

0.5326

96.6

1.2933

1.8440

0.5506

96.3

η1

0.8060

1.0162

0.2102

97.1

0.8294

1.0478

0.2184

97.0

η2

0.8158

1.0327

0.2169

96.8

0.8363

1.0663

0.2300

97.0

S3

ϑ11

0.3284

0.4120

0.0836

96.7

0.3186

0.4059

0.0872

97.2

ϑ12

0.3390

0.5242

0.1852

99.2

0.3345

0.5006

0.1661

98.3

ϑ21

0.7778

2.2237

1.4459

93.4

1.0286

2.1988

1.1702

95.4

ϑ22

0.7882

2.2249

1.4366

93.4

1.0504

2.2068

1.1564

95.7

η1

0.7944

1.0166

0.2223

98.7

0.8127

1.0251

0.2124

97.5

η2

0.7972

1.0133

0.2161

98.5

0.8235

1.0312

0.2077

97.9

S4

ϑ11

0.3324

0.4289

0.0965

96.6

0.3208

0.4174

0.0966

96.6

ϑ12

0.3545

0.4657

0.1112

96.6

0.3452

0.4584

0.1132

97.4

ϑ21

1.1546

1.5660

0.4114

96.6

1.1225

1.5279

0.4054

97.2

ϑ22

1.1005

1.5032

0.4027

96.4

1.0727

1.4812

0.4085

97.8

η1

0.7766

0.9645

0.1880

98.1

0.7912

0.9778

0.1866

97.0

η2

0.8002

0.9975

0.1972

98.1

0.8190

1.0153

0.1964

97.6

S5

ϑ11

0.3642

0.4748

0.1106

97.3

0.3435

0.4567

0.1132

96.9

ϑ12

0.3642

0.4748

0.1106

97.3

0.3476

0.4609

0.1133

97.9

ϑ21

1.5671

2.1644

0.5973

96.9

1.5175

2.1039

0.5865

96.5

ϑ22

1.5672

2.1644

0.5973

96.9

1.5225

2.1222

0.5997

97.4

η1

0.8366

1.0637

0.2270

98.5

0.8704

1.1006

0.2301

97.4

η2

0.8366

1.0637

0.2270

98.5

0.8705

1.1098

0.2394

98.0

Table 15. Est. and St.Ers of the ML and BE using MCMC for the illustrative example

Parm.

MLE

BE MCMC: SEL

BE MCMC: LINEX

Est.

St.Er

Est.

St.Er

Est.: $v=-1.5$

Est.: $v=1.5$

ϑ11

0.37645

0.11855

0.34314

0.09630

0.33636

0.32027

ϑ12

0.40752

0.12840

0.37572

0.11769

0.36586

0.36269

ϑ21

1.90398

0.56882

2.11279

0.71536

1.83632

1.63905

ϑ22

1.75120

0.47933

1.80192

0.57464

1.61408

1.60377

η1

1.26986

0.35908

1.20095

0.33957

1.12060

1.02945

η2

1.41021

0.35270

1.40928

0.34697

1.32175

1.30302

Table 16. Interval estimates and ILs values of the ML and BE using MCMC for the illustrative example

Parm.

Asy-CI

HPD

Lower

Upper

AIL

Lower

Upper

AIL

ϑ11

0.14410

0.60880

0.46470

0.16710

0.53427

0.36716

ϑ12

0.15587

0.65918

0.50331

0.16813

0.62265

0.45452

ϑ21

0.78911

3.01884

2.22973

1.01234

3.00616

1.99382

ϑ22

0.81172

2.69068

1.87896

0.82916

2.00659

1.17743

η1

0.56609

1.97364

1.40755

0.53728

1.82395

1.28667

η2

0.71893

2.10150

1.38257

0.71233

2.08963

1.37730

Figure 1. Convergence of MCMC estimates for ϑ11 and ϑ12 for n=80 and m=60

Figure 2. Convergence of MCMC estimates for ϑ21 and ϑ22 for n=80 and m=60

Nonetheless, the CIs of HPD is better than ACIs for MLE, this is because, the AIL of HPD is less than AIL for MLE. in addition to The BEs of the considered parameters based on LINEX loss function $(\tau=1.5)$ are smaller than that based on SE loss function.

