Inference for the Exponential Distribution under Generalized Progressively Hybrid Censored Data from Partially Accelerated Life Tests with a Time Transformation Function

Abstract

In this article, the tampered failure rate model is used in partially accelerated life testing. A non-decreasing time function, often called a $‘‘$time transformation function", is proposed to tamper the failure rate under design conditions. Different types of the proposed function, which have sufficient conditions in order to be accelerating functions, are investigated. A baseline failure rate of the exponential distribution is considered. Some point estimation methods, as well as approximate confidence intervals, for the parameters involved are discussed based on generalized progressively hybrid censored data. The determination of the optimal stress change time is discussed under two different criteria of optimality. A real dataset is employed to explain the theoretical outcomes discussed in this article. Finally, a Monte Carlo simulation study is carried out to examine the performance of the estimation methods and the optimality criteria.

1. Introduction

In the modern era, the products of manufacturing have attained a high quality and have become reliable. Thus, it is very hard to acquire sufficient failure information for products under normal functioning conditions. Moreover, testing their lifetimes under these conditions takes a long time and is very expensive. To overcome these problems, accelerated life tests (ALTs) or partially ALTs (PALTs) are applied to obtain sufficient information about the products’ failure data quickly and in a short time. In ALTs, products are tested under higher-than-regular levels of stress to obtain early failure times. In PALTs, products are tested under both regular and accelerated conditions. The stress factors may be pressure, vibration, temperature, cycling rate, voltage, or any other factor that immediately affects the lifetime of the products. The information obtained from the accelerated or partially accelerated tests is used to predict the failure behavior of the products under normal conditions. Various techniques can be used to implement stress in ALTs. Common techniques are the progressive-stress, step-stress, and constant-stress techniques. For a good survey on ALTs, see [1,2,3,4,5,6].

In a PALT experiment, a random sample of units is put through a life test at a normal stress level, and then the stress is raised after a pre-fixed time or a pre-fixed number of failed units. The experiment continues under accelerated conditions until the units fail or the experiment’s time is finished. For more details, see [4] and [7,8,9,10,11,12,13].

Three major models relate the distribution under accelerated stress to that under normal stress. The models are (i) the tampered random variable (TRV) model suggested in [14], (ii) the cumulative exposure (CE) model given by [15], and (iii) the tampered failure rate (TFR) model suggested in [16].

Several authors used the TFR model in ALTs; Wang and Fei [17] studied the conditions under which the coincidence of the TRV, TFR, and CE models occurred. They also generalized the TFR model from the step-stress ALT setting to the progressive-stress ALT setting (see also [10,18,19,20,21]).

In our paper, we focus on the TFR model. As in Bhattacharyya and Soejoeti [16], the stress is raised after a change time point $\tau$ by multiplying the initial failure rate function (FRF) $h\left(t\right)$ by an unknown factor $\lambda$ (which may depend on $\tau$). If $h^{☆}\left(t\right)$ is the total FRF in the PALT, then their proposed TFR model is defined by

$$h^{☆}\left(t\right)=\left\{\begin{array}{cc}h\left(t\right),& 0<t\le \tau ,\\ \lambda h\left(t\right),& \tau <t<\infty ,(\lambda >1).\end{array}\right.$$

(1)

Censoring schemes have a major role in lifetime and reliability studies. In many practical experiments that depend on the lifetimes of items, the experiment may be finished before all of the items have failed due to the experimental time considered and the associated cost. In such situations, only a part of the failure information of the items is recorded, and the data obtained are called censored data.

Type-I and type-II censoring schemes are two of the most common censoring schemes implemented in life tests. Epstein [22] suggested a mixture of type-I and type-II censoring schemes called the hybrid censoring scheme. In many situations, it is planned in advance to remove items before failure at various steps of the experiment, but the above models of censoring schemes do not have the flexibility to allow for items to be removed from the experiment at various steps other than the experiment’s end point. To solve this problem, Cohen [23] introduced a type-II progressive censoring scheme as a generalization for the above models of censoring schemes.

The type-II progressive censoring scheme can be implemented as follows: Assume that a random sample of n items is subjected to a life test experiment and that the number of observed failures $m(<n)$ is predetermined before beginning the experiment with a pre-assigned progressive censoring scheme ($R_1,R_2,\cdots ,R_m$). At the first failure time $T_{1:n}$, $R_1$ operating items are selected at random and excluded from the experiment. At the next failure time $T_{2:n}$, $R_2$ operating items are selected at random and excluded from the experiment. The procedure is continued in the same manner until the last failure time $T_{m:n}$ occurs, at which all of the remaining operating items $R_m=n-m-{\sum }_{i=1}^{m-1}{R}_i$ are excluded from the experiment. Then, the experiment is stopped at $T_{m:n}$.

A major drawback of the type-II progressive censoring scheme is that if the items are of a high quality and are reliable, the experimental time may be quite long. Kundu and Joarder [24] attempted to address this drawback in a modified scheme known as a type-I progressive hybrid censoring scheme, in which $n,m,$ and ($R_1,R_2,\cdots ,R_m$), as well as the experimental time $\eta$, were assigned before beginning the experiment. In this situation, the experiment was finished at time $T^{*}=$min$\left\{T_{m:n},\eta \right\}$. Except at the end time point, this scheme is the same as the type-II progressive censoring scheme.

One limitation of the type-I progressive hybrid censoring scheme is that the practical sample size is random and can be extremely small. Hence, statistical inference techniques can either be invalid or may be less successful. A new generalization of progressive censoring, which is called the generalized type-I progressive hybrid censoring scheme, was suggested by Cho et al. [25] to avoid defects that appeared in the type-I progressive hybrid censoring scheme. This censoring scheme ensures the satisfaction of at least a constant number of observed items $k(<m<n)$ to provide the efficiency required for statistical analysis, and it also controls the total time of the experiment to stay close to the optimum time $\eta$ if the number of observed failures is very small up to time $\eta$. In this situation, the experiment is finished at the time $T^{*}=\max \{T_{k:n},\min \left\{T_{m:n},\eta \right\}\}$, and all of the remaining operating items are excluded from the experiment.

Several authors have discussed statistical inference based on the generalized type-I progressive hybrid censoring scheme. For example, Cho et al. [25,26] developed the exact likelihood inference technique for an exponential distribution and an entropy estimation technique for a Weibull distribution. Koley and Kundu [27] and Wang et al. [28] discussed the maximum likelihood and Bayes estimation techniques for exponential and Weibull competing risk models. Zhang and Shi [29] discussed the statistical prediction problem of unobserved failure times based on the simple step-stress ALT with the generalized exponential competing risk model.

What is new in the current article is the implementation of PALTs when the failure rate is accelerated under design conditions through a non-decreasing time function, which is often called a “time transformation function". Three different types of time functions are considered. The sufficient condition under which the time function can be an accelerating function is discussed in detail with each type. Based on generalized progressively hybrid censored data, some point and interval estimations for the parameters involved are discussed. The problem of determining the optimal stress change time is studied through two different criteria of optimality.

The remaining sections of the article are laid out as follows: In Section 2, the model is described with generalized type-I progressive hybrid censoring. Some estimation methods are discussed in Section 3. The optimal stress change time is verified in Section 4 based on two optimality criteria. An application to a real dataset is introduced in Section 5. Simulation studies are presented in Section 6, followed by concluding remarks in Section 7.

2. Model Description and Generalized Type-I Progressive Hybrid Censoring

DeGroot and Goel [14] considered a PALT in which a test unit is first run at normal conditions and, if it does not fail for a specified time $\tau$, then it is run at accelerated conditions until failure occurs.

According to the TFR model (1), we assume that the n units to be tested are initially under normal conditions up to a fixed time $\tau$, and if they do not fail by this time, they are placed under accelerated conditions until failure occurs or the experimental time goes to its end—whichever is realized first. The acceleration occurs by multiplying the FRF of the units under normal conditions by a factor $\lambda (>1)$.

What is new here is the choice of $\lambda$ to be a non-negative increasing function of time t, $\lambda \equiv \lambda \left(t\right)$, ($t>\tau$), that is continuous and differentiable. Therefore, Model (1) becomes

$$h^{☆}\left(t\right)=\left\{\begin{array}{cc}{h}_{(}t),& 0<t\le \tau ,\\ \lambda \left(t\right)h_{(}t),& \tau <t<\infty .\end{array}\right.$$

(2)

According to Model (2), the accelerated conditions increase over time, causing failures to occur more quickly than those previously considered when $\lambda$ is considered constant. This also saves time and cost in the experiments.

In lifetime testing experiments and reliability studies, the exponential distribution (ED) is one of the most commonly discussed distributions due to its simplicity and easy mathematical manipulations, as it produces a simple, elegant, and closed-form solution to many problems. We assume that the random variable T follows the ED with scale parameter $\beta (>0)$. Then, the probability density function (PDF), cumulative distribution function (CDF), and FRF of T are given, respectively, by

$$f\left(t\right)=\frac{1}{\beta }exp\left[-\frac{t}{\beta }\right],0<t<\infty ,$$

$$F\left(t\right)=1-exp\left[-\frac{t}{\beta }\right],0<t<\infty ,$$

$$h\left(t\right)=\frac{1}{\beta },0<t<\infty .$$

(3)

According to Model (2), the corresponding CDF of the total lifetime T in the PALT is given by

$$F^{*}\left(t\right)=\left\{\begin{array}{c}{F}_1\left(t\right)=1-\exp \left[-{\int }_0^{t}h\left(t\right)\mathrm{d}t\right],0<t\le \tau ,\hfill \\ F_2\left(t\right)=1-\exp \left[-{\int }_0^{\tau }h\left(t\right)\mathrm{d}t-{\int }_{\tau }^t\lambda \left(t\right)h\left(t\right)\mathrm{d}t\right],\tau <t<\infty .\hfill \end{array}\right.$$

(4)

In the following, based on the FRF (3), the function $F_2\left(t\right)$ defined in the CDF (4) will be deduced for three different cases of $\lambda \left(t\right)\equiv \lambda (t;\alpha )$, where $\alpha$ is a positive parameter.

