Statistical Analysis of Chen Distribution Under Improved Adaptive Type-II Progressive Censoring

In reliability research and lifetime test experiments, most of the experiments take a long time to terminate, but considering the cost and time of the experiment, the failure time of all individuals cannot be observed, only the exact time of failure of a few experimental individuals can be observed, and the factors that may lead to individual failures need to be considered, the following research on censored data is reasonable and necessary. The classical censoring schemes includes type-I and type-II censoring, which are the most basic censoring schemes. They can be described as: suppose there are $n$ independent identical units placed in a lifetime experiment, type-I censoring requires that the experiment be terminated at a prefixed time point $T$, while type-II censoring requires that he experiment is terminated when the predetermined number of failed individuals $m<n$ is observed. The hybrid censoring scheme is a mixture of type-I censoring and type-II censoring. Epstein (1954) proposed a type-I hybrid censoring scheme for the first time. Chen & Bhattacharyya (1987) derived the exact distribution of the maximum likelihood estimator of the mean of an exponential distribution and an exact lower confidence bound for the mean based on a hybrid censored sample. Childs et al. (2003) propose a hybrid censoring scheme which guarantees at least a fixed number of failures in a life testing experiment and the exacted distribution of maximum likelihood estimate with exponential distribution as well as exact lower confidence bound for the mean ia studied. Kundu (2007) presents the statistical inference on Weibull parameters when the data are hybrid censored. The disadvantage of the above three censoring schemes is that these cells are not removed at all time points except at the termination of the experiment. To solve this problem, a type-II progressive censoring scheme is used, which is a scheme that combines type-II progressive censoring and type-I censoring. Refer to the type-II progressive censoring scheme proposed by Balakrishnan & Aggarwala (2000). The type-II progressive censoring scheme can be described as: consider $n$ identical units placed in a lifetime test experiment, let $X_{1},X_{2},\cdots,X_{n}$ be the corresponding failure time and $X_{1}<X_{2}<\cdots<X_{n}$, let $m$ is the predetermined number of failures, when the first failure has occurred, record the failure time as $X_{1:m:n}$, at this time there are $R_{1}$ units from the remaining $n-1$ units randomly removed. Similarly, when the second failure time $X_{2:m:n}$ is observed, $R_{2}$ units are randomly removed from the remaining $n-2-R_{1}$ units, and so on, at the $m$-$th$ failure time $X_{m:m:n}$, all remaining $n-m-R_{1}-R_{2}-\cdots-R_{m-1}$ units are removed. In the progressive censoring scheme, $R_{1},R_{2},\cdots,R_{m}$ are predetermined prior to the study and are not changed during the experiment. Childs et al. (2008) proposed a type-II progressive hybrid censoring scheme. If $X_{m:m:n}>T$, the experiment is terminated at $X_{m:m:n}$, otherwise, the experiment terminated at $T$. Dey & Dey (2014) takes into account the estimation for the unknown parameter of the Rayleigh distribution under type-II progressive censoring scheme. Tomer & Panwar (2015) considered point and interval estimation for the Maxwell distribution of type-I progressive hybrid censored data.$\ddot{o}$.G$\ddot{u}$r$\ddot{u}$nl$\ddot{u}$ Alma & Belaghi (2016) discussed the analysis of progressive type-II progressive hybrid censored data when the lifetime distribution of the individual item is the normal and extreme value distributions. Cramer et al. (2016) considered the exponential distribution under this censoring scheme. Zhang & Gui (2019) studied the inference of reliability of generalized Rayleigh distribution based on the progressively type-II censored data. Zhang & Gui (2020) develop a goodness of fit test process for Pareto distribution based on progressive type-II censoring scheme.

