Statistical inference for a step-stress partially-accelerated life test model with an adaptive Type-I progressively hybrid censored data from Weibull distribution

Abstract

In this paper, a new censoring scheme, namely, adaptive Type-I progressively hybrid censoring scheme under a step-stress partially accelerated test model is introduced. It has some advantages over the progressively hybrid censoring schemes already discussed in the literature. Based on this censoring scheme, the maximum likelihood estimations of Weibull distribution parameters and the acceleration factor are considered. The biases and mean squared errors of the maximum likelihood estimators of the model parameters are computed to evaluate their performances in the presence of censoring scheme developed in this paper through a Monte Carlo simulation study. The results obtained under the adaptive Type-I progressively hybrid censoring scheme are compared with those produced under the non-adaptive Type-I progressively hybrid censoring scheme using three different progressive censoring schemes. Moreover, the confidence intervals lengths and their associated coverage probabilities are obtained for both adaptive and non-adaptive Type-I progressively hybrid censoring scheme. In addition, Optimum test plans are also developed to improve/guarantee the quality of the statistical inference.

1 Introduction

Under continuous quality and reliability improvement of products, it is more difficult to obtain failure information under normal (use) condition. So, in order to quickly obtain information on the failure-times of these products, all of test units or some of them are run under higher than normal stress-levels. If the experimenter puts all test units under such stresses, the test is called accelerated life test (ALT) but if he puts some of them then the test is called partially accelerated life test (PALT). The information obtained from the test performed in accelerated environment is used to predict actual product performance in usual environment.

As Nelson (1990) indicates, the stress can be applied in various ways, commonly used methods are step-stress and constant-stress. Under step-stress PALT (SSPALT), a test item is first run at use condition and, if it does not fail for a specified time say $\tau$, then it is run at accelerated condition (stress) until failure occurs or the observation is censored. But the constant-stress PALT runs each item at either use condition or accelerated condition only, i.e. each unit is run at a constant-stress level until the test is terminated.

The objective of a PALT is to collect more failure data in a limited time without necessarily using high stresses to all test units. SSPALT is used when the stress-levels are increased in stepwise approach for each of the items on a life test. SSPALT is sometimes preferred over constant stress PALT because it quickly yields failures without necessarily shocking the units on test with an initially high stress and thereby avoiding additional, unrelated failure modes. SSPALT is particularly useful in new-product development when the appropriate stress levels for a constant stress PALT are unknown.

This paper considers SSPALT using progressively hybrid censoring scheme. The work on progressively hybrid censoring schemes has become quite popular in life testing and reliability studies. Kundu and Joarder (2006) and Childs et al. (2008) have considered a Type-I progressively hybrid censoring scheme (Type-I PHCS) in the context of life testing experiment is terminated at time min {Y$_{m:m:n},\,\eta$}; where Y$_{m:m:n}$ is the time of $m$th failure out of $m$ failures using $n$ tested units and $\eta$ is a pre-specified censoring time. Ng et al. (2009) and Lin et al. (2009) have investigated point and interval estimations for exponential and Weibull lifetimes under an adaptive Type-II progressively hybrid censoring scheme (Type-II APHCS). Lin and Huang (2012) studied point and interval estimations for exponential lifetimes under an adaptive Type-I progressively hybrid censoring scheme (Type-I APHCS).

Regarding PALT, it has been studied under conventional Type-I and Type-II censoring schemes by several authors, for example, see Goel (1971), DeGroot and Goel (1979), Bhattacharya and Soejoeti (1989), Bai and Chung (1992), Bai et al. (1993a), Abdel-Ghaly et al. (2002, 2003a, b), Abdel-Ghani (1998, 2004), Ismail (2004), Aly and Ismail (2008), Ismail and Sarhan (2009), Ismail (2010 and 2013). Also, SSPALT has been studied under hybrid censoring, see Ismail (2012a). In addition, Ismail (2012b) has considered SSPALT using progressive Type-II censoring scheme.

Specifically, based on Type-I PHCS there are some studies under ALT, for example, see Lin et al. (2012), Lin and Huang (2012) and Ling et al. (2009). While under PALT there was only one study had been made by Ismail (2013) using Type-I PHCS. Now, in this paper we focus on statistical inference of SSPALT model under Type-I APHCS assuming Weibull lifetimes. This new scheme under SSPALT will be described in the next Section.

The rest of this paper is arranged as follows. In Sect. 2 the model and the test method are described. In Sect. 3 the maximum likelihood estimations (point and interval) of Weibull distribution parameters and the acceleration factor are considered for both Type-I APHCS and Type-I PHCS. Sect. 4 considers optimum test plans under different censoring schemes using both Type-I APHCS and Type-I PHCS. Sect. 5 contains the simulation results and discussion. Concluding remarks and further studies are given in Sect. 6.

2 Description of the model

Assume that the random variable Y representing the lifetime of a product has Weibull distribution (WD) with the shape and scale parameters as $\alpha$ and $\lambda$ respectively. So, the probability density function (pdf) of Y is

$$\begin{aligned} f_Y (y;\alpha ,\lambda )=\frac{\alpha }{\lambda } \left( {\frac{\hbox {y}}{\lambda }} \right) ^{\alpha -1}\,e^{-(y/\lambda )^{\alpha }}; \quad y>0,\,\alpha >0,\,\lambda >0 \end{aligned}$$

(1)

WD is one of the most common distributions which are used to analyze several lifetime data. Its hazard function can be increasing, decreasing and constant depending on the shape parameter value. Thus, this distribution has lots of flexibility compared to other distributions.

The survival function of WD with pdf in Eq. (1) takes the form

$$\begin{aligned} S(y)=\exp \{-\,(y/\lambda )^{\alpha }\}, \end{aligned}$$

(2)

The corresponding failure rate function is given by

$$\begin{aligned} h(y)=\frac{\alpha }{\lambda }\left( {\frac{y}{\lambda }} \right) ^{\alpha -1}. \end{aligned}$$

(3)

The pdf of $Y$under SSPALT model can be written as

$$\begin{aligned} f(y)=\left\{ {{\begin{array}{ll} 0, &{}\quad y \le 0, \\ f_1 (y)\equiv f_Y (y;\alpha ,\lambda ), &{} \quad 0<y\le \tau \\ f_2 (y), &{} \quad y>\tau ,\\ \end{array} }} \right. \end{aligned}$$

(4)

where

$$\begin{aligned} f_2 (y)\equiv f_Y (y;\alpha ,\lambda ,\beta )=\beta \frac{\alpha }{\lambda }\left( {\frac{\tau +\beta (y-\tau )}{\lambda }} \right) ^{\alpha -1}\exp \{-\,([\tau +\beta (y-\tau )]/\lambda )^{\alpha }\},\nonumber \\ \end{aligned}$$

(5)

which is obtained by the transformation-variable technique using the density in Eq. (1) and the model proposed by DeGroot and Goel (1979) which is given by:

$$\begin{aligned} Y=\left\{ {{\begin{array}{ll} T &{} \quad \hbox {if}\quad T \le \tau \\ \tau +\beta ^{-1}\hbox {(T-}\tau ) &{} \quad \hbox {if}\quad T >\tau , \\ \end{array} }} \right. \end{aligned}$$

(6)

where $T$ is the lifetime of the unit under use condition,$\tau$is the stress change time and $\beta$ is the acceleration factor.