• Case A: If $\lambda \left(t\right)={(t-\tau +1)}^{\alpha }$, then $F_2\left(t\right)$, $\tau <t<\infty$, is deduced as follows

  $$\begin{array}{cc}\hfill F_2\left(t\right)& =1-exp\left[-{\int }_0^{\tau }\frac{1}{\beta }dt-{\int }_{\tau }^t\frac{1}{\beta }{(t-\tau +1)}^{\alpha }dt\right]\hfill \\ \hfill & =1-exp\left[-\frac{1}{\beta }\left(\frac{\tau (\alpha +1)+{(t-\tau +1)}^{\alpha +1}-1}{\alpha +1}\right)\right]\hfill \\ \hfill & =1-exp\left[-\frac{{\psi }_1\left(t\right)}{\beta }\right],\hfill \end{array}$$

  (5)

  where

  $${\displaystyle {\psi }_1\left(t\right)=\frac{\tau (\alpha +1)+{(t-\tau +1)}^{\alpha +1}-1}{\alpha +1}.}$$

  (6)
  The sufficient condition for the function ${\psi }_1\left(t\right)$ to be an accelerating function is (see Mann et al. [30])

  $${\psi }_1\left(t\right)>t⟺{(t-\tau +1)}^{\alpha +1}>(\alpha +1)(t-\tau )+1.$$

  (7)
• Case B: If $\lambda \left(t\right)=e^{\alpha (t-\tau )}$, then, as shown in Case A, $F_2\left(t\right)$, $\tau <t<\infty$, can be written in the form

  $$\begin{array}{cc}\hfill F_2\left(t\right)& =1-exp\left[-\frac{{\psi }_2\left(t\right)}{\beta }\right],\hfill \end{array}$$

  (8)

  where

  $${\psi }_2\left(t\right)=\tau +\frac{e^{\alpha (t-\tau )}-1}{\alpha }.$$

  (9)
  The sufficient condition for the function ${\psi }_2\left(t\right)$ to be an accelerating function is

  $${\psi }_2\left(t\right)>t⟺e^{\alpha (t-\tau )}>\alpha (t-\tau )+1.$$

  (10)
• Case C: If $\lambda \left(t\right)=ln\left(e\left[\alpha \right(t-\tau )+1]\right)$, then, as shown in Case A, $F_2\left(t\right)$, $\tau <t<\infty$, can be written in the form

  $$\begin{array}{cc}\hfill F_2\left(t\right)& =1-exp\left[-\frac{{\psi }_3\left(t\right)}{\beta }\right],\hfill \end{array}$$

  (11)

  where

  $${\psi }_3\left(t\right)=\tau +\left(t-\tau +\frac{1}{\alpha }\right)ln\left(\alpha (t-\tau )+1\right).$$

  (12)
  The sufficient condition for the function ${\psi }_3\left(t\right)$ to be an accelerating function is

  $${\psi }_3\left(t\right)>t⟺ln\left(\alpha (t-\tau )+1\right)>\frac{\alpha (t-\tau )}{\alpha (t-\tau )+1}.$$

  (13)

Hence, based on the FRF (3), the CDF (4) of the total lifetime T in the PALT for Cases A, B, and C can be rewritten in the form

$$F^{*}\left(t\right)=\left\{\begin{array}{cc}{F}_1\left(t\right)=1-exp\left[{\displaystyle -\frac{t}{\beta }}\right],\hfill & 0<t\le \tau ,\\ F_2\left(t\right)=1-exp\left[{\displaystyle -\frac{\psi \left(t\right)}{\beta }}\right],\hfill & \tau <t<\infty ,\end{array}\right.$$

(14)

where

$$\psi \left(t\right)=\left\{\begin{array}{cc}{\displaystyle {\psi }_1\left(t\right)=\frac{\tau (\alpha +1)+{(t-\tau +1)}^{\alpha +1}-1}{\alpha +1}}\hfill & \mathrm{for}\mathrm{Case}\mathrm{A},\\ {\displaystyle {\psi }_2\left(t\right)=\tau +\frac{e^{\alpha (t-\tau )}-1}{\alpha }}\hfill & \mathrm{for}\mathrm{Case}\mathrm{B},\\ {\psi }_3\left(t\right)=\tau +\left(t-\tau +\frac{1}{\alpha }\right)ln\left(\alpha (t-\tau )+1\right)\hfill & \mathrm{for}\mathrm{Case}\mathrm{C}.\end{array}\right.$$

(15)

The corresponding PDF is given by

$$f^{*}\left(t\right)=\left\{\begin{array}{cc}{\displaystyle f_1\left(t\right)=\frac{1}{\beta }exp\left[-\frac{t}{\beta }\right],}\hfill & 0<t\le \tau ,\\ {\displaystyle f_2\left(t\right)=\frac{1}{\beta }{\psi }^{{}^{\prime }}\left(t\right)exp\left[-\frac{\psi \left(t\right)}{\beta }\right],}\hfill & \tau <t<\infty ,\end{array}\right.$$

(16)

where

$${\psi }^{{}^{\prime }}\left(t\right)=\frac{\mathrm{d}\psi \left(t\right)}{\mathrm{d}t}=\left\{\begin{array}{cc}{\psi }_1^{{}^{\prime }}\left(t\right)={(t-\tau +1)}^{\alpha }\hfill & \mathrm{for}\mathrm{Case}\mathrm{A},\\ {\psi }_2^{{}^{\prime }}\left(t\right)=e^{\alpha (t-\tau )}\hfill & \mathrm{for}\mathrm{Case}\mathrm{B},\\ {\psi }_3^{{}^{\prime }}\left(t\right)=1+ln\left(\alpha (t-\tau )+1\right)\hfill & \mathrm{for}\mathrm{Case}\mathrm{C}.\end{array}\right.$$

(17)

Remark 1.

It can be noticed that for the three Cases A, B, and C, $\lambda \left(t\right)=1$ at $t=\tau$, and hence, the FRF under accelerated conditions tends to the FRF under normal conditions, similarly to what happens in Model (1).

Based on Equation (3), the total FRF in Model (2) is plotted in Figure 1 for three cases of $\lambda \left(t\right)$, where it can be noticed that the FRF increases over time after the change time $\tau$, and further increases also happen by increasing the value of parameter $\alpha$.

2.1. Generalized Type-I Progressive Hybrid Censoring

In a PALT, the generalized type-I progressive hybrid censoring can be described as follows:

• Suppose that a set n of randomly chosen units is subjected to a lifetime testing experiment.
• The units are initially tested at normal stress conditions up to a fixed time $\tau$. If they do not fail by this time, then they are tested under accelerated conditions.
• The associated lifetimes of the units, $T_1,T_2,\cdots ,T_n$, have the same distribution as that of the CDF (14) and PDF (16).
• Suppose that the experimental time $\eta$ and the integers $k,m$ and ($R_1,R_2,\cdots ,R_m$) are assigned before beginning the experiment, such that $0<\tau <\eta <\infty$, $0<k<m\le n$.
• At the first failure time $T_{1:n}$, $R_1$ operating units are selected at random and excluded from the experiment. At the next failure time $T_{2:n}$, $R_2$ operating units are selected at random and excluded from the experiment, and this technique continues. Finally, the experiment is finished at the time $T^{*}=\max \{T_{k:n},\min \left\{T_{m:n},\eta \right\}\}$, and all of the remaining operating units $R_{\mathfrak{F}}$ are excluded from the experiment. The values of the final censoring number $R_{\mathfrak{F}}$ are given in

  Table 1. Values of $T^{*}$, $\mathfrak{B}$, $\xi$, and $R_{\mathfrak{F}}$ for Cases 1–5.

  	Case 1	Cases 2 and 3	Cases 4 and 5
  $T^{*}$	$T_{k:n}$	$\eta$	$T_{m:n}$
  $\mathfrak{B}$	k	D	m
  $\xi$	0	1	0
  $R_{\mathfrak{F}}$	$R_k=R_k^{*}=n-k-{\sum }_{i=1}^{k-1}{R}_i,$	$R_{\eta }^{*}=n-D-{\sum }_{i=1}^D{R}_i$	$R_m=n-m-{\sum }_{i=1}^{m-1}{R}_i$
  	${\sum }_{i=D+1}^{k-1}{R}_i=0$

  .
• Let r($r^{*}$) be the number of units that fail under normal (accelerated) stress conditions before (after) time $\tau$, and let D be the number of units that fail before time $\eta$. Then, the experimental end time $T^{*}$ is given by

  $$T^{*}=\left\{\begin{array}{cc}{T}_{k:n}\hfill & \mathrm{if}\tau <\eta <T_{k:n}<T_{m:n},\\ \eta \hfill & \mathrm{if}\left\{\begin{array}{c}\tau <T_{k:n}\le \eta <T_{m:n},\\ T_{k:n}\le \tau <\eta <T_{m:n}.\end{array}\right.\\ T_{m:n}\hfill & \mathrm{if}\left\{\begin{array}{c}\tau <T_{k:n}<T_{m:n}\le \eta ,\\ T_{k:n}\le \tau <T_{m:n}\le \eta ,\\ T_{k:n}<T_{m:n}\le \tau <\eta .\end{array}\right.\end{array}\right.$$
  One of the next six cases may be noticed with the following observations:
  • Case 1: If the experimental time $\eta$ is reached before the k-th failure time $T_{k:n}$ occurs and $D(<k)$ failures occur up to time $\eta$, $\tau <\eta <T_{k:n}<T_{m:n}$, then we will not eliminate any operating units from the experiment directly following the $(D+1)$-th, ..., $(k-1)$-th failure times and will eliminate all of the remaining operating units $R_k^{*}=n-k-{\sum }_{i=1}^{k-1}{R}_i$ from the experiment at the k-th failure time, thereby stopping the experiment at $T^{*}=T_{k:n}$, where $R_{D+1}=\cdots =R_{k-1}=0$; see Figure 2. In this case, we permit the experiment to continue after experimental time $\eta$ is reached to ensure that at least the k-th failure time $T_{k:n}$ occurs. In this case, $r^{*}=k-r$ and the following observations will be noticed:

    $$T_{1:n}<\cdots <T_{r:n}\le \tau <T_{r+1:n}<\cdots <T_{D:n}\le \eta <T_{D+1:n}<\cdots <T_{k:n}.$$
  • Case 2: If the k-th failure time $T_{k:n}$ occurs between times $\tau$ and $\eta$, $\tau <T_{k:n}\le \eta <T_{m:n}$, and $D(\ge k)$ failures occur up to time $\eta$, then all of the remaining operating units $R_{\eta }^{*}$=$n-D-{\sum }_{i=1}^D{R}_i$ are eliminated from the experiment, thereby stopping the experiment at $T^{*}=\eta$; see Figure 3. In this case, $r^{*}=D-r$ and the following observations will be noticed:

    $$T_{1:n}<\cdots <T_{r:n}\le \tau <T_{r+1:n}<\cdots <T_{k:n}<\cdots <T_{D:n}\le \eta .$$
  • Case 3: If the k-th failure time $T_{k:n}$ occurs before time $\tau$, $T_{k:n}\le \tau <\eta <T_{m:n}$, and $D(\ge k)$ failures occur up to time $\eta$, then all of the remaining operating units $R_{\eta }^{*}$=$n-D-{\sum }_{i=1}^D{R}_i$ are eliminated from the experiment, thereby stopping the experiment at $T^{*}=\eta$; see Figure 4. In this case, $r^{*}=D-r$ and the following observations will be noticed:

    $$T_{1:n}<\cdots <T_{k:n}<\cdots <T_{r:n}\le \tau <T_{r+1:n}<\cdots <T_{D:n}\le \eta .$$
  • Case 4: If the k-th failure time $T_{k:n}$ occurs after time $\tau$ and the m-th failure time $T_{m:n}$ occurs before time $\eta$, $\tau <T_{k:n}<T_{m:n}\le \eta$, then all of the remaining operating units $R_m$=n−m−${\sum }_{i=1}^{m-1}$$R_i$ are eliminated from the experiment, thereby stopping the experiment at $T^{*}=T_{m:n}$; see Figure 5. In this case, $r^{*}=m-r$ and the following observations will be noticed:

    $$T_{1:n}<\cdots <T_{r:n}\le \tau <T_{r+1:n}<\cdots <T_{k:n}<\cdots <T_{m:n}\le \eta .$$
  • Case 5: If the k-th failure time $T_{k:n}$ occurs before time $\tau$ and the m-th failure time $T_{m:n}$ occurs between times $\tau$ and $\eta$, $T_{k:m:n}\le \tau <T_{m:n}\le \eta$, then all of the remaining operating units $R_m$=n−m−${\sum }_{i=1}^{m-1}$$R_i$ are eliminated from the experiment, thereby stopping the experiment at $T^{*}=T_{m:n}$; see Figure 6. In this case, $r^{*}=m-r$ and the following observations will be noticed:

    $$T_{1:n}<\cdots <T_{k:n}<\cdots <T_{r:n}\le \tau <T_{r+1:n}<\cdots <T_{m:n}\le \eta .$$
  • Case 6: If the m-th failure time $T_{m:n}$ occurs before time $\tau$, $T_{k:n}<T_{m:n}\le \tau <\eta$, then all of the remaining operating units $R_m$=n−m−${\sum }_{i=1}^{m-1}$$R_i$ are eliminated from the experiment, thereby stopping the experiment at $T^{*}=T_{m:n}$; see Figure 7. In this case, $r=m$, $r^{*}=0$, and the following observations will be noticed:

    $$T_{1:n}<\cdots <T_{k:n}<\cdots <T_{m:n}\le \tau <\eta .$$

3. Estimation Methods

This section deals with procedures of obtaining maximum likelihood estimates (MLEs) and percentile estimates (PEs) of the underlying parameters $\alpha$ and $\beta$ based on data collected from the PALT under generalized type-I progressive hybrid censoring.