However, a problem with this type of censoring scheme is that the number of effective units is random. If the effective sample obtained is very small or close to $0$, it will make the statistical inference process infeasible or the experimental efficiency is very low. Therefore, to avoid this problem, Ng et al. (2009) proposed for the first time a new scheme called adaptive type-II progressive censoring (AT-II PCS) which is a mixture of type-I censoring and type-II progressive censoring. Under this censoring scheme, the number of failed units $m$ to be observed and the progressive censoring scheme $R_{1},R_{2},\cdots,R_{m}$ are given in advance, but in the process of the experiment, some $R_{i}$ values may change according to the situation, so this censoring scheme can try to strike a balance between the total experiment time, the number of failed units, and the validity of the statistical analysis. The detailed description is as follows: if $X_{m:m:n}<T$, the experiment ends at point $X_{m:m:n}$, and the remaining $R_{m}$ units are all removed. Otherwise, if before time point $T$, the number of failed individuals is $j(0<j<m)$ and $X_{j:m:n}<T<X_{j+1:m:n}$. Let $R_{j+1}=\cdots=R_{m-1}=0$, The test continued until the failure of the m-th unit was observed and $R_{m}=n-m-\sum_{i=1}^{j}R_{i}\quad(i=1,\cdots,j)$, the experiment ends at point $X_{m:m:n}$. Here $T$, $m$, $R_{1}$, $R_{2}$, $\cdots$, $R_{m}$ $(R_{1}+R_{2}+\cdots+R_{m}+m=n)$ are all preset before the experiment. The value of time $T$ plays a very important role in determining the value of $R_{i}$. When $T\to\infty$, it is obvious that time is not the main concern of the experimenter. So the censoring scheme will degenerate into a general type-II progressive censoring, if $T\to 0$, the censoring scheme degenerates into the traditional type-II censoring scheme. Some general statistical properties of an adaptive type-II progressive censoring scheme are investigated by Ye et al. (2014). Nassar & Abo-Kasem (2017) describes the estimation of the inverse Weibull parameters under the AT-II PCS. El-Din et al. (2017) studied the estimation of generalized exponential distribution based on AT-II PCS. Panahi & Moradi (2020) discussed the problem of estimating parameters of the inverted exponentiated Rayleigh distribution based on AT-II PCS. Mohan & Chacko (2021) studied the estimation of parameters for a two parameter Kumaraswamy-exponential distribution based on AT-II PCS. An adaptive type-II progressive hybrid censored sampling with random removals is considered by Elshahhat & Nassar (2022). Lv et al. (2022) studied the statistical inference of Gompertz distribution based on AT-II PCS. In AT-II PCS, the time of the experiment is not an important consideration. We focus on being able to observe enough $m$ failure units. If the test unit is a high-reliability product, the duration of the experiment will be very long, while AT-II PCS does not guarantee a satisfactory, appropriate experimental total test time. But in many practical situations, the time of the experiment is inevitably an important factor to consider.

To remedy this deficiency, Yan et al. (2021) proposed an improved censoring scheme called improved adaptive Type-II progressive censoring scheme(IAT-II PCS). One of its advantages is that it is guaranteed to end within the time specified in the experiment. A detailed description of IAT-II PCS is as follows: suppose there are $n$ units in the experiment. Assuming that they are independent and identically distributed, the failure unit $m$ to be observed and the censoring scheme $R=(R_{1},R_{2},...,R_{m})$ are per-specified before the experiment. In addition, two time thresholds $T_{1}$ and $T_{2}$ $(T_{1},T_{2}>0)$ are per-specified with $T_{1}<{T_{2}}$, where time $T_{1}$ is a warning about the test time, and $T_{2}$ is the maximum time allowed for the experiment, which means that when the experiment is carried out to time $T_{1}$, the experiment needs to be accelerated. In order to ensure that as many failed individuals as possible can be observed when the experiment is terminated at $T_{2}$, no experimental units will be censored when the experiment is accelerated. Of course, under this censoring scheme, when the experiment is carried out for enough time, the experiment is allowed to observe less than $m$ failure individuals. The experiment has the following three cases.

The censoring scheme and the end position of the experiment in these three cases are as follows

In real life, there are many hazard rate functions used to model real life data, the most popular failure rate functions are constant, increasing or decreasing failure rate functions. For example Weibull, gamma and exponentiated exponential, among others. But there are other different forms of the hazard function, such as unimodal, bathtub-shaped, increasing-decreasing-increasing, etc. These distribution models described above also do not fit reasonably non-monotonic hazard rates, especially bathtub shape hazard rates that are common in reliability and other fields. An example of a bathtub-shaped failure rate is that the failure rate of newly installed electronic equipment is relatively high, failures often occur, production performance drops to a minimum level for a short period of time, and then the equipment will stabilize after a few days or months of use, remaining at this level for several years, and then the equipment underwent various forms of overhaul or technical transformation to restore production performance. Yan et al. (2021) used the Burr-XII distribution in the IAT-II PC proposed in 2021, and its hazard rate function could not fit the data of the bathtub curve. In recent years, a number of probability distributions have been introduced to model the risk rate of having a bathtub shape, which can be found in the literature. In this paper, we mainly focus on a two-parameter bathtub distribution proposed by Chen (2000). the probability density function $(PDF)$ and cumulative distribution function $(CDF)$ of $X$ are given by

The reliability and the hazard rate functions hazard function of $X$ are as follows

where $\alpha$ and $\beta$ are the unknown scale and shape parameters, respectively, and when $\beta<1$, the hazard rate function is bathtub-shape.

The rest of this paper is organized as follows: In Section $2$, maximum likelihood estimate (MLE) on unknown parameters is derived and the approximate confidence intervals is proposed. In Section $3$, Bayesian estimation is performed with gamma prior, and Importance sampling and Metropolis-Hastings algorithm are used. Next, based on the Monte Carlo method, a large number of simulations are carried out for parameter estimations and interval estimations in Section $4$. A real data set has been used to evaluated the estimation methods in Section $2$ and $3$. Finally, in Section $6$, we put forward some concluding conclusions.