Figure 3. Convergence of MCMC estimates for η1 and η2 for n=80 and m=60

Figure 4. Convergence of MCMC estimates of ϑ11 and ϑ12 for the illustrative example

Figure 5. Convergence of MCMC estimates of ϑ21 and ϑ22 for the illustrative example

Figure 6. Convergence of MCMC estimates of η1 and η2 for the illustrative example

7. Conclusion

In this article, established on a cumulative exposure model under several progressively censoring type II, we studied a simple SS-ALT model with two independent failures competing for risks from extension Weibull distribution. We have derived MLEs and asymptotic confidence interval estimates for the unknown parameters of extension Weibull distribution. Also, we computed BEs and the corresponding HPD interval estimates under informative priors based on two different types of loss functions LINEX and squared error loss functions. We have then performed a simulation study to assess the performance of all these procedures and an explanatory instance has been offered to demonstrate all the methods of inference developed in this paper. An approximate CI and Bayesian credible interval for the unknown parameters are discussed when the sample size increases. Established on the outcomes of the simulation study, our recommendation for the BEs performs better than MLEs in terms of Average estimate and MSE. The MLEs and BEs for fixed sample size and pre-fixed number of failures increase, the MSEs and Average estimate for unknown parameter decrease. The MLE and Bayesian estimates for Scheme 5 have best properties than other Schemes. Nevertheless, MSE for BE under LINEX with $(\tau=1.5)$ is smaller than other BE methods for all parameters expected. Our recommendation for credible/confidence intervals, the average lengths of ACIs and credible intervals become smaller as the pre-fixed number of failures increases.

  References

[1] Rao, B.R. (1992). Equivalence of the tampered random variables and tampered failure rate models in ALT for a class of life distribution having the setting the clock back to zero property. Communication in Statistics-Theory and Methods, 21(3): 647-664. https://doi.org/10.1080/03610929208830805

[2] Pham, H. (2003). Handbook of Reliability Engineering. Springer London. https://doi.org/10.1007/b97414 

[3] Nelson, W. (1990). Accelerated Testing: Statistical Models Test Plans and Data Analyses. John Wiley and Sons, INC., Hoboken, New Jersey. https://download.e-bookshelf.de/download/0000/5714/39/L-G-0000571439-0002358027.pdf.

[4] Mohie El-Din, M.M., Abu-Youssef, S.E., Ali, N.S.A., Abd El-Raheem, A.M. (2016). Parametric inference on step-stress accelerated life testing for the extension of exponential distribution under progressive type-II censoring. Communications for Statistical Applications and Methods, 23(4): 269-285. https://dx.doi.org/10.5351/CSAM.2016.23.4.269

[5] Han, D., Bai, T. (2019). On the maximum likelihood estimation for progressively censored lifetimes from constant-stress and step-stress accelerated tests. Electronic Journal of Applied Statistical Analysis, 12(2): 392-404. https://dx.doi.org/10.1285/i20705948v12n2p392

[6] Abd-Elfattah, A.M., Hassan, A.S., Nassr, S.G. (2008). Estimation in step-stress partially accelerated life tests for the burr type XII distribution using type I censoring. Statistical Methodolgy, 5(6): 502-514. https://doi.org/10.1016/j.stamet.2007.12.001

[7] Mohie El-Din, M.M., Abd El-Raheem, A.M., Abd El-Azeem, S.O. (2021). On progressive-stress accelerated life testing for power generalized Weibull distribution under progressive type-II censoring. Journal of Statistics Applications & Probability Letters, 5(3): 131-143. http://dx.doi.org/10.18576/jsapl/050303

[8] Hafez, E.H., Riad, F.H., Mubarak, S.A.M., Mohamed, M.S. (2020). Study on Lindley distribution accelerated life tests: Application and numerical simulation. Symmetry, 12(12): 1-18. https://doi.org/10.3390/sym12122080

[9] Nassr, S.G., Elharoun, N.M. (2019). Inference for exponentiated Weibull distribution under constant stress partially accelerated life tests with multiple censored. Communications for Statistical Applications and Methods, 26(2): 131-148. https://doi.org/10.29220/CSAM.2019.26.2.131

[10] Mohie El-Din, M.M., Abd El-Raheem, A.M., Abd El-Azeem, S.O. (2021). On step-stress accelerated life testing for power generalized Weibull distribution under progressive type-II censoring. Annals of Data Science, 8: 629-644. https://doi.org/10.1007/s40745-020-00270-4

[11] Mahto, A.K., Dey, S., Tripathi, Y.M. (2020). Statistical inference on progressive-stress accelerated life testing for the logistic exponential distribution under progressive type II censoring. Quality and Reliability Engineering International, 36(1): 112-124. https://doi.org/10.1002/qre.2562