In this paper, we introduce a new progressively hybrid censoring scheme under SSPALT called Type-I APHCS. It guarantees the termination of the life testing experiment at a fixed time $\eta$ and results a higher efficiency in estimations as compared with Type-I PHCS discussed in the literature by Ismail (2013). This new censoring scheme, Type-I APHCS, can be described as follows. Assume $n$ identical units are placed on a test with progressive censoring scheme ($R_{1}$,$R_{2}$,..., $R_{m})$, 1  $\le \,m\, \le \,n,$ and the experiment is terminated at time $\eta$, where $\eta \epsilon (0, \infty )$ and integers $R_{i}$’s are fixed in advance. At the time of the first failure Y$_{1:m:n},\,R_{1}$ of the remaining units are randomly removed. Similarly, at the time of the second failure Y$_{2:m:n},\,R_{2}$ of the remaining units are randomly removed and so on. Let $K$ indicate the number of failures that occur before time $\eta$. If the $m$th failure Y$_{m:m:n}$ occurs before time $\eta$, the process will not stop at the time Y$_{m:m:n, }$but continue to observe additional failures (without any further withdrawals) up to time $\eta$. Thus, at time $\eta$ all the remaining $\mathop R\nolimits _K^*=n-K-\sum _{i=1}^K {\mathop {\,R}\nolimits _i }$units are removed and the experiment is terminated. It can be noted that the progressive censoring scheme in this case will become ($R_{1}$, $R_{2}$, ..., $R_{m},\,R_{m+1}$,..., $R_{K})$, where $R_{m}=R_{m+1}=R_{K}$ = 0. Clearly, there is an advantage that more than $m$ failures may be observed which will significantly increase the efficiency/quality of the statistical inference. The process when Y$_{m:m:n} \quad <\,\eta$ will have a progressive censoring scheme as ($R_{1}$,$R_{2}$,..., $R_{K})$. Let $n_{u}$ be the number of units that fail before time $\tau ,\,n_{a}$ be the number of units that fail before time $\eta$ at accelerated condition. That is, the total number of failures that can occur before the time $\eta$ is $K=n_{u}+n_{a}$.

3 Maximum likelihood estimation

This section discusses the process of obtaining the maximum likelihood estimates (MLEs) of the parameters $\alpha ,\,\lambda$ and $\beta$ based on two sets of data obtained from Type-I PHC and Type-I APHC. Both point and interval estimations of the parameters are considered for each data set.

3.1 Estimation based on Type-I APHC

3.1.1 Point estimation

Here, the ML estimates of SSPALT model parameters are given based on the observed Type-I APHC data from WD.

Now, suppose that a Type-I APHCS from Weibull sample with a given progressive censoring scheme ($R_{1}$, $R_{2}$,..., $R_{m})$ is observed. We indicate the resulting failure times by Y$_{1:m:n}$, Y$_{2:m:n}$,..., Y$_{m:m:n}$, Y$_{m+1:n}$, ..., Y$_{K:n}$ when Y$_{m:m:n} \,>\,\eta$, and Y$_{1:m:n}$, Y$_{2:m:n}$, ..., Y$_{K:m:n}$ when Y$_{m:m:n} \,< \eta$. Then, the likelihood function of the given sample is given by

$$\begin{aligned} L(\alpha ,\lambda ,\,\beta )\propto \prod _{i=1}^{\mathop n\nolimits _u } {f_1 (\mathop y\nolimits _i ).[\mathop s\nolimits _1 (\mathop y\nolimits _i )]^{\mathop R\nolimits _i }.[\mathop s\nolimits _1 (\tau )]^{\mathop {}\nolimits _{\mathop n\nolimits _\tau } }}. \prod _{i=\mathop n\nolimits _u +1}^K {f_2 (\mathop y\nolimits _i ).[\mathop s\nolimits _2 (\mathop y\nolimits _i )]^{\mathop R\nolimits _i }.[\mathop s\nolimits _2 (\eta )]^{\mathop R\nolimits _K^*}},\nonumber \\ \end{aligned}$$

(7)

where

$$\begin{aligned}&\mathop s\nolimits _1 (y)=\exp \{-\,(y/\lambda )^{\alpha }\}, \mathop s\nolimits _2 (y)=\exp \{-\,[(\tau +\beta (y-\tau ))/\lambda ]^{\alpha }\},\\&\quad \mathop n\nolimits _\tau =n-nu-\sum _{i=1}^{nu} {\mathop R\nolimits _i } \end{aligned}$$

and $\mathop R\nolimits _K^*=n-K-\sum _{i=1}^K {\mathop R\nolimits _i }$ with $R_{m}=R_{m+1}=R_{K}$ = 0 if $K\ge m$. For simplicity of notation, we use $y_\mathrm{i}$ instead of $y_{i:m:n}$ or $y_{\hbox {i:}n}$ in the remaining discussion.

From Eq. (7) the natural logarithm of the likelihood function given $K \ge 1$is

$$\begin{aligned} ln\,L(\alpha ,\lambda ,\,\beta )&= K\,ln\,\alpha \,-\,K\alpha \,ln\,\lambda \,\!+\!n_a \,ln\,\beta +(\alpha -1) \left[ {\sum _{i=1}^{n_u } {ln\,y_\mathrm{i} } \!+\!\sum _{i=n_u +1}^K {ln\,\psi _\mathrm{i} }} \right] \nonumber \\&-\,\frac{1}{\lambda ^{\alpha }}\left\{ \sum _{i=1}^{n_u } {y_i^\alpha } \!+\!\sum _{i=1}^{n_u } {\mathop R\nolimits _i y_i^\alpha } +\sum _{i=n_u +1}^K {\psi _i^\alpha } \!+\!\sum _{i=n_u +1}^K {\mathop R\nolimits _i \,\psi _i^\alpha } +n_u \mathop n\nolimits _\tau \,\tau ^{\alpha }\right. \nonumber \\&\left. +\,n_a \,\psi _\eta ^\alpha \mathop R\nolimits _K^*\right\} ,\, \end{aligned}$$

(8)

where $\psi _\mathrm{i} =\tau +\beta (y_\mathrm{i} -\tau )$and $\psi _\eta =\tau +\beta (\eta -\tau )$.