Let $X=X_{1:m:n}^{R},X_{2:m:n}^{R},\dots,X_{k_{1}:m:n}^{R},\dots,X_{k_{2}:m:n}^{R},\dots,X_{m:m:n}^{R}$ is an IAT-II PC sample from a lifetime test of size $m$ from a sample of size $n$ , where lifetimes have Chen distribution with pdf,cdf as given by (7). With predetermined number of removal of units from experiment, say $R=(R_{1},R_{2},\dots,R_{k_{1}},\dots,R_{k_{2}},\dots,R_{m})$. The parameter $x_{i:m:n}^{R}$ (simplified as $x_{i}$ in later equation, $i=1,2,\dots,m$) is used to represent the observed values of IAT-II PC sample. on this basis, the corresponding likelihood function is given by

where $D_{2}$, $D_{1}$, $B$, $C$ are shown in Table (1)

Then, from (7), (11), The likelihood function of $\alpha$ and $\beta$ is given by

Then, we propose several statistical inference methods.

This section is devoted to obtain the Bayes estimators of the parameters $\alpha$ and $\beta$ of Chen distribution based on improved adaptive asymptotic type-II censored data. The Bayesian estimates are obtained using symmetric as well as asymmetric loss function such as squared error loss function(SEL), LINEX loss function(LL) and entropy loss(EL) function. We assume the parameters $\alpha$ and $\beta$ independent and have gamma prior distributions with the following prior distribution. $G(a,b)$ and $G(c,d)$ distribution with PDFs respectively as

Therefore, the joint prior distribution of $\alpha$ and $\beta$ is given by

Subsequently, the joint posterior distribution of $\alpha$ and $\beta$ becomes

Given $\alpha$, $\beta$, directly calculate the joint posterior distribution $\pi(\alpha,\beta\mid X)$ as follow

Under the square error loss function (SEL), the Bayes estimate for any parameter $\mu$ is given by

For LINEX loss function(LL), the Bayes estimate for any parameter $u$ is given by

For entropy loss function(EL), the Bayesian estimates obtained by minimizing the risk function are

Where $g$ and $q$ are known constants, $\eta$ represents any one of the unknown parameters. Obviously, (25) (26) (27) cannot get an explicit solution. Therefore, the Markov chain Carlo method(MCMC) is used to generate samples from the posterior density function and in turn to compute the Bayesian estimation of the unknown parameters. The following two technique are introduced to calculate bayes estimation.

In this section, the simulation study is carried out Monte Carlo simulation to evaluate the performance of the proposed methods. The Biases and mean square errors(MSE) of the MLE and Bayesian estimates for $\alpha$ and $\beta$ are evaluated under the IAT-II PCS and and in $R$ program. The Bayesian estimates are computed by using Importance sampling and M-H technique. Under different combination of $(n,m)$ and different censoring schemes $(T_{1},T_{2},R_{1},R_{2},\cdots,R_{m})$. the point and interval estimation results are assessed on the basis of mean Bias and MSE respectively.

All the estimates are to compute arbitrary the unknown parameter values $\alpha=0.2$ and $\beta=0.5)$. Accordingly hyperparameters in gamma prior are assigned as $a=2$, $b=2,$ and $c=2$, $d=2$. Here we consider four different censoring schemes, namely:

In our study, two expected time on test $(T_{1},T_{2})$ are (0.4,4) and (1,7). We consider three different values of $(n,m)$, namely (15,5), (20,10) and (30,15). For evaluating Bayes estimators under LINEX loss function, we take $g=1$ and for entropy loss function, we take $q=1$. Based on the simulation study we have the following conclusion.

From the simulate results in Table 2 and 3, one can observe following conclusions for MLE. From these Tables the following conclusion are made:

1. The estimation of MLE is evaluated by Bias and MSE, among which MSE is better than Bias.

2. In most cases, for different censoring schemes, Scheme 4 is uniform censoring scheme and others are non-uniform schemes. It can be seen from the table that the estimation effect of uniform censoring scheme is better than that of other non-uniform censoring schemes.

3. Compared with Table 2 and Table 3, in most cases, when the time threshold becomes larger, the Bias and MSE of related MLE becomes larger, the estimation effect of $(T_{1},T_{2})=(1,7)$ is poor. Therefore, it is essential to set a reasonable time threshold in the experiment.

Table 4 and 5 present the coverage probability and average length of two-side $95.5\%$ confidence intervals of parameters constructed by maximum likelihood method (using fisher observation information matrix). From these Tables the following conclusion are made:

1. The truth value of $\alpha=0.2$ and $\beta=0.5$ is in the middle of the each interval and can be well covered.

2. The asymptotic confidence interval of $\beta$ obtained by fisher observation information matrix, in most case, the average length of the interval is getting longer when the time threshold increase. Meanwhile, different censoring schemes $R$ had no significant effect on the $\alpha$ and $\beta$ results of interval estimation.

3. The average length of the interval about $\alpha$ is significantly different under the two time thresholds $(0.4,4)$ and $(1,7)$.

4. The asymptotic confidence interval of $\alpha$, in most case, are more ideal than that of $\beta$.