3.1. Maximum Likelihood Estimation

The likelihood function based on generalized type-I progressively hybrid censored data from the PALT presented in Cases 1–6, which were given in the previous section, is given by

• For Cases 1–5:

  $$\begin{array}{cc}\hfill \mathbb{L}(\alpha ,\beta ;\mathbf{t})& \left[\prod _{j=1}^r{f}_1\left(t_{j:n}\right){\left[1-F_1\left(t_{j:n}\right)\right]}^{R_j}\right]\left[\prod _{j=r+1}^{\mathfrak{B}}{f}_2\left(t_{j:n}\right){\left[1-F_2\left(t_{j:n}\right)\right]}^{R_j}\right]{\left[1-F_2\left(\eta \right)\right]}^{\xi R_{\eta }^{*}},\hfill \end{array}$$

  (18)

  where $\mathbf{t}=(t_{1:n},\cdots ,t_{\mathfrak{B}:n})$, and the values of the experimental end time $T^{*}$, $\mathfrak{B}$, $\xi$, and the final censored number $R_{\mathfrak{F}}$ for Cases 1–5 are presented in Table 1. It can be noticed that, with respect to Case 1, some values of $R_i$, $i=1,2,\cdots ,m$ may be changed during the test rather than being fixed before beginning the experiment.
• For Case 6:

  $$\begin{array}{cc}\hfill \mathbb{L}(\alpha ,\beta ;\mathbf{t})& \prod _{j=1}^m{f}_1\left(t_{j:n}\right){\left[1-F_1\left(t_{j:n}\right)\right]}^{R_j},\hfill \end{array}$$

  (19)

  where $\mathbf{t}=(t_{1:n},\cdots ,t_{m:n})$.

Generally, maximizing the natural logarithm of the likelihood function to determine the MLEs of the underlying parameters is easier than maximizing the likelihood function. Using the CDF (14) and PDF (16), the natural logarithm of the likelihood function takes the form:

• For Cases 1–5:

  $$\begin{array}{cc}\hfill \mathcal{L}=ln\mathbb{L}(\alpha ,\beta ;\mathbf{t})& -\mathfrak{B}ln\beta -\sum _{j=1}^r\frac{1+R_j}{\beta }{t}_{j:n}+\sum _{j=r+1}^{\mathfrak{B}}\left[ln{\psi }^{{}^{\prime }}\left(t_{j:n}\right)-\frac{1+R_j}{\beta }\psi \left(t_{j:n}\right)\right]-\xi \frac{R_{\eta }^{*}}{\beta }\psi \left(\eta \right),\hfill \end{array}$$

  (20)

  where $\psi (.)$ and ${\psi }^{{}^{\prime }}(.)$ are given by (15) and (17), respectively.
• For Case 6:

  $$\begin{array}{cc}\hfill \mathcal{L}=ln\mathbb{L}(\alpha ,\beta ;\mathbf{t})& -mln\beta -\sum _{j=1}^m\frac{1+R_j}{\beta }{t}_{j:n}.\hfill \end{array}$$

  (21)

Based on Equations (20) and (21), the MLEs $(\widehat{\alpha },\widehat{\beta })$ of $(\alpha ,\beta )$ can be obtained as follows:

• For Cases 1–5, the MLEs of $\alpha$ and $\beta$ can be calculated by equating to zero the first partial derivatives of (20) with respect to $\alpha$ and $\beta$. Then, the likelihood equations take the forms

  $$\begin{array}{c}\hfill \frac{\partial \mathcal{L}}{\partial \beta }=0=\frac{-\mathfrak{B}}{\beta }+\sum _{j=1}^r\frac{1+R_j}{{\beta }^2}{t}_{j:n}+\sum _{j=r+1}^{\mathfrak{B}}\frac{1+R_j}{{\beta }^2}\psi \left(t_{j:n}\right)+\xi \frac{R_{\eta }^{*}}{{\beta }^2}\psi \left(\eta \right),\end{array}$$

  (22)

  $$\begin{array}{c}\hfill \frac{\partial \mathcal{L}}{\partial \alpha }=0=\sum _{j=r+1}^{\mathfrak{B}}\left[\frac{1}{{\psi }^{{}^{\prime }}\left(t_{j:n}\right)}W\left(t_{j:n}\right)-\frac{1+R_j}{\beta }Q\left(t_{j:n}\right)\right]-\xi \frac{R_{\eta }^{*}}{\beta }Q\left(\eta \right),\end{array}$$

  (23)

  where

  $$W\left(t\right)=\frac{d{\psi }^{{}^{\prime }}\left(t\right)}{d\alpha }=\left\{\begin{array}{cc}{(t-\tau +1)}^{\alpha }ln(t-\tau +1)\hfill & \mathrm{for}\mathrm{Case}\mathrm{A},\\ (t-\tau )e^{\alpha (t-\tau )}\hfill & \mathrm{for}\mathrm{Case}\mathrm{B},\\ {\displaystyle \frac{t-\tau }{\alpha (t-\tau )+1}}\hfill & \mathrm{for}\mathrm{Case}\mathrm{C},\end{array}\right.$$

  (24)
  $$\begin{array}{cc}\hfill Q\left(t\right)& =\frac{d\psi \left(t\right)}{d\alpha }=\left\{\begin{array}{cc}{\displaystyle \frac{(\alpha +1){(t-\tau +1)}^{\alpha +1}ln(t-\tau +1)-{(t-\tau +1)}^{\alpha +1}+1}{{(\alpha +1)}^2}}\hfill & \mathrm{for}\mathrm{Case}\mathrm{A},\\ {\displaystyle \frac{(\alpha (t-\tau )-1)e^{\alpha (t-\tau )}+1}{{\alpha }^2}}\hfill & \mathrm{for}\mathrm{Case}\mathrm{B},\\ {\displaystyle \frac{\alpha \left(t-\tau \right)-ln\left(\alpha (t-\tau )+1\right)}{{\alpha }^2}}\hfill & \mathrm{for}\mathrm{Case}\mathrm{C}.\end{array}\right.\hfill \end{array}$$

  (25)
  From (22), the MLE of $\beta$ can be obtained as a function of $\alpha$ as follows:

  $$\begin{array}{c}\hfill \widehat{\beta }\left(\alpha \right)={\mathfrak{B}}^{-1}\left(\sum _{j=1}^r\left(1+R_j\right)t_{j:n}+\sum _{j=r+1}^{\mathfrak{B}}(1+R_j)\psi \left(t_{j:n}\right)+\xi R_{\eta }^{*}\psi \left(\eta \right)\right).\end{array}$$

  (26)
  By substituting $\widehat{\beta }\left(\alpha \right)$ in (23) and solving the likelihood equation ${\displaystyle \frac{\partial \mathcal{L}}{\partial \alpha }=0}$ with respect to $\alpha$ by using any numerical iteration method, the MLE of $\alpha$ can be obtained.
• For Case 6, no failures were detected under accelerated stress conditions, so the MLE of $\alpha$ does not exist, but the MLE of $\beta$ can be calculated by equating to zero the first partial derivative of (21) with respect to $\beta$. Then, the MLE of $\beta$ is given by

  $$\begin{array}{c}\hfill \widehat{\beta }=\frac{1}{m}\sum _{j=1}^m\left(1+R_j\right)t_{j:n}.\end{array}$$

  (27)

Remark 2.

It can be noticed that:

• In Cases 1,2, and 4, if $r=0$, no failures were detected under normal stress conditions; then, the MLE of β does not exist.
• In Case 3, if $D=r$, no failures were detected under accelerated stress conditions; then, the MLE of α does not exist.

Based on the common asymptotic normality theory of MLEs, we can consider that ${\displaystyle \frac{\widehat{\alpha }-\alpha }{\sqrt{\mathrm{Var}\left(\widehat{\alpha }\right)}}}$ and ${\displaystyle \frac{\widehat{\beta }-\beta }{\sqrt{\mathrm{Var}(\widehat{\beta })}}}$ can be approximated by a standard normal distribution, i.e.,

$${\displaystyle \frac{\widehat{\alpha }-\alpha }{\sqrt{\mathrm{Var}\left(\widehat{\alpha }\right)}}\sim N(0,1)\mathrm{and}\frac{\widehat{\beta }-\beta }{\sqrt{\mathrm{Var}(\widehat{\beta })}}\sim N(0,1),}$$

where $\mathrm{Var}\left(\widehat{\alpha }\right)$ and $\mathrm{Var}(\widehat{\beta })$ are the variances of $\widehat{\alpha }$ and $\widehat{\beta }$, which can be obtained from the inverse of the local Fisher information matrix (FIM), i.e.,

$$\begin{array}{cc}\hfill \mathbb{V}={\mathbb{F}}^{-1}=& \left(\begin{array}{cc}\mathrm{Var}\left(\widehat{\alpha }\right)& \mathrm{Cov}(\widehat{\alpha },\widehat{\beta })\\ \mathrm{Cov}(\widehat{\alpha },\widehat{\beta })& \mathrm{Var}(\widehat{\beta })\end{array}\right),\hfill \end{array}$$

(28)

where

$$\begin{array}{cc}\hfill \mathbb{F}=-& \left(\begin{array}{cc}{\displaystyle \frac{{\partial }^2\widehat{\pounds }}{\partial {\alpha }^2}}& {\displaystyle \frac{{\partial }^2\widehat{\pounds }}{\partial \alpha \partial \beta }}\\ {\displaystyle \frac{{\partial }^2\widehat{\pounds }}{\partial \beta \partial \alpha }}& {\displaystyle \frac{{\partial }^2\widehat{\pounds }}{\partial {\beta }^2}}\end{array}\right),\hfill \end{array}$$

(29)

where the caret $\widehat{}$ denotes that the derivative is evaluated at $(\widehat{\alpha },\widehat{\beta })$. The second partial derivatives of the natural logarithm of the likelihood function with respect to $\alpha$ and $\beta$ can be obtained without difficulty.

Suppose that ${\nu }_1=\alpha$ and ${\nu }_2=\beta$. Then, for $i=1,2$, a $100(1-\gamma )\%$ normal approximation confidence interval (NACI) for ${\nu }_i$ can be defined as

$$\left(\max \{0,\widehat{{\nu }_i}-z_{\gamma /2}\sqrt{\mathrm{Var}\left(\widehat{{\nu }_i}\right)}\},\widehat{{\nu }_i}+z_{\gamma /2}\sqrt{\mathrm{Var}\left(\widehat{{\nu }_i}\right)}\right),$$

where $\widehat{{\nu }_i}$ is the MLE of ${\nu }_i$ and $z_{\gamma /2}$ is the upper $\gamma /2$ percentile of the $N(0,1)$ distribution.