By equating the first partial derivatives of $ln\,L$ to zero with respect to $\alpha ,\,\lambda$ and $\beta$, the resulting three equations are

$$\begin{aligned} \frac{\partial lnL}{\partial \alpha }&= \frac{k}{\alpha }-k\,ln\,\lambda +\sum _{i=1}^{n_u } {ln\,y_\mathrm{i} +\sum _{i=n_u +1}^k {ln\,\psi _\mathrm{i} } }-\frac{1}{\lambda ^{\alpha }}\left\{ \sum _{i=1}^{n_u } y_i^\alpha ln\,y_\mathrm{i} +\sum _{i=1}^{n_u } {\mathop R\nolimits _i y_i^\alpha ln\,y_\mathrm{i}}\right. \nonumber \\&\left. {\mathop +\sum _{i=n_u +1}^k {\psi _i^\alpha ln\,\psi _\mathrm{i} } }+\sum _{i=n_u +1}^k {\mathop R\nolimits _i \psi _i^\alpha ln\,\psi _\mathrm{i} } \right. \nonumber \\&\left. +\, n_u \,\mathop n\nolimits _\tau \mathop \tau \nolimits ^\alpha ln\,\tau +\,n_a \,\mathop R\nolimits _K^*\psi _\eta ^\alpha ln\,\psi _\eta -\,\psi _\alpha \,ln\,\lambda \right\} =0, \end{aligned}$$

(9)

where

$$\begin{aligned} \psi _\alpha =\sum _{i=1}^{n_u } {y_i^\alpha } +\sum _{i=1}^{n_u } {\mathop R\nolimits _i y_i^\alpha } +\sum _{i=n_u +1}^K {\psi _i^\alpha } +\sum _{i=n_u +1}^K {\mathop R\nolimits _i \,\psi _i^\alpha } +\,n_u \mathop n\nolimits _\tau \,\tau ^{\alpha }\,+\,n_a \,\psi _\eta ^\alpha \mathop R\nolimits _K^*\end{aligned}$$

$$\begin{aligned} \frac{\partial lnL}{\partial \lambda }=-\frac{k\alpha }{\lambda }+\frac{\alpha \psi _\alpha }{\lambda ^{\alpha +1}}\,=0, \end{aligned}$$

(10)

$$\begin{aligned} \frac{\partial lnL}{\partial \beta }=\frac{n_a }{\beta }+(\alpha -1)\sum _{i=n_u +1}^k {\frac{y_i -\tau }{\psi _i }} -\frac{\alpha }{\lambda ^{\alpha }}\left\{ \sum _{i=n_u +1}^k {(y_i -\tau )} \,\psi _i^{\alpha -1} +\sum _{i=n_u +1}^k\right. \nonumber \\ \left. {\mathop R\nolimits _i (y_i -\tau )} \,\psi _i^{\alpha -1}+\,n_a \mathop R\nolimits _K^*\,(\eta -\tau )\psi _\eta ^{\alpha -1} \right\} =0. \end{aligned}$$

(11)

From Eq. (10) we can obtain $\hat{{\lambda }}$ as a function of $\alpha$ and $\beta$ as

$$\begin{aligned} \hat{{\lambda }}= \left( {\frac{\psi _\alpha }{k\,}} \right) ^{1/\alpha }. \end{aligned}$$

(12)

Now, the system is reduced to two non-linear likelihood equations in $\alpha$ and $\beta$. It can be solved iteratively using an iterative method such as Newton-Raphson to obtain the ML estimates of $\alpha$ and $\beta$. Therefore, the ML estimate of $\lambda$ can be easily obtained from Eq. (12).

3.1.2 Interval estimation

In this subsection, the approximate confidence intervals of the parameters are constructed based on the asymptotic distribution of the ML estimators of the elements of the vector of unknown parameters $\Omega = (\alpha ,\,\lambda ,\,\beta$). It is known that the asymptotic distribution of the ML estimators of $\Omega$ is given by; see Miller (1981),

$$\begin{aligned} \left( {(\hat{{\alpha }}-\alpha ),(\hat{{\lambda }}-\lambda ),(\hat{{\beta }}-\beta )) \rightarrow \,N\,(0,\,{\mathbf{I}}^{-1}(\alpha ,\,\lambda ,\,\beta )} \right) \end{aligned}$$

where ${\mathbf{I}}^{-1}(\alpha ,\lambda ,\beta )$ is the variance-covariance matrix of the unknown parameters $\Omega$ = ($\alpha ,\,\lambda ,\,\beta$). The elements of the 3 $\times$ 3 matrix ${\mathbf{I}}^{-1}$ , $I_{ij}$ ($\alpha ,\,\lambda ,\,\beta$), i, j = 1, 2, 3 ; can be approximated by $I_{ij}(\hat{{\alpha }},\hat{{\lambda }},\hat{{\beta }})$, where

$$\begin{aligned} I_{ij} (\hat{{\Omega }})=-\frac{\partial ^{2}lnL(\Omega )}{\partial \Omega _i \partial \Omega _j }\left. \right| _{\Omega =\hat{{\Omega }}} \end{aligned}$$

Now, we get the following

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \alpha ^{2}}=-\frac{k}{\alpha ^{2}}-\frac{(\psi _{\alpha .} -\psi _{\alpha ..} ln\lambda )}{\lambda ^{\alpha }}, \end{aligned}$$

(13)

where

$$\begin{aligned} \psi _{\alpha .}&= \sum _{i=1}^{n_u } {y_i^\alpha (ln\,y_\mathrm{i} )^{2}+\sum _{i=1}^{n_u } {\mathop R\nolimits _i y_i^\alpha (ln\,y_\mathrm{i} )^{2}{+}\sum _{i=n_u +1}^k {\psi _i^\alpha (ln\psi _\mathrm{i}}})^{2}{+}\sum _{i=n_u +1}^k {\mathop R\nolimits _i \psi _i^\alpha (ln\psi _\mathrm{i})^{2}}} \\&+\,n_u \,\mathop n\nolimits _\tau \mathop \tau \nolimits ^\alpha (ln\,\tau )^{2}+\,n_a \,\mathop R\nolimits _K^*\psi _\eta ^\alpha (ln\,\psi _\eta )^{2}-(\psi _{\alpha ..} +\psi _\alpha ln\,\lambda )ln\,\lambda , \\ \end{aligned}$$

and

$$\begin{aligned} \psi _{\alpha ..}&= \sum _{i=1}^{n_u } {y_i^\alpha ln\,y_\mathrm{i} +\sum _{i=1}^{n_u } {\mathop R\nolimits _i y_i^\alpha ln\,y_\mathrm{i} +\sum _{i=n_u +1}^k {\psi _i^\alpha ln\,\psi _\mathrm{i} } } +\sum _{i=n_u +1}^k {\mathop R\nolimits _i \psi _i^\alpha ln\,\psi _\mathrm{i} } } \, \\&+\,n_u \,\mathop n\nolimits _\tau \mathop \tau \nolimits ^\alpha ln\,\tau +\,n_a \,\mathop R\nolimits _K^*\psi _\eta ^\alpha ln\,\psi _\eta -\,\psi _\alpha \,ln\,\lambda \end{aligned}$$