[12] Nassar, M., Nassr, S.G., Dey, S. (2017). Analysis of burr type XII distribution under step stress partially accelerated life tests with type I and adaptive type II progressively hybrid censoring schemes. Annals of Data Sciences, 4(2): 227-248. https://doi.org/10.1007/s40745-017-0101-8

[13] Wang, B.X., Yu, K., Sheng, Z. (2014). New inference for constant-stress accelerated life tests with Weibull distribution and progressively type-II censoring. IEEE Transactions on Reliability, 63(3): 807-815. https://doi.org/10.1109/TR.2014.2313804

[14] Mohie El-Din, M.M., Abu-Youssef, S.E., Ali, N.S.A., Abd El-Raheem, A.M. (2015). Estimation in step-stress accelerated life tests for power generalized Weibull distribution with progressive censoring. Advances in Statistics, 2015: 319051. http://dx.doi.org/10.1155/2015/319051  

[15] Riad, F.H., Hafez, E.H., Mubarak, A.M. (2021). Study on step-stress accelerated life testing for the Burr-XII distribution using cumulative exposure model under progressive type-II censoring with real data example. Journal of Statistics Application & Probability, 10(1): 35-44. http://dx.doi.org/10.18576/jsap/100104  

[16] Balakrishnan, N., Aggarwala, R. (2000). Progressive Censoring: Theory, Methods and Applications. Boston: Birkhäuser. https://doi.org/10.1007/978-1-4612-1334-5

[17] Cramer, E., Schmiedt, A.B. (2011). Progressively type-II censored competing risks data from Lomax distributions. Computational Statistics & Data Analysis, 55(3): 1285-1303. https://doi.org/10.1016/j.csda.2010.09.017

[18] Pareek, B., Kundu, D., Kumar, S. (2009). On progressively censored competing risks data for Weibull distributions. Computational Statistics & Data Analysis, 53(12): 4083-4094. https://doi.org/10.1016/j.csda.2009.04.010

[19] Sarhan, A.M., Alameri, M., Al-Wasel, I. (2008). Analysis of progressive censoring competing risks data with binomial removals. International Journal of Mathematical Analysis, 2(17-20): 965-976. http://www.m-hikari.com/ijma/ijma-password-2008/ijma-password17-20-2008/index.html, accessed on Sep. 23, 2022.

[20] Han, D., Kundu, D. (2015). Inference for a step-stress model with competing risks for failure from the generalized exponential distribution under type-I censoring. IEEE Transactions on Reliability, 64(1): 31-43. https://doi.org/10.1109/TR.2014.2336392

[21] Crowder, M.J. (1991). On the identifiability crises in competing risks analysis. Scandinavian Journal of Statistics, 18(3): 223-233. https://www.jstor.org/stable/4616205, accessed on Oct. 10, 2022.

[22] Kalbfleisch, J.D., Prentice, R.L. (2002). The Statistical Analysis of Failure Time Data. New York, John Wiley & Sons. doi:10.1002/9781118032985

[23] Klein, J.P., Basu, A.P. (1981). Weibull accelerated life tests when there are competing causes of failure. Communication in Statistics-Theory and Methods, 10(20): 2073-2100. https://doi.org/10.1080/03610928108828174

[24] Klein, J.P., Basu, A.P. (1982). Accelerated life testing under competing exponential failure distributions. IAPQR Trans, 7: 1-20. https://apps.dtic.mil/sti/citations/ADA107626.

[25] Xu, A., Tang, Y. (2011). Objective Bayesian analysis of accelerated competing failure models under Type-I censoring. Computational Statistics & Data Analysis, 55(10): 2830-2839. https://doi.org/10.1016/j.csda.2011.04.009

[26] Shi, Y.M., Jin, L., Wei, C., Yue, H.B. (2013). Constant stress accelerated life test with competing risks under progressive type II hybrid censoring. Advanced Materials Research, 712-715: 2080-2083. https://doi.org/10.4028/www.scientific.net/AMR.712-715.2080

[27] Wu, M., Shi, Y., Sun, Y. (2014). Inference for accelerated competing failure models from Weibull distribution under Type-I progressive hybrid censoring. Journal of Computational and Applied Mathematics, 263: 423-431. https://doi.org/10.1016/j.cam.2013.12.048