Sometimes, the lower bound of the NACI may have a negative value for the positive parameter. So, Meeker and Escobar [31] suggested using a log transformation confidence interval (LTCI) for this parameter. Based on the normal approximation of a log-transformed MLE, ${\displaystyle \frac{ln\widehat{{\nu }_i}-ln{\nu }_i}{\sqrt{\mathrm{Var}(ln\widehat{{\nu }_i})}}}$, $i=1,2$, can be approximated to a standard normal distribution, i.e.,

$${\displaystyle \frac{ln\widehat{{\nu }_i}-ln{\nu }_i}{\sqrt{\mathrm{Var}(ln\widehat{{\nu }_i})}}\sim N(0,1),}$$

where ${\displaystyle \mathrm{Var}(ln\widehat{{\nu }_i})=\frac{\mathrm{Var}\left(\widehat{{\nu }_i}\right)}{{\widehat{{\nu }_i}}^2}}$.

Therefore, a $100(1-\gamma )\%$ LTCI for ${\nu }_i$ can be defined as

$$\left(\widehat{{\nu }_i}exp\left[-{\displaystyle z_{\gamma /2}\frac{\sqrt{\mathrm{Var}\left(\widehat{{\nu }_i}\right)}}{\widehat{{\nu }_i}}}\right],\widehat{{\nu }_i}exp\left[{\displaystyle z_{\gamma /2}\frac{\sqrt{\mathrm{Var}\left(\widehat{{\nu }_i}\right)}}{\widehat{{\nu }_i}}}\right]\right).$$

3.2. Percentile Estimation

A percentile method for estimating unknown parameters was proposed by Kao [32]. It is simply natural to estimate the unknown parameter by adjusting a straight line to the theoretical points obtained by the CDF and the percentile points of the sample if data were collected from a closed-form CDF. The empirical CDF applied in this process can be described as

$$\widehat{G}\left(t_{j:n}\right)=1-\prod _{l=1}^j(1-{\widehat{\mathbf{p}}}_l),j=1,\cdots ,\mathfrak{B},$$

where $\mathfrak{B}$ is defined as in Table 1 and

$${\displaystyle {\widehat{\mathbf{p}}}_l=\frac{1}{n-\left[{\sum }_{i=1}^{l-1}{R}_i\right]-l+1},l=1,\cdots ,\mathfrak{B}.}$$

Based on generalized type-I progressive hybrid censoring under the PALT, the PEs of the parameters $\alpha$ and $\beta$ can be obtained as follows:

• For Cases 1–5: It is possible to obtain the PEs $\check{\alpha }$ and $\check{\beta }$ of $\alpha$ and $\beta$ by minimizing the next quantity with respect to $\alpha$ and $\beta$.
  • Case A:

    $$\begin{array}{cc}\hfill {{\rm Y}}(\alpha ,\beta ;\mathbf{t})=& \sum _{i=1}^r{\left(t_{i:n}+\beta ln(1-{\mathrm{\Omega }}_i)\right)}^2+\sum _{i=r+1}^{\mathfrak{B}}(t_{i:n}-\tau +1\hfill \\ \hfill & -{\left[1-\beta (\alpha +1)ln(1-{\mathrm{\Omega }}_i)-\tau (\alpha +1)\right]}^{\frac{1}{\alpha +1}}{)}^2.\hfill \end{array}$$

    (30)
  • Case B:

    $$\begin{array}{cc}\hfill {{\rm Y}}(\alpha ,\beta ;\mathbf{t})=& \sum _{i=1}^r{\left(t_{i:n}+\beta ln(1-{\mathrm{\Omega }}_i)\right)}^2+\sum _{i=r+1}^{\mathfrak{B}}(t_{i:n}-\tau \hfill \\ \hfill & -\frac{1}{\alpha }ln\left[1-\alpha (\tau +\beta ln[1-{\mathrm{\Omega }}_i])\right]{)}^2.\hfill \end{array}$$

    (31)
  • Case C:

    $$\begin{array}{cc}\hfill {{\rm Y}}(\alpha ,\beta ;\mathbf{t})=& \sum _{i=1}^r{\left(t_{i:n}+\beta ln(1-{\mathrm{\Omega }}_i)\right)}^2+\sum _{i=r+1}^{\mathfrak{B}}(\tau +\beta ln(1-{\mathrm{\Omega }}_i)\hfill \\ \hfill & +\left(t_{i:n}-\tau +\frac{1}{\alpha }\right)ln\left[\alpha (t_{i:n}-\tau )+1\right]{)}^2,\hfill \end{array}$$

    (32)
  where

  $$\begin{array}{c}\hfill {\mathrm{\Omega }}_i=\frac{\widehat{G}\left(t_{i-1:n}\right)+\widehat{G}\left(t_{i:n}\right)}{2}.\end{array}$$

  (33)
  Minimizing the ${{\rm Y}}$ quantity can be done by solving ${\displaystyle \frac{\partial {{\rm Y}}}{\partial \alpha }=0}$ and ${\displaystyle \frac{\partial {{\rm Y}}}{\partial \beta }=0}$ with respect to $\alpha$ and $\beta$.
• For Case 6, the PE of $\alpha$ does not exist, but the PE of $\beta$ can be obtained by minimizing the next quantity with respect to $\beta$:

  $$\begin{array}{cc}\hfill {{\rm Y}}(\alpha ,\beta ;\mathbf{t})=& \sum _{i=1}^m{\left(t_{i:n}+\beta ln(1-{\mathrm{\Omega }}_i)\right)}^2,\hfill \end{array}$$

  (34)

  where ${\mathrm{\Omega }}_i$ is given by (33).

4. Common Optimality Criteria

The current section aims to study the problem of determining the optimal time, $\tau$, of the increase in the stress level under generalized type-I progressively hybrid censored data supposing the ED as a lifetime model. The criteria of A-optimality and D-optimality, which allow the optimal value of $\tau$ to be determined, are considered; see, for example, [33,34,35].

• A-optimality criterion
  This criterion is concerned with the maximization of the trace of the FIM, which provides an overall information measure based on the marginal information about the parameters. It is preferred when the estimates are, at most, moderately correlated. Therefore, the optimal value ${\tau }_A^{*}$ of $\tau$ can be obtained by maximizing the trace of the FIM (29) of MLEs $\widehat{\alpha }$ and $\widehat{\beta }$, i.e.,

  $$\mathrm{Maximize}\left\{tr\right(\mathbb{F}\left)\right\}.$$
• D-optimality criterion
  This criterion is concerned with maximizing the determinant of the FIM, which provides an overall measure of variability by taking into account the correlation between the estimates. It is preferred that it is applied when the estimates are highly correlated. It is also pertinent for the construction of the joint confidence region for the parameters. Therefore, the optimal value ${\tau }_D^{*}$ of $\tau$ can be obtained by maximizing (minimizing) the determinant of the FIM (inverse of the FIM) ((29) and (28)) of the MLEs $\widehat{\alpha }$ and $\widehat{\beta }$, i.e.,

  $$\mathrm{Maximize}\left\{\mathbb{F}\right\}=\mathrm{Minimize}\left\{{\mathbb{F}}^{-1}\right\}.$$

The maximization of the $tr\left(\mathbb{F}\right)$ [$\mathbb{F}$] can be achieved by solving the equation ${\displaystyle \frac{\partial tr\left(\mathbb{F}\right)}{\partial \tau }=0}$$\left[{\displaystyle \frac{\partial \mathbb{F}}{\partial \tau }=0}\right]$ with respect to $\tau$. The solution cannot be obtained in a closed form. So, an iterative method is needed in order to obtain a numerical solution.

5. Real Dataset

In this section, we consider a real dataset in order to explain the procedures of inference presented in the last sections. The real dataset was collected from an ALT experiment by Zhu [36] as follows: A sample of $n=64$ light bulbs was subjected to a life test at a stress level of 2.25 V, and then the sample subjected to accelerated conditions with a stress level 2.44 V at the time point of $\tau =96$ hours. The stress level of 2.25 V is considered to correspond to the normal stress conditions. The failure times of $m=53$ light bulbs were noticed, $R_m=11$ light bulbs were removed from the experiment, and then the experiment was finished. The dataset is shown in

Table 2. The failure times of 64 light bulbs from a PALT experiment with $\tau =96.0$.

Normal stress	12.07, 14.0, 17.95, 19.5, 22.1, 23.11, 24.0, 24.0, 25.1, 26.46, 26.58, 26.9, 28.06, 34.0, 36.13, 36.64,
(before $\tau =96.0$)	40.85, 41.11, 42.63, 44.1, 46.3, 52.51, 54.0, 58.09, 62.68, 64.17, 72.25, 73.13, 83.63, 86.9, 90.09,
	91.22, 91.56, 94.38
Accelerated stress	97.71, 101.53, 102.1, 105.1, 105.11, 109.2, 112.11, 114.4, 117.9, 119.58, 120.2,
(after $\tau =96.0$)	121.9, 122.5, 123.6, 126.5, 126.95, 129.25, 130.1, 136.31

. One logical question now is that of whether or not the CDF (14), with the ED as a lifetime distribution under normal stress conditions, fits the real dataset that appears in Table 2. For this reason, the Kolmogorov–Smirnov (K-S) test statistic and its corresponding p-value were used.

Table 3. MLEs of the parameters $(\alpha ,\beta )$, the K-S statistic, and the p-value.

	Parameters	K-S	p-Value
Case A	$\widehat{\alpha }=0.406382,\widehat{\beta }=132.597$	0.0797349	0.88902758
Case B	$\widehat{\alpha }=0.04765,\widehat{\beta }=125.266$	0.0866267	0.82128778
Case C	$\widehat{\alpha }=0.455233,\widehat{\beta }=130.959$	0.0812119	0.87567578

shows the results, in which it can be noticed that all of the p-values are greater than 0.05, so the ED fits the given real dataset well. This is shown also by drawing the empirical CDFs of the given real dataset together with the CDFs (14) in Cases A, B, and C; see Figure 8.

We consider $n=64,m=53,k=30,R_1=R_2=\cdots =R_{m-1}=0,R_m=11,\tau =96,$ and three values of $\eta$ 116 (Scheme I), 125 (Scheme II), and 140 (Scheme III) in order to apply the generalized type-I progressive hybrid censoring to the real dataset; see

Table 4. The samples under generalized type-I progressive hybrid censoring considering $\tau =96.0$.