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \alpha \partial \lambda }=-\frac{k}{\lambda }+\frac{1}{\lambda ^{\alpha +1}} \left\{ {(1-\alpha \,ln\,\lambda )\,\psi _\alpha +\alpha \,(\psi _{\alpha ..} +\psi _\alpha \,ln\,\lambda )} \right\} , \end{aligned}$$

(14)

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \alpha \partial \beta }&= \sum _{i=n_u +1}^k {\frac{y_\mathrm{i} -\tau }{\psi _i }} -\frac{1}{\lambda ^{\alpha }}\left\{ \sum _{i=n_u +1}^k {(y_i -\tau )} [\alpha \,\psi _i^{\alpha -1} ln\psi _i +\psi _i^\alpha /\psi _i ]\right. \nonumber \\&\left. +\sum _{i=n_u +1}^k {\mathop R\nolimits _i (y_i -\tau )} [\alpha \,\psi _i^{\alpha -1} ln\psi _i +\psi _i^\alpha /\psi _i ]\right. \nonumber \\&\left. +\, n_a \,\mathop R\nolimits _K^*\,(\eta -\tau )[\alpha \,\psi _\eta ^{\alpha -1} ln\psi _\eta +\psi _\eta ^\alpha /\psi _\eta ]\right. \nonumber \\&\left. -\, \alpha [\sum _{i=n_u +1}^k {(y_i -\tau )\psi _i^{\alpha -1} +}\sum _{i=n_u +1}^k {\mathop R\nolimits _i (y_i -\tau )\psi _i^{\alpha -1}}\right. \nonumber \\&\left. {\mathop +\,n_a \,\mathop R\nolimits _K^*\,(\eta -\tau )\psi _\eta ^{\alpha -1} } ]\,ln\lambda \right\} , \end{aligned}$$

(15)

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \lambda ^{2}}=\frac{k\alpha }{\lambda ^{2}}-\frac{\alpha (\alpha +1)\psi _\alpha }{\lambda ^{\alpha +2}}\,, \end{aligned}$$

(16)

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \lambda \partial \beta }&= \frac{\alpha ^{2}}{\lambda ^{\alpha +1}}\left\{ \sum _{i=n_u +1}^k (y_i -\tau )\psi _i^{\alpha -1}\right. \nonumber \\&\left. + \sum _{i=n_u +1}^k {\mathop R\nolimits _i (y_i -\tau )\psi _i^{\alpha -1} +n_a \,\mathop R\nolimits _K^*\,(\eta -\tau )\psi _\eta ^{\alpha -1} } \right\} , \end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \beta ^{2}}&=-\frac{n_a }{\beta ^{2}}-(\alpha -1)\sum _{i=n_u +1}^k \frac{(y_\mathrm{i} -\tau )^{2}}{\psi _i^2 }-\frac{\alpha (\alpha -1)}{\lambda ^{\alpha }}\left\{ \sum _{i=n_u +1}^k {(y_i -\tau )^{2}} \,\psi _i^{\alpha -2}\right. \nonumber \\&\left. \quad +\sum _{i=n_u +1}^k {\mathop R\nolimits _i (y_i -\tau )^{2}} \,\psi _i^{\alpha -2} +\,n_a \mathop R\nolimits _K^*\,(\eta -\tau )^{2}\psi _\eta ^{\alpha -2} \right\} , \end{aligned}$$

(18)

Thus, the approximate 100(1 - $\gamma )$  % two sided confidence intervals for $\alpha ,\,\lambda$ and $\beta$ are, respectively, given by

$$\begin{aligned} \hat{{\alpha }}\pm Z_{\gamma /2} \,\sqrt{\hbox {I}_{11}^{-1} (\hat{{\alpha }},\hat{{\lambda }},\hat{{\beta }})}, \hat{{\lambda }}\pm Z_{\gamma /2} \,\sqrt{\hbox {I}_{\mathrm{22}}^{-1} (\hat{{\alpha }},\hat{{\lambda }},\hat{{\beta }})}\hbox { and }\hat{{\beta }}\pm Z_{\gamma /2} \,\sqrt{\hbox {I}_{33}^{-1} (\hat{{\alpha }},\hat{{\lambda }},\hat{{\beta }})}.\nonumber \\ \end{aligned}$$

(19)

where Z$_{\gamma /2 }$ is the upper ($\gamma$/2)th percentile of a standard normal distribution.

3.2 Estimation based on Type-I PHC

The SSPALT Type-I PHC scheme can be applied as follows. Assume that $n$ identical units are set on a life test. Each of the $n$ units is first operated under use condition. This use condition level is switched to an accelerated condition at time $\tau$at which $r$ living units of the residual units are randomly withdrawn and the test is continued. It is observed that $\tau$ and $r$ are predetermined. If the $m^{th}$ failure ($m > n$) happens at a time $y_{m:n}$ before a preset $\eta \,>\,\tau$, the experiment ends at the time point $y_{m:n}$. But if $y_{m:n} \,>\,\eta$, then all the surviving units are removed and the experiment finishes at the time $\eta$. The end time of the Type-I PHC scheme is at most $\eta$. Suppose that $n_{u}$ be the number of units that fail before the time $\tau ,\,n_{a}$ be the number of units that fail before the time $\eta$ at accelerated condition and $n_{f}$ be the number of units that fail before the experiment ends. Thus, we have

$$\begin{aligned} n_{f} =\left\{ {\begin{array}{ll} n_{u} + n_{a} =m,\,&{} \hbox { if }\,\tau <y_{m:n} \le \eta \\ n_{u} + n_{a} <m, &{} \hbox { if } y_{m:n} >\eta \\ \end{array}} \right. \end{aligned}$$

(20)

Based on Type-I PHC scheme, we can notice the following kinds of observations:

Set 1::

$\mathop y\nolimits _{1:n} \!<\!\mathop {\cdots <y}\nolimits _{\mathop n\nolimits _u :n} \le \tau <\mathop y\nolimits _{\mathop n\nolimits _u +1:n} <\mathop {\cdots <y}\nolimits _{m:n} \le \eta , \qquad \,\,\,\, \hbox { if }\quad \tau \!<\!\mathop y\nolimits _{m:n} \le \eta$

Set 2::

$\mathop y\nolimits _{1:n} <\mathop {\cdots <y}\nolimits _{\mathop n\nolimits _u :n} \le \tau <\mathop y\nolimits _{\mathop n\nolimits _u +1:n} <\mathop {\cdots <y}\nolimits _{\mathop n\nolimits _u +\mathop n\nolimits _a :n} \le \eta , \, \mathop { \hbox {if }\quad y}\nolimits _{m:n} >\eta$

3.2.1 Point estimation

Using the observed progressively Type-I hybrid censored data from WD, we introduce the likelihood function under SSPALT for the two sets of data specified above as follows to obtain the ML estimates of the unknown parameters.

