Bayesian Analysis in Partially Accelerated Life Tests for Weighted Lomax Distribution

: Accelerated life testing has been widely used in product life testing experiments because it can quickly provide information on the lifetime distributions by testing products or materials at higher than basic conditional levels of stress, such as pressure, temperature, vibration, voltage, or load to induce early failures. In this paper, a step stress partially accelerated life test (SS-PALT) is regarded under the progressive type-II censored data with random removals. The removals from the test are considered to have the binomial distribution. The life times of the testing items are assumed to follow length-biased weighted Lomax distribution. The maximum likelihood method is used for estimating the model parameters of length-biased weighted Lomax. The asymptotic confidence interval estimates of the model parameters are evalu-ated using the Fisher information matrix. The Bayesian estimators cannot be obtained in the explicit form, so the Markov chain Monte Carlo method is employed to address this problem, which ensures both obtaining the Bayesian observed failures, and censoring schemes, in most cases. Further performance comparison is conducted by the experiments with real data.

In many cases, when the life data are analyzed, all units in the sample may not fail. This type of data is called censored or incomplete data. The most common censoring schemes are the type-I censored scheme (or time censored scheme) and type-II censored scheme (or failure-censored scheme). These two censoring schemes do not allow for units to be removed from the experiments while they are still alive. In [2], a more general censoring scheme was proposed, which is known as progressive censoring; this scheme allows units to be removed from the test. Progressive censoring is useful in a life-testing experiment because its ability to remove live units from the experiment saves both time and money. The PALTs have been widely studied for step-stress schemes of type-I, type-II, and progressive type-II censoring (PTIIC) schemes [3][4][5][6][7][8][9][10][11][12].
In this paper, a PTIIC with random removal is proposed to provide more economical and less time-consuming testing. The PTIIC scheme is designed as follows. Suppose n units are placed on a life test, and an experimenter decides beforehand m, which represents the number of units to be failed. Then, at the time of the first failure, which is denoted as t(1), r 1 of the remaining (n − 1) surviving units are randomly removed from the experiment. The test continues until the mth failure occurs, and at that moment, all the remaining surviving units r m = n − m − r 1 − r 2 . . . − r m − 1 are removed from the experiment. Many authors have studied the PTIIC application using different lifetime distributions [13][14][15].
On another statistical level, the Lomax distribution is an important heavy-tail probability distribution for lifetime analysis, and it has been often used in many different fields, including business, economics, and actuarial modeling [16][17][18][19][20][21]. The cumulative distribution function (CDF) and the probability density function (PDF) of the Lomax distribution with the shape parameter θ and scale parameter λ are, respectively, given by: g (x; λ, θ) , x, λ, θ > 0.
The weighted distributions arise in the context of unequal probability sampling. The weighted distributions have prominent importance in reliability, biomedicine, ecology, and other fields. The unified concept of the length-biased distribution can be used in the development of proper models for lifetime data. The length-biased distribution represents a special case of the more general form known as weighted distribution. In [22], a length-biased weighted Lomax (LBWL) distribution was introduced, and its CDF and PDF are, respectively, given by: Statistical properties and applications of the LBWL distribution to real data have been presented in [22]. In [23], the estimation of stress strength reliability of the LBWL distribution in the presence of outliers was studied and discussed.
In view of the importance of the weighted distributions and SS-PALT in reliability studies, this paper applies the SS-PALT to items whose lifetimes under design conditions are assumed to follow the LBWL distribution under a PTIIC scheme with random removals. The removals from the test are considered to obey the binomial distribution. The maximum likelihood (ML) and approximate confidence intervals (CIs) of the estimators are presented. The Bayesian estimators, percentile bootstrap CIs, and bootstrap-t CIs are obtained. The Monte Carlo simulations, as well as experiments with real data, are performed to verify the theoretical analysis results.
The rest of the paper is organized as follows. Test procedure and the assumptions of the SS-PALT model are presented in Section 2. In Section 3, the ML and approximate CI estimators of the model parameters are provided. The Bayesian estimators of the model parameters are introduced in Section 4. The bootstrap CI estimates for ML and Bayesian estimation are given in Section 5. In Section 6, the simulation results are presented to illustrate accuracy of the estimates. The experiment with real data is introduced in Section 7. Finally, concluding remarks are given in Section 8.