Scheme I
Normal stress	12.07, 14.00, 17.95, 19.50, 22.10, 23.11, 24.00, 24.0, 25.10, 26.46, 26.58, 26.90, 28.06, 34.00,
	36.13, 36.64, 40.85, 41.11, 42.63, 44.10, 46.30, 52.51, 54.00, 58.09, 62.68, 64.17, 72.25,
	73.13, 83.63, 86.90, 90.09, 91.22, 91.56, 94.38
Accelerated stress	97.71, 101.53, 102.1, 105.1, 105.11, 109.2, 112.11, 114.40
($\eta =116.0$)
Scheme II
Normal stress	12.07, 14.00, 17.95, 19.50, 22.10, 23.11, 24.00, 24.00, 25.10, 26.46, 26.58, 26.9, 28.06, 34.00,
	36.13, 36.64, 40.85, 41.11, 42.63, 44.10, 46.30, 52.51, 54.00, 58.09, 62.68, 64.17, 72.25,
	73.13, 83.63, 86.9, 90.09, 91.22, 91.56, 94.38
Accelerated stress	97.71, 101.53, 102.10, 105.10, 105.11, 109.20, 112.11, 114.40, 117.90, 119.58, 120.20,
($\eta =125.0$)	121.90, 122.50, 123.60
Scheme III
Normal stress	12.07, 14.00, 17.95, 19.50, 22.10, 23.11, 24.00, 24.00, 25.10, 26.46, 26.58, 26.9, 28.06, 34.00,
	36.13, 36.64, 40.85, 41.11, 42.63, 44.10, 46.30, 52.51, 54.00, 58.09, 62.68, 64.17, 72.25,
	73.13, 83.63, 86.9, 90.09, 91.22, 91.56, 94.38
Accelerated stress	97.71, 101.53, 102.10, 105.10, 105.11, 109.20, 112.11, 114.40, 117.90, 119.58, 120.20,
($\eta =140.0$)	121.90, 122.50, 123.60, 126.50, 126.95, 129.25, 130.10, 136.31

. The MLEs, PEs, NACIs, and LTCIs of the parameters $\alpha$ and $\beta$ were calculated, and the optimal values of $\tau$ under two optimality criteria—${\tau }_D^{*}$ and ${\tau }_A^{*}$—of increasing stress level were calculated, as presented in

Table 5. MLEs, PEs, NACIs, and LTCIs of the parameters $\alpha$ and $\beta$ and the optimal values, ${\tau }_D^{*}$ and ${\tau }_A^{*}$, for Cases A, B, and C.

		PE($\mathit{\alpha }$)	MLE($\mathit{\alpha }$)	NACI($\mathit{\alpha }$)	LTCI($\mathit{\alpha }$)	${\mathit{\tau }}_{\mathit{D}}^{*}$
	$\mathit{\eta }$	PE($\mathit{\beta }$)	MLE($\mathit{\beta }$)	NACI($\mathit{\beta }$)	LTCI($\mathit{\beta }$)	${\mathit{\tau }}_{\mathit{A}}^{*}$
Case A	116	0.45343	0.30888	(0.0, 0.63271)	(0.10826, 0.88126)	71.3397
		133.038	130.799	(87.3089, 174.288)	(93.8, 182.391)	71.3397
	125	0.40396	0.39898	(0.17329, 0.62467)	(0.22661, 0.70245)	71.3397
		132.899	132.476	(88.3358, 176.616)	(94.9364, 184.859)	71.3397
	140	0.42124	0.40638	(0.21944, 0.59332)	(0.25654, 0.64374)	71.3397
		133.063	132.597	(88.551, 176.642)	(95.1195, 184.84)	71.3397
Case B	116	0.11873	0.05355	(0.0, 0.11652)	(0.016528, 0.17353)	114.783
		132.822	127.842	(86.3012, 169.382)	(92.375, 176.926)	114.881
	125	0.07414	0.05983	(0.02756, 0.09209)	(0.03489, 0.10259)	122.951
		132.046	128.837	(87.345, 170.329)	(93.3633, 177.79)	123.277
	140	0.06431	0.04765	(0.02615, 0.06915)	(0.03035, 0.07482)	131.446
		131.521	125.266	(85.9392, 164.593)	(91.5141, 171.466)	132.369
Case C	116	0.42073	0.22842	(0.0, 0.83665)	(0.01593, 3.27483)	97.3663
		131.48	131.674	(87.1202, 176.227)	(93.875, 184.692)	97.4677
	125	0.37920	0.39619	(0.0, 1.24462)	(0.046546, 3.37232)	97.1638
		131.313	131.282	(87.3779, 175.186)	(93.9649, 183.419)	97.1568
	140	0.53392	0.45523	(0.0, 1.36748)	(0.06137, 3.37696)	97.212
		132.409	130.959	(87.365, 174.554)	(93.8786, 182.686)	97.206

.

6. Simulation Study

As it is theoretically difficult to assess the efficiency of estimation methods, a Monte Carlo simulation was used to overcome this difficulty. In the current section, through a Monte Carlo simulation, we conduct a numerical study in order to assess the efficiency and performance of the estimation methods and optimality criteria considered in the last sections. The following steps were used:

• Assign values of the sample size n, stress change time $\tau$, and parameters $(\alpha ,\beta )$, and assign values of $k,m,\eta$, and $(R_1,R_2,\cdots ,R_m)$.
• Generate n observations $(V_1,V_2,\cdots ,V_n)$ from the Uniform(0,1) distribution.
• We can obtain the observations $(t_1,t_2,\cdots ,t_n)$ from the PALT with the CDF (14) as follows:

  $$t_i=\left\{\begin{array}{cc}{x}_{1,i},& x_{1,i}\le \tau ,\\ x_{2,i},& x_{1,i}>\tau ,\end{array}\right.$$

  where $i=1,\cdots ,n$,

  $$x_{1,i}=-\beta ln(1-V_i),$$

  and

  $$x_{2,i}=\left\{\begin{array}{cc}\tau -1+{\left[1-\beta (\alpha +1)ln(1-V_i)-\tau (\alpha +1)\right]}^{\frac{1}{\alpha +1}}\hfill & \mathrm{for}\mathrm{Case}\mathrm{A},\\ \tau +\frac{1}{\alpha }ln\left[1-\alpha (\tau +\beta ln(1-V_i))\right]\hfill & \mathrm{for}\mathrm{Case}\mathrm{B},\\ \mathrm{Solution}\left\{\left(x_{2,i}-\tau +\frac{1}{\alpha }\right)ln\left(\alpha (x_{2,i}-\tau )+1\right)+\tau +\beta ln(1-V_i)=0\right\}\hfill & \mathrm{for}\mathrm{Case}\mathrm{C}.\end{array}\right.$$
• Apply the generalized type-I progressive hybrid censoring to the random sample obtained in Step 3, as shown in Section 2.1.
• Find the value of r, which is the number of observations with values $\le \tau$, as follows:

  $$t_{r:n}\le \tau <t_{r+1:n}.$$
• Compute the MLEs, PEs, NACIs, and LTCIs of $(\alpha ,\beta )$, as shown in Section 3.
• Compute the optimal values, ${\tau }_D^{*}$ and ${\tau }_A^{*}$, of the stress change time, as shown in Section 4.
• Replicate Steps 2–7 $\mathbb{M}$(= 5000) times.
• Calculate the mean of the estimates, mean squared error (MSE), and relative absolute bias (RAB) of $\widehat{\nu }$ over $\mathbb{M}$ samples as follows:

  $$\overline{\widehat{\nu }}=\frac{1}{\mathbb{M}}\sum _{i=1}^{\mathbb{M}}{\widehat{\nu }}_i,\mathrm{RAB}\left(\widehat{\nu }\right)=\frac{1}{\mathbb{M}}\sum _{i=1}^{\mathbb{M}}\frac{{\widehat{\nu }}_i-\nu }{\nu },\mathrm{MSE}\left(\widehat{\nu }\right)=\frac{1}{\mathbb{M}}\sum _{i=1}^{\mathbb{M}}{(\widehat{{\nu }_i}-\nu )}^2,$$

  where $\widehat{\nu }$ is an estimate of $\nu$.
• Using Step 9, calculate the mean of the estimates of the parameters $(\alpha ,\beta )$ with their MSEs and RABs.
• Calculate the average of the RABs (ARAB) and MSEs (AMSE) as follows:

  $$\begin{array}{c}\hfill \mathrm{ARAB}=\frac{\mathrm{RAB}(\widehat{\alpha })+\mathrm{RAB}(\widehat{\beta })}{2},\mathrm{AMSE}=\frac{\mathrm{MSE}(\widehat{\alpha })+\mathrm{MSE}(\widehat{\beta })}{2}.\end{array}$$
• Evaluate the 95% NACIs and LTCIs of the parameters $(\alpha ,\beta )$, and then evaluate their average interval lengths (AILs) and coverage probabilities (COVPs). In addition, also evaluate the average of the AILs (AAIL) as follows:

  $$\begin{array}{c}\hfill \mathrm{AAIL}=\frac{\mathrm{AIL}(\widehat{\alpha })+\mathrm{AIL}(\widehat{\beta })}{2}.\end{array}$$

The next three CSs are used in the generation of the samples:

• CS1:

  $$\begin{array}{cc}\hfill R_i& =n-m,i=1,\hfill \\ \hfill R_i& =0,\mathrm{otherwise}.\hfill \end{array}$$
• CS2:

  $$\begin{array}{cc}\hfill R_i& =\frac{n-m}{2},i=1,\frac{m}{2},\hfill \\ \hfill R_i& =0,\mathrm{otherwise}.\hfill \end{array}$$
• CS3:

  $$\begin{array}{cc}\hfill R_i& =1,i=1,\cdots ,n-m,\hfill \\ \hfill R_i& =0,\mathrm{otherwise}.\hfill \end{array}$$

The results of the calculations are recorded in

Table 6. PEs and MLEs of $\alpha$ and $\beta$ with their MSEs, RABs, AMSEs, and ARABs based on 5000 simulations. The population parameter values are $\alpha =2.0$ and $\beta =0.55$ for Case A.