The likelihood function of the data set 1 is presented by

$$\begin{aligned} L(\alpha ,\lambda ,\,\beta )\propto \prod _{i=1}^{\mathop n\nolimits _u } {f_1 (\mathop y\nolimits _i ).[{\varvec{S}}_1 (\tau )]^{r}} . \prod _{i=\mathop n\nolimits _u +1}^m {f_2 (\mathop y\nolimits _i ).[{\varvec{S}}_2 (\mathop y\nolimits _{m:n} )]^{n-m-r}} \, \end{aligned}$$

(21)

where

$$\begin{aligned} \mathop s\nolimits _1 (y)=\exp \{-\,(y/\lambda )^{\alpha }\} \end{aligned}$$

and

$$\begin{aligned} \mathop s\nolimits _2 (y)=\exp \{-\,[(\tau +\beta (y-\tau ))/\lambda ]^{\alpha }\}, \end{aligned}$$

For the data set 2 the likelihood function is given by

$$\begin{aligned} L(\alpha ,\lambda ,\,\beta )\propto \prod _{i=1}^{\mathop n\nolimits _u } {f_1 (\mathop y\nolimits _i ).[{\varvec{S}}_1 (\tau )]^{r}} \hbox { . }\prod _{i=\mathop n\nolimits _u +1}^{\mathop n\nolimits _u +\mathop n\nolimits _a } {f_2 (\mathop y\nolimits _i ).[{\varvec{S}}_2 (\eta )]^{n-(\mathop n\nolimits _u +\mathop n\nolimits _a )-r}} \, \end{aligned}$$

(22)

To get the ML estimates of the model parameters, the natural logarithm of the likelihood functions for both data set 1 and data set 2 are, respectively, as follows

$$\begin{aligned} lnL(\alpha ,\lambda ,\,\beta )&= m\,ln\,\alpha \,-\,m\alpha \,ln\,\lambda \,+n_a \,ln\,\beta \nonumber \\&+\,(\alpha -1) \left\{ \sum _{i=1}^{n_u } {ln\,y_\mathrm{i} } +\sum _{i=n_u +1}^m {ln[\tau +\beta (y_\mathrm{i} -\tau )]} \right\} \nonumber \\&-\,\frac{1}{\mathop \lambda \nolimits ^\alpha }\left\{ \sum _{i=1}^{n_u } {y_i^\alpha } +\sum _{i=n_u +1}^m {[\tau +\beta (y_\mathrm{i} -\tau )]} ^{\alpha } \right. \nonumber \\&\left. +\, r\,n_u \,\tau ^{\alpha }+(n-m-r) n_a [\tau +\beta (y_{m:n} -\tau )]^{\alpha }\right\} ,\, \end{aligned}$$

(23)

and

$$\begin{aligned} lnL(\alpha ,\lambda ,\,\beta )&= (n_u +n_a )ln\,\alpha \,-\,(n_u +n_a )\alpha \,ln\,\lambda \\&+\, n_a \,ln\,\beta +(\alpha -1)\left\{ \sum _{i=1}^{n_u } {ln\,y_\mathrm{i} }+\sum _{i=n_u +1}^{n_u +n_a } {ln[\tau +\beta (y_\mathrm{i} -\tau )]} \right\} \\&-\,\frac{1}{\lambda ^{\alpha }}\{\sum _{i=1}^{n_u } {y_i^\alpha } +\sum _{i=n_u +1}^{n_u +n_a } {[\tau +\beta (y_\mathrm{i} -\tau )]} ^{\alpha }+r\,n_u \,\tau ^{\alpha } \\&+\,[n-(n_u +n_a )-r]n_a [\tau +\beta (\eta -\tau )]^{\alpha }\}. \end{aligned}$$

We shall consider only the case of data set 1 to perform the needed statistical inference. Equating the partial derivatives of $ln\,L$ to zero with respect to $\alpha ,\,\lambda$ and $\beta$, the resulting three equations are:

$$\begin{aligned} \frac{\partial lnL}{\partial \alpha }&= \frac{m}{\alpha }-mln\,\lambda +\sum _{i=1}^{n_u } {ln\,y_\mathrm{i} +\sum _{i=n_u +1}^m {ln\,\psi _\mathrm{i} } } -\frac{1}{\lambda ^{\alpha }}\nonumber \\&\times \left\{ \left[ \sum _{i=1}^{n_u } {y_i^\alpha ln\,y_\mathrm{i} +\sum _{i=n_u +1}^m {\psi _i^\alpha } ln\,\psi _\mathrm{i} } \,+r\,n_u \,\tau ^{\alpha }ln\,\tau +(n-m-r)n_a \,\psi _m^\alpha \,ln\psi _m \right] \right. \nonumber \\&\left. -\left\{ \sum _{i=1}^{n_u } {y_i^\alpha } +\sum _{i=n_u +1}^m {\psi _i^\alpha } +r\,n_u \,\tau ^{\alpha }+(n-m-r)n_a \,\psi _m^\alpha \right\} ln\,\lambda \,\right\} =0, \end{aligned}$$

(24)

where $\psi _\mathrm{i} =\tau +\beta (y_\mathrm{i} -\tau )$ and $\psi _m =\tau +\beta (y_{m:n} -\tau )$.

$$\begin{aligned} \frac{\partial lnL}{\partial \lambda }=-\frac{m\alpha }{\lambda }\!+\!\frac{\alpha }{\lambda ^{\alpha +1}}\left\{ \sum _{i=1}^{n_u } {y_i^\alpha } +\sum _{i=n_u +1}^m {\psi _i^\alpha } \!+\!r\,n_u \,\tau ^{\alpha }+(n-m-r)n_a \,\psi _m^\alpha \right\} =0, \end{aligned}$$

(25)

$$\begin{aligned} \frac{\partial lnL}{\partial \beta }&= \frac{n_a }{\beta }+(\alpha -1)\sum _{i=n_u +1}^m {\frac{y_i -\tau }{\psi _i }} -\frac{\alpha }{\lambda ^{\alpha }}\nonumber \\&\times \left\{ \sum _{i=n_u +1}^m {(y_i -\tau )} \,\psi _i^{\alpha -1} +(n-m-r)n_a (y_{m:n} -\tau )\,\psi _m^{\alpha -1} \right\} =0. \end{aligned}$$

(26)

From Eq. (25) we can obtain $\hat{{\lambda }}$ as a function of $\alpha$ and $\beta$ as

$$\begin{aligned} \hat{{\lambda }}=\left\{ {\frac{\sum _{i=1}^{n_u } {y_i^\alpha } +\sum _{i=n_u +1}^m {\psi _i^\alpha } +r\,n_u \,\tau ^{\alpha }+(n-m-r)n_a \,\psi _m^\alpha }{m}} \right\} ^{1/\alpha }. \end{aligned}$$

(27)

Now, the system is reduced to two non-linear likelihood equations in $\alpha$ and $\beta$ . It can be solved iteratively using an iterative method such as Newton-Raphson to obtain the ML estimates of $\alpha$ and $\beta$. Therefore, the ML estimate of $\lambda$ can be easily obtained from Eq. (27).