[28] Yousef, M.M., Alsultan, R., Nassr, S.G. (2022). Parameter inference on partially accelerated life testing for the inversed Kumaraswamy distribution based on type-II progressive censoring data. Mathematical Biosicences and Engineering, 20(2): 1674-1694. https://doi.org/10.3934/mbe.2023076

[29] Wu, S.J., Huang, S.R. (2017). Planning two or more level constant-stress accelerated life tests with competing risks. Reliability Engineering & System Safety, 158: 1-8. https://doi.org/10.1016/j.ress.2016.09.007

[30] Balakrishnan, N., Han, D. (2008). Exact inference for a simple step-stress model with competing risks for failure from exponential distribution under Type-II censoring. Journal of Statistical Planning and Inference, 138(12): 4172-4186. https://doi.org/10.1016/j.jspi.2008.03.036

[31] Han, D., Balakrishnan, N. (2010). Inference for a simple step-stress model with competing risks for failure from exponential distribution under time Constraint. Computational Statistics & Data Analysis, 54(9): 2066-2081. https://doi.org/10.1016/j.csda.2010.03.015

[32] Ganguly, A., Kundu, D. (2015). Analysis of simple step stress model in presence of competing risks. Journal of Statistical Computation and Simulation, 86(10): 1989-2006. doi:10.1080/00949655.2015.1096362

[33] David, H.A., Moeschberger, M.L. (1978). The theory of competing risks. London, Griffin. https://catalogue.nla.gov.au/Record/2560111?lookfor=subject:%22Competing%20risks.%22&offset=1&max=4, accessed on Jun. 20, 2022.

[34] Sarhan, A.M., Hamilton, D.C., Smith, B. (2010). Statistical analysis of competing risks models. Reliability Engineering and System Safety, 95(9): 953-962. https://doi.org/10.1016/j.ress.2010.04.006

[35] Abu El Azm, W.S., Aldallal, R., Aljohani, H.M., Nassr, S.G. (2022). Estimations of Competing lifetime data from inverse Weibull distribution under adaptive progressively hybrid censored. Mathematical Biosciences and Engineering, 19(6): 6252-6276. https://doi.org/10.3934/mbe.2022292

[36] Crowder, M.J. (2001). Classical Competing Risks. Chapman & Hall. https://www.routledge.com/Classical-Competing-Risks/Crowder/p/book/9780429119217, accessed on Sep. 23, 2022.

[37] Nassr, S.G., Abu El Azm, W.S., Almetwally, E.M. (2021). Statistical inference for the extended weibull distribution based on adaptive type-II progressive hybrid censored competing risks data. Thailand Statistician, 19(3): 547-564. https://ph02.tci-thaijo.org/index.php/thaistat, accessed on Oct. 20, 2022.

[38] Liu, F., Shi, Y. (2017). Inference for a simple step-stress model with progressively censored competing risks data from Weibull distribution. Communications in Statistics-Theory and Methods, 46(14): 7238-7255. https://doi.org/10.1080/03610926.2016.1147585

[39] Hassan, A.S., Nassr, S.G., Pramanik, S., Maiti, S.S. (2020). Estimation in constant stress partially accelerated life tests distribution based on censored competing risks data. Annals of Data Science, 7(1): 45-62. https://doi.org/10.1007/s40745-019-00226-3

[40] Samanta, D., Gupta, A., Kundu, D. (2019). Analysis of Weibull step-stress model in presence of competing risk. IEEE Transactions on Reliability, 68(2): 420-438. https://doi.org/10.1109/TR.2019.2896319

[41] Fan, T., Wang, Y. (2021). Comparison of optimal accelerated life tests with competing risks model under exponential distribution. Quality and Reliability Engineering International, 37: 902-919. https://doi.org/10.1002/qre.2772

[42] Chen, Z. (2000). A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function. Statistics & Probability Letters, 49(2): 155-161. https://doi.org/10.1016/S0167-7152(00)00044-4

[43] Gilks, W.R., Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Journal of the Royal Statistical Society. Series C (Applied Statistics), 41(2): 337-348. https://doi.org/10.2307/2347565

[44] Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. (1953). Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6): 1087-1091. https://doi.org/10.1063/1.1699114

[45] Balakrishnan, N., Sandu, R.A. (1995). A simple simulation algorithm for generating progressive Type-II censored samples. The American Statistician, 49(2): 229-230. https://www.tandfonline.com/doi/abs/10.1080/00031305.1995.10476150.

[46] Chen, M.H., Shao, Q.M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8(1): 69-92. https://doi.org/10.1080/10618600.1999.10474802