					PE				MLE
					$\overline{\widehat{\mathit{\alpha }}}$	MSE($\widehat{\mathit{\alpha }}$)	RAB($\widehat{\mathit{\alpha }}$)	AMSE	$\overline{\check{\mathit{\alpha }}}$	MSE($\check{\mathit{\alpha }}$)	RAB($\check{\mathit{\alpha }}$)	AMSE
n	m	k	$\mathit{\eta }$	CS	$\overline{\widehat{\mathit{\beta }}}$	MSE($\widehat{\mathit{\beta }}$)	RAB($\widehat{\mathit{\beta }}$)	ARAB	$\overline{\check{\mathit{\beta }}}$	MSE($\check{\mathit{\beta }}$)	RAB($\check{\mathit{\beta }}$)	ARAB
30	18	12	0.75	1	4.10970	11.2033	1.30263	5.63093	3.37393	3.42300	0.70773	1.75625
					0.64754	0.05853	0.29228	0.79746	0.70101	0.08950	0.37164	0.53968
				2	4.33833	12.4739	1.38312	6.26307	3.47865	3.93665	0.75631	1.99988
					0.64156	0.05224	0.27201	0.82757	0.67197	0.06311	0.31505	0.53568
				3	4.19268	11.1990	1.30426	5.62179	3.56698	4.29929	0.79475	2.17842
					0.63390	0.04462	0.26054	0.78240	0.66843	0.05754	0.30622	0.55049
			1.25	1	2.68433	2.59585	0.59042	1.33347	2.84073	2.30459	0.5663	1.18466
					0.63422	0.07108	0.32037	0.45539	0.65492	0.06473	0.31353	0.43992
				2	2.76357	2.94032	0.61863	1.49434	2.97708	2.92565	0.62315	1.48793
					0.62260	0.04835	0.28770	0.45316	0.63643	0.05022	0.28241	0.45278
				3	2.8303	3.23319	0.66009	1.6401	3.01581	3.08544	0.63716	1.56479
					0.62345	0.04701	0.28397	0.47203	0.63185	0.04414	0.27165	0.45441
	24	12	0.75	1	2.97613	3.89284	0.75303	1.97052	2.95832	2.20531	0.55766	1.13000
					0.61994	0.04820	0.27166	0.51234	0.65119	0.05468	0.28791	0.42278
				2	2.95310	3.74576	0.73898	1.89329	3.02478	2.44235	0.58229	1.24426
					0.60994	0.04081	0.24948	0.49423	0.64388	0.04617	0.26949	0.42589
				3	2.95322	3.63054	0.73559	1.83620	2.94569	2.12297	0.54747	1.08389
					0.61535	0.04187	0.26235	0.49897	0.64359	0.04482	0.27197	0.40972
			1.25	1	2.55345	1.79452	0.49628	0.91790	2.61782	1.65911	0.49061	0.85252
					0.61176	0.04127	0.26281	0.37955	0.62809	0.04592	0.27096	0.38078
				2	2.53464	1.82050	0.49915	0.92836	2.66263	1.85759	0.50809	0.94796
					0.60397	0.03621	0.24803	0.37359	0.61612	0.03833	0.25178	0.37993
				3	2.50760	1.65639	0.48453	0.84743	2.56043	1.54207	0.47052	0.78880
					0.60903	0.03848	0.25598	0.37026	0.61692	0.03554	0.24879	0.35965
	30	–	–	–	2.30212	1.24452	0.41972	0.63620	2.42642	1.15576	0.41019	0.59154
					0.58749	0.02788	0.22429	0.32200	0.59864	0.02731	0.21961	0.31490
60	36	24	0.75	1	3.43922	7.39602	1.01964	3.71142	2.53677	1.29372	0.43634	0.66028
					0.60888	0.02682	0.20950	0.61457	0.60662	0.02685	0.21441	0.32537
				2	2.64437	2.32974	0.58995	1.17467	2.60190	1.41379	0.45254	0.71662
					0.58736	0.01959	0.18838	0.38916	0.59668	0.01944	0.18739	0.31997
				3	2.70948	2.58468	0.62010	1.30140	2.66926	1.54932	0.47151	0.78424
					0.58763	0.01812	0.18414	0.40212	0.59808	0.01916	0.18572	0.32861
			1.25	1	2.33366	1.01848	0.38467	0.52067	2.33813	0.96605	0.37642	0.49470
					0.58622	0.02286	0.20352	0.29410	0.59120	0.02335	0.20379	0.29010
				2	2.40504	1.12983	0.40460	0.57439	2.37395	1.06426	0.39390	0.54091
					0.58864	0.01895	0.18748	0.29604	0.58188	0.01756	0.18222	0.28806
				3	2.43026	1.19981	0.41538	0.60900	2.43005	1.19989	0.41316	0.60876
					0.58722	0.01819	0.18599	0.30068	0.58375	0.01763	0.18145	0.29731
	48	24	0.75	1	2.39614	1.54233	0.49143	0.78028	2.34300	0.97986	0.38883	0.49868
					0.58246	0.01823	0.18235	0.33689	0.58855	0.01750	0.17509	0.28196
				2	2.42587	1.57380	0.49546	0.79447	2.35547	0.97236	0.38550	0.49363
					0.57768	0.01514	0.16779	0.33163	0.58533	0.01490	0.16574	0.27562
				3	2.39619	1.54925	0.48748	0.78304	2.32671	0.91357	0.38036	0.46465
					0.58120	0.01682	0.17773	0.33260	0.58321	0.01573	0.17248	0.27642
			1.25	1	2.27817	0.73893	0.32926	0.37744	2.23473	0.69062	0.32600	0.35332
					0.58029	0.01594	0.17331	0.25128	0.58016	0.01601	0.17269	0.24934
				2	2.27104	0.76043	0.33433	0.38763	2.25127	0.72031	0.32935	0.36714
					0.57929	0.01484	0.16856	0.25145	0.57680	0.01396	0.16228	0.24581
				3	2.25994	0.69591	0.32225	0.35576	2.22000	0.64545	0.31412	0.32984
					0.58023	0.01560	0.17289	0.24757	0.57510	0.01424	0.16552	0.23982
	60	–	–	–	2.15387	0.55668	0.28837	0.28452	2.20042	0.51189	0.27649	0.26211
					0.57066	0.01236	0.15467	0.22152	0.57498	0.01233	0.15267	0.21458

,

Table 7. AILs and COVPs (in %) of the 95% CIs of $\alpha$ and $\beta$ based on 5000 simulations. The population parameter values are $\alpha =2.0$ and $\beta =0.55$ for Case A.

					NACI			LTCI			Optimal Value
					COVP$\left(\mathit{\alpha }\right)$	AIL$\left(\mathit{\alpha }\right)$		COVP$\left(\mathit{\alpha }\right)$	AIL$\left(\mathit{\alpha }\right)$
n	m	k	$\mathit{\eta }$	CS	COVP$\left(\mathit{\beta }\right)$	AIL$\left(\mathit{\beta }\right)$	AAIL	COVP$\left(\mathit{\beta }\right)$	AIL$\left(\mathit{\beta }\right)$	AAIL	${\mathit{\tau }}_{\mathit{D}}^{*}$	${\mathit{\tau }}_{\mathit{A}}^{*}$
30	18	12	0.75	1	94.14	6.47228	3.76255	80.62	9.21429	5.18726	0.56556	0.57287
					98.72	1.05283		97.12	1.16024
				2	92.14	6.56681	3.73274	78.30	9.28831	5.12874	0.48625	0.47228
					98.68	0.89867		96.74	0.96916
				3	91.42	6.69255	3.78226	77.18	9.41054	5.17359	0.47352	0.45495
					98.64	0.87197		97.06	0.93665
			1.25	1	95.06	5.04383	2.98203	84.28	6.93378	3.9673	0.58531	0.60134
					98.48	0.92023		97.34	1.00083
				2	92.62	5.24578	3.03094	81.82	7.16377	4.01912	0.51999	0.51414
					98.18	0.81611		96.90	0.87447
				3	92.86	5.38945	3.09228	81.96	7.41417	4.13191	0.5002	0.49022
					97.92	0.79511		97.1	0.84964
	24	12	0.75	1	94.98	5.59843	3.21471	84.56	8.13174	4.51119	0.53359	0.53728
					98.38	0.83100		96.84	0.89064
				2	93.7	5.64222	3.21152	83.26	8.11909	4.47503	0.49546	0.49248
					98.48	0.78083		96.72	0.83096
				3	95.36	5.59081	3.19587	85.04	8.07499	4.465	0.53904	0.53693
					98.48	0.80092		97.1	0.85502
			1.25	1	93.90	4.32257	2.53875	84.90	5.72211	3.26230	0.57303	0.58748
					97.106	0.75494		96.10	0.80248
				2	93.64	4.41189	2.56065	84.02	5.83504	3.29259	0.5255	0.53127
					97.00	0.70941		96.26	0.75014
				3	95.00	4.30250	2.51391	87.02	5.75165	3.26019	0.57194	0.57988
					97.18	0.72532		97.06	0.76872
	30	–	–	–	95.18	3.80526	2.21832	88.00	4.73587	2.69875	0.56485	0.57587
					97.06	0.63138		96.68	0.66162
60	36	24	0.75	1	95.22	4.60444	2.60893	88.54	6.66823	3.65434	0.52015	0.52334
					97.76	0.61343		96.78	0.64044
				2	94.76	4.68530	2.61318	87.06	6.74321	3.65160	0.44714	0.44359
					97.66	0.54107		96.32	0.56000
				3	94.34	4.77857	2.65535	86.54	6.82727	3.68865	0.45700	0.44899
					97.32	0.53212		96.28	0.55003
			1.25	1	94.86	3.51239	2.03809	88.58	4.30683	2.44629	0.56400	0.57334
					96.80	0.56379		96.24	0.58575
				2	94.80	3.60637	2.05704	88.28	4.48194	2.50303	0.48178	0.48230
					95.76	0.50771		95.98	0.52412
				3	94.24	3.70235	2.10175	86.90	4.60024	2.55853	0.48829	0.48411
					96.02	0.50114		95.12	0.51681
	48	24	0.75	1	96.04	4.03729	2.2744	89.80	5.71223	3.12012	0.51759	0.52259
					97.32	0.51152		96.34	0.52801
				2	95.86	4.07491	2.27961	90.28	5.74677	3.12258	0.48015	0.48366
					97.94	0.48431		96.80	0.49840
				3	96.54	4.03397	2.2648	90.72	5.70665	3.10876	0.51893	0.52096
					96.86	0.49563		96.56	0.51087
			1.25	1	95.62	3.04419	1.76014	89.98	3.52464	2.00718	0.56559	0.57375
					95.96	0.47609		95.68	0.48972
				2	95.02	3.08273	1.7684	89.84	3.57207	2.01903	0.52023	0.52555
					95.60	0.45407		95.48	0.46600
				3	96.10	3.04957	1.75644	91.22	3.52236	1.99921	0.56692	0.57116
					96.18	0.4633		95.98	0.47606
	60	–	–	–	95.24	2.68636	1.55267	91.24	2.93992	1.68416	0.56996	0.57573
					95.68	0.41897		94.88	0.42839

,

Table 8. PEs and MLEs of $\alpha$ and $\beta$ with their MSEs, RABs, AMSEs, and ARABs based on 5000 simulations. The population parameter values are $\alpha =1.9$ and $\beta =0.55$ for Case B.