3.2.2 Interval estimation

In this subsection, the approximate confidence intervals of the parameters are derived based on the asymptotic distribution of the ML estimators of the elements of the vector of unknown parameters $\Omega$ = ($\alpha ,\,\lambda ,\,\beta$).

Now, we obtain the second partial derivatives of $ln\,L$ with respect to $\alpha ,\,\lambda$ and $\beta$ as follows

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \alpha ^{2}}&= -\frac{m}{\alpha ^{2}}-\frac{1}{\lambda ^{\alpha }}\left\{ \left[ \sum _{i=1}^{n_u } {y_i^\alpha (ln\,y_\mathrm{i} )^{2}+\sum _{i=n_u +1}^m {\psi _i^\alpha } (ln\,\psi _\mathrm{i} } )^{\hbox {2}}\,+r\,n_u \,\tau ^{\alpha }(ln\,\tau )^{2}\right. \right. \nonumber \\&\left. \left. +\, (n-m-r)n_a \,\psi _m^\alpha (ln\psi _m )^{2}\right] -\left[ \sum _{i=1}^{n_u } y_i^\alpha ln\,y_\mathrm{i} \right. \right. \nonumber \\&\left. \left. +\sum _{i=n_u +1}^m {\psi _i^\alpha } ln\,\psi _\mathrm{i} \,+r\,n_u \,\tau ^{\alpha }ln\,\tau +(n-m-r)n_a \,\psi _m^\alpha \,ln\psi _m \right] \right\} \,ln\,\lambda \,\, \end{aligned}$$

(28)

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \alpha \partial \lambda }&= -\frac{m}{\lambda }+\frac{1}{\lambda ^{\alpha +1}}\{1+\alpha \,[[\sum _{i=1}^{n_u } {y_i^\alpha ln\,y_\mathrm{i} +\sum _{i=n_u +1}^m {\psi _i^\alpha } ln\,\psi _\mathrm{i} } \,\nonumber \\&+\, r\,n_u \,\tau ^{\alpha }ln\,\tau +(n-m-r)n_a \,\psi _m^\alpha \,ln\psi _m ] \nonumber \\&-\left\{ \sum _{i=1}^{n_u } {y_i^\alpha } +\sum _{i=n_u +1}^m {\psi _i^\alpha } +r\,n_u \,\tau ^{\alpha }+(n-m-r)n_a \,\psi _m^\alpha \}\,ln\,\lambda \,]\right\} , \end{aligned}$$

(29)

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \alpha \partial \beta }&= \sum _{i=n_u +1}^m {\frac{y_\mathrm{i} -\tau }{\psi _i }} -\frac{1}{\lambda ^{\alpha }}\nonumber \\&\times \left\{ \left[ \left\{ \sum _{i=n_u +1}^m {(y_i -\tau )} \,\psi _i^{\alpha -1} +(n-m-r)n_a (y_{m:n} -\tau )\,\psi _m^{\alpha -1} \right\} \right. \right. \nonumber \\&\left. \left. +\,\alpha \,\left\{ \sum _{i=n_u +1}^m {(y_i -\tau )} \,\psi _i^{\alpha -1} \,ln\psi _i +(n-m-r)n_a (y_{m:n} -\tau )\,\psi _m^{\alpha -1} \,ln\psi _m \right\} \right] \right. \nonumber \\&\left. -\, \alpha \,\left\{ \sum _{i=n_u +1}^m {(y_i -\tau )} \,\psi _i^{\alpha -1} +(n-m-r)n_a (y_{m:n} -\tau )\,\psi _m^{\alpha -1} \right\} \,ln\lambda \right\} , \end{aligned}$$

(30)

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \lambda ^{2}}=\frac{m\alpha }{\lambda ^{2}}-\frac{\alpha \,(\alpha +1)}{\lambda ^{\alpha +2}}\left\{ \sum _{i=1}^{n_u } {y_i^\alpha } +\sum _{i=n_u +1}^m {\psi _i^\alpha } +r\,n_u \,\tau ^{\alpha }+(n-m-r)n_a \,\psi _m^\alpha \right\} \,, \end{aligned}$$

(31)

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \lambda \partial \beta }=\frac{\alpha ^{2}(\alpha -1)}{\lambda ^{\alpha +1}}\left\{ \sum _{i=n_u +1}^m {(y_i -\tau )^{2}} \,\psi _i^{\alpha -2} +(n-m-r)n_a (y_{m:n} -\tau )^{2}\psi _m^{\alpha -2} \right\} \!, \end{aligned}$$

(32)

$$\begin{aligned} \frac{\partial ^{2}lnL}{\partial \beta ^{2}}&= -\frac{n_a }{\beta ^{2}}-(\alpha -1)\sum _{i=n_u +1}^m {\frac{(y_\mathrm{i} -\tau )^{2}}{\psi _i^2 }} \,-\frac{\alpha (\alpha -1)}{\lambda ^{\alpha }}\,\nonumber \\&\times \left\{ \sum _{i=n_u +1}^m {(y_i -\tau )^{2}} \,\psi _i^{\alpha -2} +(n-m-r)n_a (y_{m:n} -\tau )^{2}\psi _m^{\alpha -2} \right\} .\qquad \end{aligned}$$

(33)

Thus, the approximate 100(1 - $\gamma )$  % two sided confidence intervals for $\alpha ,\,\lambda$ and $\beta$ are, respectively, given by

$$\begin{aligned} \hat{{\alpha }}\pm Z_{\gamma /2} \,\sqrt{\hbox {I}_{11}^{-1} (\hat{{\alpha }},\hat{{\lambda }},\hat{{\beta }})}\hbox {, }\hat{{\lambda }}\pm Z_{\gamma /2} \,\sqrt{\hbox {I}_{\hbox {22}}^{-1} (\hat{{\alpha }},\hat{{\lambda }},\hat{{\beta }})}\hbox { and }\hat{{\beta }}\pm Z_{\gamma /2} \,\sqrt{\hbox {I}_{33}^{-1} (\hat{{\alpha }},\hat{{\lambda }},\hat{{\beta }})}. \end{aligned}$$

(34)

where Z$_{\gamma /2 }$is the upper ($\gamma$/2)th percentile of a standard normal distribution.

4 Optimum test plans

The main purpose of this section is to choose the optimal stress-change time $\tau ^{*}$ for both Type-I PHC and Type-I APHC under different progressive censoring schemes. In step-stress setting, the experimenter is often interested in estimating the mean life at use condition with maximum precision. The mean lifetime is an important characteristic in reliability analysis. In practice, the optimum test plans are important for improving precision in parameter estimation and thus improving the quality of the statistical inference. So, these optimum test plans are more useful for estimating the life distribution at design stress. One selection criterion, the D-optimality criterion, is proposed which enables the experimenter to determine the optimal value of $\tau$.

The D-optimality criterion is based on the determinant of Fisher’s information matrix F. It has been extensively used in the context of planning life test. If one is more interested in estimation with high precision, a more reasonable criterion should be D-optimality, which takes into account the overall parameter space. It can be constructed in terms of the generalized asymptotic variance (GAV) of the MLEs of the model parameters. This GAV is proportional to reciprocal of the determinant of Fisher-information matrix; see Bai et al. (1993b). So that maximizing this determinant is equivalent to minimizing GAV. The criterion function is then defined by

$$\begin{aligned} GAV(\hat{{\alpha }},\hat{{\lambda }},\hat{{\beta }})=\frac{1}{\left| \varvec{F }\right| } \end{aligned}$$

(35)

Hence, the optimal stress-change time $\tau \,^{*}$ is chosen so that GAV is minimized.

It is noted that the D-optimality criterion is based on the information matrix F $.$ This criterion has been extensively used in the design selection process for designed experiments. The two approaches of schemes; Type-I PHC and Type-I APHC are compared with each other under different progressive censoring schemes in terms of the optimal GAV of the MLEs of the model parameters.

5 Simulation studies

In this section simulation studies are conducted to discuss the performance of the ML estimators in terms of their biases and mean square errors (MSEs) for different choices of parameter values and different choices of $n,\,m, \tau$and $\eta$ values based on two different types of progressively hybrid censoring schemes which are Type-I PHC and Type-I APHC schemes. Also, 95  % asymptotic confidence intervals based on the asymptotic distribution of the ML estimators are constructed and their lengths are computed and presented with associated coverage probabilities.

Three different progressive censoring schemes are considered:

Scheme 1::

$R_\mathrm{1} = \cdots =R_{m-1} =0\,and\,R_m =n-m;$

Scheme 2::

$R_\mathrm{1} =n-m\,and\,R_\mathrm{2} =\cdots =R_m =0;$ and

Scheme 3::

$R_\mathrm{1} =\cdots =R_{m-1} =1\,and\,R_m =n-2m+1.$

For each setting under both Type-I PHC and Type-I APHC schemes, the biases and MSEs based on 10,000 simulations are computed and reported in Tables 1, 2, 3, 4, 5, 6. In addition, 95  % asymptotic confidence intervals lengths are computed and presented with associated coverage probabilities in Tables 7, 8, 9, 10, 11, 12.

The simulation study is carried out according to the following algorithm

• (1) Specify the values of $n,\,m,\tau$ and $\eta$.

• (2) Specify the values of the parameters $\alpha ,\,\lambda$ and $\beta$.

• (3) Generate a random sample of size $n$ from the random variable $Y$ given by Eq. (6) and sort it. The Weibull random variable can be easily generated. For example, if $U$ represents a uniform random variable from [0, 1], then $Y=-\lambda \,[ln\,(1-U)]^{1/\alpha }$has Weibull distribution with pdf given by Eq. (1) $if\hbox { y}\le \tau$. But $if\hbox { y}>\tau$ then $Y=\tau +\{-\lambda \,[ln\,(1-U)]^{1/\alpha }-\tau \}/\beta$has Weibull distribution with pdf given by Eq. (5).

• (4) Use the model given by Eq. (4) to generate progressively hybrid censored data for given $n,\,m,\tau ,\,\eta$ ($\eta \,>\tau ),\,\alpha ,\,\lambda$ and $\beta$.

• (5) Use the progressively hybrid censored data to compute the MLEs of the model parameters. Newton-Raphson method is applied for solving the nonlinear system to obtain the MLEs of the parameter$s$.

• (6) Replicate the steps 3–5 10,000 times.

• (7) Compute the average values of biases and MSEs associated with the MLEs of the parameters.

• (8) Compute the average values of intervals lengths (ILs) as well as the associated coverage probabilities with each parameter using confidence level 1$-\gamma$ = 0.95.

• (9) Steps 1–8 are done with different values of $n,\,m,\tau ,\,\eta$ ($\eta \,>\,\tau ),\,\alpha ,\,\lambda$ and $\beta$.

Conducting the above algorithm under both Type-I PHC and Type-I APHC schemes, the average values of biases and MSEs are obtained using 10,000 replications to avoid randomness. The results reported in Tables 1, 2, 3, 4, 5, 6 are based on different values of $n,\,m,\tau ,\,\eta$ ($\eta \,>\,\tau ),\,\alpha ,\,\lambda$ and $\beta$ to investigate the performance of the MLEs of the model parameters. Also, the average values of ILs as well as the corresponding coverage probabilities with each parameter using confidence level 1 $-\gamma$ = 0.95 are computed and the results are presented in Tables 7, 8, 9, 10, 11, 12.

Table 1 Average values of the biases and MSEs of the MLEs based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda ,\,\beta$,$\tau$ and $\eta$ set at 0.4, 0.7, 1.2, 2 and 5, respectively

Table 2 Average values of the biases and MSEs of the MLEs based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda$ , $\beta$,$\tau$ and $\eta$ set at 0.4, 0.7, 1.2, 2 and 10, respectively

Table 3 Average values of the biases and MSEs of the MLEs based on both Type-I PHC and Type-I APHC when  $\alpha ,\,\lambda$ , $\beta$, $\tau$ and $\eta$ set at 0.4, 0.7, 1.2, 7 and 10, respectively

Table 4 Average values of the biases and MSEs of the MLEs based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda$, $\beta$, $\tau$ and $\eta$ set at 1.4, 0.7, 1.2, 2 and 10, respectively

Table 5 Average values of the biases and MSEs of the MLEs based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda$, $\beta$, $\tau$ and $\eta$ set at 1.4, 0.7, 1.2, 2 and 5, respectively