					PE				MLE
					$\overline{\widehat{\mathit{\alpha }}}$	MSE($\widehat{\mathit{\alpha }}$)	RAB($\widehat{\mathit{\alpha }}$)	AMSE	$\overline{\check{\mathit{\alpha }}}$	MSE($\check{\mathit{\alpha }}$)	RAB($\check{\mathit{\alpha }}$)	AMSE
n	m	k	$\mathit{\eta }$	CS	$\overline{\widehat{\mathit{\beta }}}$	MSE($\widehat{\mathit{\beta }}$)	RAB($\widehat{\mathit{\beta }}$)	ARAB	$\overline{\check{\mathit{\beta }}}$	MSE($\check{\mathit{\beta }}$)	RAB($\check{\mathit{\beta }}$)	ARAB
30	18	12	0.75	1	2.82931	3.54341	0.80975	1.78568	2.90194	2.37936	0.60538	1.22738
					0.59976	0.02794	0.22394	0.51684	0.68174	0.07540	0.34360	0.47449
				2	3.16299	5.37988	0.90301	2.71953	2.99101	2.78859	0.63979	1.42086
					0.64115	0.05919	0.29807	0.60054	0.65172	0.05313	0.29298	0.46639
				3	3.13278	4.77410	0.86444	2.41437	3.03398	2.95214	0.65726	1.49995
					0.62655	0.05464	0.28358	0.57401	0.64625	0.04777	0.28181	0.46954
			1.25	1	2.39827	1.93518	0.52975	0.99867	2.56056	1.83094	0.51694	0.94684
					0.6198	0.06216	0.30402	0.41688	0.63873	0.06274	0.31018	0.41356
				2	2.49453	2.23036	0.55891	1.13844	2.62990	2.17056	0.55417	1.10792
					0.61438	0.04651	0.28040	0.41965	0.62341	0.04528	0.27137	0.41277
				3	2.51084	2.35251	0.58018	1.19797	2.67482	2.30001	0.56872	1.17173
					0.61479	0.04343	0.27009	0.42513	0.62354	0.04346	0.2708	0.41976
	24	12	0.75	1	2.66153	2.84045	0.67903	1.44386	2.55870	1.51688	0.48784	0.78104
					0.61423	0.04727	0.26993	0.47448	0.63323	0.04519	0.26975	0.37880
				2	2.65765	2.80670	0.66891	1.42267	2.57206	1.52838	0.48807	0.78438
					0.60499	0.03863	0.24942	0.45916	0.62852	0.04037	0.25546	0.37176
				3	2.66155	2.70155	0.66215	1.37114	2.54656	1.48692	0.47915	0.76343
					0.60974	0.04073	0.25680	0.45948	0.62569	0.03995	0.25838	0.36877
			1.25	1	2.27670	1.27666	0.43552	0.65768	2.34350	1.13894	0.42183	0.58874
					0.60144	0.03869	0.25497	0.34525	0.61286	0.03854	0.25427	0.33805
				2	2.25654	1.23947	0.43502	0.63557	2.37776	1.24784	0.43577	0.64098
					0.59460	0.03167	0.23929	0.33716	0.60409	0.03411	0.24156	0.33866
				3	2.26794	1.2673	0.42851	0.65210	2.34428	1.15663	0.42587	0.59623
					0.60239	0.0369	0.25062	0.33956	0.60774	0.03583	0.24687	0.33637
	30	–	–	–	2.16628	0.93180	0.37558	0.47932	2.25434	0.85314	0.36844	0.44084
					0.58901	0.02684	0.21922	0.29740	0.59955	0.02853	0.22622	0.29733
60	36	24	0.75	1	2.33399	1.57484	0.51583	0.79989	2.25629	0.91773	0.39724	0.47061
					0.58825	0.02495	0.20946	0.36264	0.59751	0.02348	0.20279	0.30001
				2	2.43163	1.79258	0.54229	0.90620	2.2866	0.97781	0.40555	0.49762
					0.58731	0.01982	0.18985	0.36607	0.58566	0.01742	0.18139	0.29347
				3	2.45710	1.81181	0.54569	0.91491	2.33607	1.08403	0.42181	0.55021
					0.58494	0.01802	0.18516	0.36543	0.58618	0.01639	0.17513	0.29847
			1.25	1	2.14646	0.71289	0.33572	0.36726	2.15126	0.67416	0.33218	0.34784
					0.58273	0.02164	0.19793	0.26682	0.58502	0.02153	0.19829	0.26523
				2	2.18068	0.83142	0.35672	0.42469	2.18173	0.75843	0.34926	0.38770
					0.58181	0.01796	0.18432	0.27052	0.57677	0.01697	0.18145	0.26536
				3	2.18179	0.80429	0.35137	0.41070	2.21906	0.83871	0.36420	0.42749
					0.58197	0.01710	0.17881	0.26509	0.57718	0.01627	0.17642	0.27031
	48	24	0.75	1	2.20066	1.11075	0.43534	0.56389	2.09036	0.67092	0.34587	0.34366
					0.57946	0.01702	0.17808	0.30671	0.57909	0.01639	0.17340	0.25963
				2	2.21804	1.16005	0.44356	0.58810	2.12635	0.70902	0.35412	0.36170
					0.57796	0.01614	0.17109	0.30732	0.57727	0.01438	0.16389	0.25900
				3	2.22030	1.08627	0.43096	0.55127	2.10469	0.67264	0.34647	0.34364
					0.57881	0.01627	0.17439	0.30268	0.57773	0.01464	0.16522	0.25585
			1.25	1	2.09235	0.46897	0.27601	0.24186	2.08515	0.46462	0.27725	0.23961
					0.57649	0.01475	0.16596	0.22099	0.57627	0.01460	0.16689	0.22207
				2	2.10153	0.52980	0.28948	0.27176	2.10875	0.53386	0.29658	0.27428
					0.57430	0.01372	0.16352	0.22650	0.57566	0.01469	0.16694	0.23176
				3	2.07216	0.48080	0.27873	0.24758	2.08643	0.46544	0.27737	0.23989
					0.57299	0.01435	0.16614	0.22243	0.57450	0.01434	0.16532	0.22135
	60	–	–	–	2.03842	0.38016	0.25164	0.19590	2.04836	0.35345	0.24393	0.18223
					0.56937	0.01163	0.15094	0.20129	0.57014	0.01100	0.14767	0.19580

,

Table 9. AILs and COVPs (in %) of the 95% CIs of $\alpha$ and $\beta$ based on 5000 simulations. The population parameter values are $\alpha =1.9$ and $\beta =0.55$ for Case B.

					NACI			LTCI			Optimal Value
					COVP$\left(\mathit{\alpha }\right)$	AIL$\left(\mathit{\alpha }\right)$		COVP$\left(\mathit{\alpha }\right)$	AIL$\left(\mathit{\alpha }\right)$
n	m	k	$\mathit{\eta }$	CS	COVP$\left(\mathit{\beta }\right)$	AIL$\left(\mathit{\beta }\right)$	AAIL	COVP$\left(\mathit{\beta }\right)$	AIL$\left(\mathit{\beta }\right)$	AAIL	${\mathit{\tau }}_{\mathit{D}}^{*}$	${\mathit{\tau }}_{\mathit{A}}^{*}$
30	18	12	0.75	1	93.16	5.35176	3.17217	81.8	7.70284	4.39454	0.53412	0.53472
					98.32	0.99258		97.12	1.08625
				2	91.92	5.47946	3.16522	80.8	7.83493	4.37456	0.49218	0.46833
					98.16	0.85099		96.84	0.9142
				3	91.62	5.58575	3.20481	80.02	8.0354	4.45869	0.49094	0.46331
					98.16	0.82387		96.78	0.88198
			1.25	1	93.16	4.21141	2.5428	84.4	5.4724	3.20957	0.58178	0.58047
					96.9	0.87419		96.68	0.94673
				2	92.56	4.36677	2.57477	83.24	5.70721	3.27176	0.53877	0.51030
					97.42	0.78277		96.48	0.8363
				3	93.02	4.49523	2.63212	83.34	5.87499	3.34729	0.53431	0.50136
					97.54	0.76901		96.92	0.8196
	24	12	0.75	1	94.44	4.63114	2.70773	86.02	6.72956	3.78325	0.53800	0.53265
					97.46	0.78432		96.86	0.83694
				2	94.58	4.66790	2.70536	85.46	6.76652	3.77721	0.51335	0.49896
					97.56	0.74282		96.84	0.78790
				3	94.28	4.62708	2.6923	86.28	6.71204	3.75891	0.5393	0.52938
					97.6	0.75751		96.84	0.80579
			1.25	1	94.10	3.56791	2.14128	86.74	4.38774	2.57227	0.59115	0.58196
					96.78	0.71464		96.06	0.75681
				2	93.90	3.6468	2.16209	86.30	4.51675	2.61545	0.56063	0.53988
					96.16	0.67738		96.16	0.71415
				3	94.54	3.57774	2.13757	86.44	4.4192	2.57818	0.59352	0.57819
					96.3	0.6974		96.54	0.73717
	30	–	–	–	94.62	3.16431	1.8919	87.94	3.70022	2.1741	0.59591	0.58135
					96.42	0.6195		96.10	0.64797
60	36	24	0.75	1	95.52	3.83408	2.21163	89.14	5.38558	2.99967	0.54185	0.53177
					97.18	0.58918		96.60	0.61376
				2	95.38	3.91135	2.21626	88.58	5.55468	3.04669	0.47757	0.45583
					97.12	0.52116		96.38	0.53869
				3	94.62	3.98578	2.24866	87.78	5.65008	3.08907	0.48471	0.45998
					96.8	0.51153		96.48	0.52806
			1.25	1	94.70	2.872	1.70744	89.62	3.31451	1.93868	0.60036	0.58081
					95.36	0.54288		95.34	0.56286
				2	94.68	2.98105	1.73734	88.86	3.45964	1.9843	0.53218	0.49806
					95.38	0.49363		95.2	0.50896
				3	94.34	3.06493	1.77565	88.24	3.586	2.0435	0.53529	0.49806
					95.36	0.48637		95.4	0.501
	48	24	0.75	1	96.60	3.36755	1.92931	91.70	4.54549	2.5258	0.54106	0.5308
					96.52	0.49107		96.46	0.50611
				2	95.88	3.40287	1.93511	90.76	4.59372	2.53703	0.5085	0.49354
					96.24	0.46735		96.02	0.48035
				3	96.42	3.3696	1.92491	91.62	4.5236	2.50896	0.54301	0.52961
					96.48	0.48022		96.4	0.49431
			1.25	1	94.66	2.47636	1.4684	91.14	2.70883	1.59087	0.60478	0.58173
					95.38	0.46044		95.34	0.47291
				2	93.82	2.5207	1.48232	89.8	2.77381	1.61447	0.56629	0.53803
					94.78	0.44394		94.8	0.45513
				3	94.48	2.48398	1.46812	90.6	2.72151	1.59283	0.60687	0.58005
					95.12	0.45226		95.26	0.46414
	60	–	–	–	95.00	2.18996	1.29749	90.96	2.32862	1.37114	0.60744	0.58175
					95.48	0.40501		95.44	0.41366

,

Table 10. PEs and MLEs of $\alpha$ and $\beta$ with their MSEs, RABs, AMSEs, and ARABs based on 5000 simulations. The population parameter values are $\alpha =1.0$ and $\beta =0.55$ for Case C.