Table 6 Average values of the biases and MSEs of the MLEs based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda$, $\beta$, $\tau$ and $\eta$ set at 1.4, 0.7, 1.2, 4 and 5, respectively

Table 7 Average confidence intervals lengths (IL) and their coverage probabilities (CP) based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda$, $\beta$, $\tau$ and $\eta$ set at 0.4, 0.7, 1.2, 2 and 5, respectively

Table 8 Average confidence intervals lengths (IL) and their coverage probabilities (CP) based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda$, $\beta$, $\tau$ and $\eta$ set at 0.4, 0.7, 1.2, 2 and 10, respectively

Table 9 Average confidence intervals lengths (IL) and their coverage probabilities (CP) based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda$, $\beta$, $\tau$ and $\eta$ set at 0.4, 0.7, 1.2, 7 and 10, respectively

Table 10 Average confidence intervals lengths (IL) and their coverage probabilities (CP) based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda$, $\beta$, $\tau$ and $\eta$ set at 1.4, 0.7, 1.2, 2 and 10, respectively

Table 11 Average confidence intervals lengths (IL) and their coverage probabilities (CP) based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda$, $\beta$, $\tau$ and $\eta$ set at 1.4, 0.7, 1.2, 2 and 5, respectively

Table 12 Average confidence intervals lengths (IL) and their coverage probabilities (CP) based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda$, $\beta$, $\tau$ and $\eta$ set at 1.4, 0.7, 1.2, 4 and 5, respectively

t is observed from Tables 1, 2, 3, 4, 5, 6 that in all cases the MLEs of the model parameters based on Type-I APHC give smaller MSEs compared to those based on Type-I PHC. In all cases the MSEs of the MLEs of the three parameters based on Type-I APHC decrease as the effective sample size ($m)$ increases. This is also true for Type-I PHC except for some cases for Scheme 2 because of the heavy censoring at the early stages of the experiment. In addition, the biases of the MLEs are all smaller under Type-I APHC. Generally, the precision of estimation under the Type-I APHC scheme is better because we have a larger number of failures to be noticed. Thus, when the time of experiment is not the major concern, the Type-I APHC scheme will be a desirable choice in order to improve the quality of the statistical inference about the model parameters. Moreover, it is shown from the results presented in Tables 7, 8, 9, 10, 11, 12 that the average ILs obtained under Type-I APHC scheme are shorter than those obtained using Type-I PHC scheme. Also, we observed that the computed coverage probabilities of the confidence intervals for each parameter under Type-I APHC scheme are very close to the nominal level. On the other hand, it was found that these coverage probabilities using Type-I PHC scheme are not satisfactory. They are considerably lower than the nominal level in general.

In addition, from Tables 1, 2, 3, 4, 5, 6 the following observations based on Type-I APHC scheme can be made.

• (1) For fixed $n$,$\tau$ and $\eta$, the MSEs decrease as $m$ increases.

• (2) For fixed $\tau$ and $\eta$, the MSEs considerably decrease as $n$ and $m$ increase at the same time.

• (3) For fixed $n,\,m$ and $\eta$, the MSEs decrease as $\tau$ decreases. This is also true in the case of Type-I PHC scheme.

• (4) For fixed $n,\,m$ and $\tau$, the MSEs decrease as $\eta$ increases. This is also true in the case of Type-I PHC scheme. It is noted that, under Type-I PHC scheme, as $\eta$ gets longer the MSEs decrease unless $\tau$ is large.

The same pattern is observed for the biases as shown form the results.

For the two approaches of censoring schemes; Type-I PHC and Type-I APHC, optimum test plans have been developed, numerically. The results of optimal stress change-time $\tau$* under different progressive censoring schemes, and the optimal GAV of the MLEs of the model parameters are given in the Tables 13, 14, 15, 16. The optimal GAV is numerically obtained with $\tau$* in place of $\tau$. Under the two approaches of schemes; Type-I PHC and Type-I APHC, the optimal GAV of the MLEs of the model parameters decreases as the sample size n increases. As indicated from the results, the optimal GAV of the MLEs of the model parameters under Type-I APHC scheme is much smaller than that obtained by Type-I PHC scheme. That is, the performance of the MLEs of the model parameters under Type-I APHC scheme is much better than that of the Type-I PHC scheme.

Table 13 Average values of optimal $\tau$ and the optimal GAV based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda ,\,\beta$, and $\eta$ set at 1.4, 0.7, 1.2, and 5, respectively

Table 14 Average values of optimal $\tau$ and the optimal GAV based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda ,\,\beta$, and $\eta$ set at 1.4, 0.7, 1.2, and 10, respectively

Table 15 Average values of optimal $\tau$ and the optimal GAV based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda ,\,\beta$, and $\eta$ set at 0.4, 0.7, 1.2, and 5, respectively

Table 16 Average values of optimal $\tau$ and the optimal GAV based on both Type-I PHC and Type-I APHC when $\alpha ,\,\lambda ,\,\beta$, and $\eta$ set at 0.4, 0.7, 1.2, and 10, respectively

6 Concluding remarks and further studies

In this paper, the maximum likelihood estimations of Weibull distribution parameters and the acceleration factor have been discussed using data obtained based on both non-adaptive and adaptive Type-I progressively hybrid censoring schemes assuming three different progressive censoring schemes. The biases and mean squared errors of the maximum likelihood estimators of the model parameters have been computed to evaluate their performances in the presence of censoring schemes developed in this paper through a Monte Carlo simulation study. Moreover, the confidence intervals lengths and their associated coverage probabilities have been obtained for both adaptive and non-adaptive Type-I progressively hybrid censoring schemes. The results obtained under the adaptive Type-I progressively hybrid censoring scheme have been compared with those produced under the non-adaptive Type-I progressively hybrid censoring scheme using the three different progressive censoring schemes. It has been observed that the results obtained under Type-I APHC scheme are better than those using Type-I PHC scheme. In most cases the results of Type-I PHC scheme are not satisfactory. Since there are more expected failures using Type-I APHC scheme than those obtained by Type-I PHC scheme, so, Type-I APHC scheme is highly recommended to use for improving the quality of the statistical inference.

Moreover, statistically optimum step-stress partially accelerated life test plans have been developed. The optimality criterion adopted is the minimization of the GAV of the MLEs of the model parameters. That is, the optimal stress-change time $\tau$* is obtained such that the GAV of the MLEs of the model parameters is minimized. Thus, the optimal design of the life tests can be considered as a technique to improve the quality of the statistical inference. The design of an optimal life test already enables us to obtain estimations of high degree of precision. This issue coincides with the note of Wu and Huang (2010). They said that “In order to obtain a precise estimate of mean life, one needs to design an optimal life test”. As a future work, the Bayesian inference in the case of SSPALT under the same censoring schemes proposed in this paper will be considered.