					PE				MLE
					$\overline{\widehat{\mathit{\alpha }}}$	MSE($\widehat{\mathit{\alpha }}$)	RAB($\widehat{\mathit{\alpha }}$)	AMSE	$\overline{\check{\mathit{\alpha }}}$	MSE($\check{\mathit{\alpha }}$)	RAB($\check{\mathit{\alpha }}$)	AMSE
n	m	k	$\mathit{\eta }$	CS	$\overline{\widehat{\mathit{\beta }}}$	MSE($\widehat{\mathit{\beta }}$)	RAB($\widehat{\mathit{\beta }}$)	ARAB	$\overline{\check{\mathit{\beta }}}$	MSE($\check{\mathit{\beta }}$)	RAB($\check{\mathit{\beta }}$)	ARAB
30	18	12	0.75	1	2.17114	3.99222	1.49238	2.02063	1.56840	1.55187	0.97416	0.78835
					0.66173	0.04904	0.30438	0.89838	0.58370	0.02484	0.21509	0.59462
				2	2.34821	4.58109	1.62639	2.31543	1.54813	1.50802	0.95967	0.76442
					0.67595	0.04977	0.30681	0.96660	0.56876	0.02081	0.20241	0.58104
				3	2.35557	4.68525	1.65054	2.36836	1.57068	1.52619	0.96485	0.77299
					0.68716	0.05148	0.31169	0.98111	0.56929	0.01980	0.19607	0.58046
			1.25	1	1.41464	1.80542	0.95059	0.91389	1.44685	1.31165	0.88403	0.66620
					0.53475	0.02235	0.21540	0.58300	0.56785	0.02075	0.20141	0.54272
				2	1.4515	1.93496	0.97806	0.97717	1.47440	1.35351	0.89526	0.68699
					0.52402	0.01938	0.2025	0.59028	0.56890	0.02047	0.19854	0.54690
				3	1.49695	2.12906	1.02507	1.07402	1.48678	1.40529	0.91533	0.71242
					0.52636	0.01899	0.20086	0.61297	0.56449	0.01955	0.19408	0.55471
	24	12	0.75	1	2.11856	3.84570	1.45668	1.94282	1.52749	1.47995	0.94928	0.74896
					0.63982	0.03994	0.26919	0.86293	0.57299	0.01796	0.18741	0.56834
				2	1.97300	3.38275	1.34404	1.70325	1.53808	1.47733	0.94485	0.74736
					0.57566	0.02374	0.21203	0.77803	0.57166	0.01739	0.18555	0.56520
				3	1.95572	3.31454	1.32478	1.66900	1.52887	1.44759	0.93795	0.73276
					0.57791	0.02346	0.20947	0.76712	0.57408	0.01793	0.18908	0.56352
			1.25	1	1.38669	1.72566	0.91878	0.87199	1.42009	1.27118	0.87035	0.64421
					0.54287	0.01832	0.19464	0.55671	0.56738	0.01724	0.18345	0.52690
				2	1.38908	1.73950	0.92703	0.87796	1.46111	1.36687	0.90733	0.69182
					0.53526	0.01641	0.18510	0.55606	0.56607	0.01676	0.18057	0.54395
				3	1.36414	1.71952	0.91591	0.86808	1.44570	1.33560	0.89572	0.67641
					0.53438	0.01665	0.18663	0.55127	0.56957	0.01722	0.18365	0.53969
	30	–	–	–	1.10612	1.15803	0.76992	0.58451	1.32290	1.14694	0.82315	0.58085
					0.51997	0.01100	0.15398	0.46195	0.56012	0.01477	0.17187	0.49751
60	36	24	0.75	1	2.06335	3.70211	1.41565	1.86252	1.50620	1.40106	0.91983	0.70743
					0.61422	0.02294	0.20854	0.81210	0.57217	0.01380	0.16241	0.54112
				2	2.25351	4.27479	1.56326	2.14736	1.49731	1.41194	0.92624	0.71181
					0.61433	0.01993	0.19675	0.88001	0.56619	0.01168	0.15238	0.53931
				3	1.93645	3.20929	1.31084	1.61172	1.49309	1.40126	0.91361	0.70647
					0.57022	0.01415	0.16772	0.73928	0.56567	0.01168	0.15344	0.53353
			1.25	1	1.34742	1.58772	0.88932	0.80006	1.37573	1.18773	0.83789	0.60038
					0.54364	0.01239	0.15992	0.52462	0.56542	0.01303	0.16160	0.49974
				2	1.37444	1.67479	0.89721	0.84317	1.39712	1.23752	0.85905	0.62439
					0.53713	0.01155	0.15651	0.52686	0.56330	0.01125	0.15131	0.50518
				3	1.40157	1.71212	0.93292	0.86176	1.40659	1.26004	0.86067	0.63556
					0.53468	0.01140	0.15780	0.54536	0.56228	0.01107	0.14951	0.50509
	48	24	0.75	1	1.78336	2.73233	1.19009	1.37272	1.47256	1.32299	0.89085	0.66688
					0.57615	0.01312	0.15946	0.67477	0.56903	0.01077	0.14675	0.51880
				2	1.80859	2.77815	1.19801	1.39538	1.46585	1.33092	0.89158	0.67070
					0.57113	0.01261	0.15559	0.67680	0.56806	0.01049	0.14412	0.51785
				3	1.76336	2.64301	1.16230	1.32788	1.46537	1.35280	0.89503	0.68157
					0.57393	0.01276	0.15685	0.65957	0.56617	0.01034	0.14459	0.51981
			1.25	1	1.34385	1.49593	0.85893	0.75335	1.3559	1.14415	0.82289	0.57745
					0.54792	0.01077	0.14955	0.50424	0.5642	0.01075	0.14599	0.48444
				2	1.35650	1.55408	0.86808	0.78207	1.35240	1.14350	0.81526	0.57663
					0.54656	0.01006	0.14439	0.50623	0.56313	0.00976	0.14033	0.47779
				3	1.32831	1.39706	0.83624	0.70369	1.36178	1.11986	0.81096	0.56509
					0.54477	0.01032	0.14578	0.49101	0.56434	0.01032	0.14414	0.47755
	60	–	–	–	1.08480	0.89976	0.69059	0.45346	1.29689	1.00335	0.75650	0.50599
					0.52858	0.00716	0.12373	0.40716	0.55882	0.00862	0.13227	0.44438

and

Table 11. AILs and COVPs (in %) of the 95% CIs of $\alpha$ and $\beta$ based on 5000 simulations. The population parameter values are $\alpha =1.0$ and $\beta =0.55$ for Case C.

					NACI			LTCI			Optimal Value
					COVP$\left(\mathit{\alpha }\right)$	AIL$\left(\mathit{\alpha }\right)$		COVP$\left(\mathit{\alpha }\right)$	AIL$\left(\mathit{\alpha }\right)$
n	m	k	$\mathit{\eta }$	CS	COVP$\left(\mathit{\beta }\right)$	AIL$\left(\mathit{\beta }\right)$	AAIL	COVP$\left(\mathit{\beta }\right)$	AIL$\left(\mathit{\beta }\right)$	AAIL	${\mathit{\tau }}_{\mathit{D}}^{*}$	${\mathit{\tau }}_{\mathit{A}}^{*}$
30	18	12	0.75	1	100.00	7.58446	4.22565	100.00	8.3369	4.64840	0.20424	0.44114
					99.14	0.86685		99.96	0.95989
				2	99.98	7.64752	4.20315	99.98	8.34642	4.58494	0.19781	0.40666
					98.16	0.75877		99.78	0.82347
				3	99.96	7.75997	4.25047	99.98	8.46083	4.62971	0.20022	0.40418
					97.60	0.74097		99.50	0.79860
			1.25	1	100.00	6.38073	3.58809	100.00	6.77272	3.82209	0.217	0.47592
					97.64	0.79546		99.68	0.87147
				2	100.00	6.54931	3.64013	100.00	6.90472	3.84629	0.20671	0.43776
					96.66	0.73094		99.36	0.78787
				3	100.00	6.53461	3.62206	100.00	6.95781	3.85946	0.20683	0.43329
					96.70	0.70950		99.30	0.76111
	24	12	0.75	1	100.00	7.25760	4.00007	100.00	7.69363	4.24681	0.20117	0.43564
					98.28	0.74254		99.60	0.79999
				2	100.00	7.28863	3.99804	100.00	7.74797	4.25314	0.19809	0.41539
					97.38	0.70745		99.32	0.75831
				3	100.00	7.22512	3.97734	100.00	7.66505	4.22436	0.20027	0.43711
					97.86	0.72955		99.42	0.78367
			1.25	1	99.98	5.98044	3.34178	99.98	6.38817	3.57219	0.21180	0.46854
					97.84	0.70312		99.36	0.75622
				2	99.98	6.11328	3.39474	99.98	6.54616	3.63442	0.20548	0.44463
					97.52	0.67620		99.06	0.72267
				3	100.00	6.02677	3.36065	100.00	6.45903	3.60106	0.21087	0.46892
					97.76	0.69453		99.30	0.74309
	30	–	–	–	98.12	5.07645	2.84264	98.12	6.39162	3.51761	0.21496	0.47672
					96.88	0.60883		98.68	0.64361
60	36	24	0.75	1	99.98	5.03231	2.82001	99.98	6.77235	3.70549	0.19640	0.43171
					97.94	0.60770		98.92	0.63864
				2	99.96	4.99889	2.76598	100.00	6.78508	3.66938	0.19302	0.38688
					97.08	0.53307		98.72	0.55368
				3	100.00	5.07471	2.79803	100.00	6.82111	3.68090	0.19267	0.39424
					97.20	0.52134		98.34	0.54069
			1.25	1	99.96	4.37391	2.47518	99.96	5.74542	3.17454	0.20554	0.46446
					97.20	0.57646		98.78	0.60367
				2	99.94	4.41337	2.46448	99.94	5.83758	3.18608	0.19541	0.41668
					96.76	0.51559		98.80	0.53458
				3	99.96	4.44709	2.47536	99.96	5.88787	3.20458	0.19678	0.42178
					96.98	0.50363		98.66	0.52129
	48	24	0.75	1	100.00	4.87704	2.70037	100.00	6.38144	3.46231	0.19400	0.42938
					98.38	0.52369		99.08	0.54318
				2	100.00	4.87786	2.68734	100.00	6.38594	3.44970	0.19267	0.40875
					97.52	0.49682		98.68	0.51346
				3	100.00	4.83923	2.67338	100.00	6.36455	3.44493	0.19513	0.43037
					97.94	0.50752		99.04	0.52531
			1.25	1	99.52	4.1727	2.3373	99.52	5.53689	3.02835	0.20409	0.46233
					96.84	0.50191		98.42	0.51981
				2	99.58	4.16144	2.31843	99.58	5.53870	3.01455	0.19897	0.43895
					97.06	0.47543		98.34	0.49040
				3	99.54	4.19359	2.34165	99.54	5.55192	3.02897	0.20262	0.46293
					97.38	0.48971		98.50	0.50602
	60	–	–	–	94.96	3.73590	2.08512	94.96	5.82791	3.13701	0.21386	0.47246
					96.74	0.43435		98.22	0.44610

, taking into account the following population parameter values: $\alpha =2.0$ (for Case A), $\alpha =1.90$ (for Case B), $\alpha =1.00$ (for Case C), and $\beta =0.55$. The following values were taken into account through the simulation analysis: $n=30,60$, $m=60\%,80\%,100\%$ of the sample size, $k=40\%$ of the sample size, $\eta =0.75,1.25$, and $\tau =0.20$.

6.1. Simulation Results

Based on the results of the calculations recorded in Table 6, Table 7, Table 8, Table 9, Table 10 and Table 11, the following points can be observed:

• The MLEs are better than the PEs through the ARABs and AMSEs.
• The NACIs are better than the LTCIs through the AAILs.
• The optimal values ${\tau }_D^{*}$ and ${\tau }_A^{*}$ are close one to each other each time for Cases A and B. However, for Case C, the optimal value ${\tau }_D^{*}$ is less than ${\tau }_A^{*}$ and approaches the proposed value $(\tau =0.2)$.
• For fixed values of m and $\eta$, by increasing n, the RABs, MSEs, MRABs, MMSEs, AIL, and AAIL decrease.
• For fixed values of n and $\eta$, by increasing m, the RABs, MSEs, MRABs, MMSEs, AIL, and AAIL decrease.
• For fixed values of n and m, by increasing $\eta$, the RABs, MSEs, MRABs, MMSEs, AIL, and AAIL decrease.
• By increasing $n,m$, or $\eta$, the COVPs are close to 95% for Cases A and B. However, for Case C, the COVPs are higher than 95% because greater values of the MSEs of the estimates than those for Cases A and B were obtained, which led to wider CIs
• For fixed values of n and $\eta$, by increasing m, the optimal values ${\tau }_D^{*}$ and ${\tau }_A^{*}$ increase.
• For fixed m and $\eta$, by increasing n, the optimal values ${\tau }_D^{*}$ and ${\tau }_A^{*}$ decrease.
• For fixed n and m, by increasing $\eta$, the optimal values ${\tau }_D^{*}$ and ${\tau }_A^{*}$ increase.

Except in rare situations, which might be consequent to fluctuations in the data, the above findings are accurate. Based on the above observations, we recommend using the accelerating function $\lambda \left(t\right)$ for the power series (Case A) or exponential (Case B) functions.

7. Concluding Remarks

We applied PALTs to units whose lifetimes under normal stress conditions are exponentially distributed and proposed the use of the TFR model when the failure rate under normal stress conditions is tampered by a non-decreasing function of time, which is sometimes called a “time transformation function”. Three different types of the proposed function were considered. The sufficient conditions for the proposed functions to be accelerating functions were discussed. Based on generalized type-I progressively hybrid censored data, we discussed the MLEs and PEs, as well as the approximate CIs, of the included parameters. The determination of the optimal stress change time was discussed for two different criteria of optimality. A real dataset was used to explain the procedures of inference presented in the paper. In the end, a Monte Carlo simulation study was carried out to examine the performance of the estimation methods and the optimality criteria. From the numerical results, we suggest the use of the ML method and the NACI to estimate the parameters included in the situations of acceleration and censoring used. In addition, the results are completed with a recommendation for the use of the power series or exponential functions as accelerating functions.