Model Description and Test Procedure
The following assumptions are adopted in this work: Assume n identical and independent units obey the LBWL distribution in the test.
The lifetime of a unit under the SS-PALT denoted as X can be expressed as: where T denotes the lifetime of an item under normal conditions, and β represents the acceleration factor. Thus, from the assumptions of the SS-PALT model, the PDF (3) of a total lifetime of the test item can be expressed as: where f 2 (x) is obtained by using the transformation-variable technique. The number of units removed at each failure time follows a binomial distribution R 1 ∼ binomial(n − m, p). For i = 1, 2, . . ., (m − 1), the probability mass function is expressed as: and

Maximum Likelihood Estimation
This section introduces the ML estimators of the population parameters and acceleration factor based on the PTIIC data with binomial removal. Moreover, approximate CIs (ACIs) of the population parameters and acceleration factor are also presented.
Let (x i , r i , δ 1i , δ 2i ) , i = 1, 2, . . . , m, denote the observation obtained from a PTIIC sample under a SS-PALT. For simplicity, in the following text, when referring to x (1) . ., m} pre-determined number of removals R = R 1 = r 1 , R 2 = r 2 , . . . , R m−1 = r m−1 , can be expressed as: Suppose that R is independent of X for all i; then, the likelihood function can be expressed as follows: where That is, Based on (11), the natural logarithm of the conditional likelihood function, denoted as lnL 1 , is given by: where m u and m a are the numbers of unites under normal and accelerated conditions, respectively, and m = m u i=1 δ 1i + m a i=1 δ 2i . Since P (R = r) does not involve parameters θ, λ and β their ML estimators, denoted asθ,λ andβ can be directly obtained by maximizing ln L 1 . Thus, the ML estimators of the population parameters can be derived by solving the following non-linear equations: Eqs. (15)- (17) have no closed-form solutions, so an iterative technique has to be employed to obtain the ML estimators of the parameters.
Similarly, since L 1 (x i , β, λ, θ|R = r) does not involve the binomial parameter p, the ML estimator of p can be directly obtained by maximizing (13). Thus, the ML estimator of p can be obtained by solving the following equation: Furthermore, the ACIs of the parameters using the PTIIC data based on the asymptotic properties of the ML estimators can be obtained. The ACIs can be approximated by numerically inverting Fisher's information matrix. Accordingly, the approximate 100(1 − α)% and the two-sided ACIs for θ, λ, and β can be, respectively, obtained as: where z α/2 is the [100(1 − α)/2] th standard normal quantile, and σ is the standard deviation of the ML estimators.

Bayesian Estimation
In this section, the Bayesian estimation of the population parameters and the acceleration factor of the LBWL distribution based on the PTIIC data with binomial removal are presented. The Bayesian estimation is considered under the squared error loss function (SELF).
Assume that the prior of θ, λ, and β has an independent gamma distribution. Then, the joint prior density of θ, λ, and β can be expressed as: To elicit the hyper-parameters of the informative priors, the same procedure as that presented in [24] is conducted. Hence, these hyper-parameters of the informative priors are obtained from the ML estimates of (θ, λ, β) by equating their values with the mean and variance using the inverse Fisher information matrix of θ ,λ,β .
By equating the mean and variance of θ ,λ,β for gamma priors, we get: Hence, the estimated hyper-parameters can be expressed as: Based on the likelihood function (11) and the joint prior density (20), the joint posterior of the SS-PALT of the LBWL distribution with parameters θ, λ, and β is given by: Therefore, the Bayesian estimators of the parameters θ, λ, and β under the SELF, which are denoted asθ,λ, andβ, can be obtained as the posterior mean, which is expressed as follows: The integrals presented in (25) are very difficult to solve analytically, so the Markov chain Monte Carlo (MCMC) method is used in this paper for a numerical evaluation. An important sub-class of the MCMC method is the Gibbs sampling and a more general Metropolis within Gibbs samplers. The MCMC method was first introduced in [25,26]. The Metropolis-Hastings (MH) algorithm and the Gibbs sampling are the two most popular variants of the MCMC method. Similar to the acceptance-rejection sampling, the MH method considers that for each iteration of the algorithm, a candidate value can be generated from a proposal distribution. The MH algorithm generates a sequence of draws from this distribution by conducting the following steps: 1. Set the initial value 0 l ; = (β, λ, θ) ; l = 1, 2, 3 satisfying the condition of π 0 l > 0.

Bootstrap Confidence Intervals
In this section, different bootstrap CIs of population parameters and acceleration factor based on the PTIIC data with binomial removal are proposed for the LBWL distribution.

Percentile Bootstrap (PB) Confidence Intervals
(i) Compute estimates of θ, λ, and β by the ML and Bayesian methods.

Bootstrap-t (BT) Confidence Interval
Steps 1 and 2 are the same as Steps (i) and (ii) in the previous BP algorithm, respectively.

Simulation Study
The performance of the proposed methods for estimation of the LBWL distribution based on the SS-PALT under PTIIC was verified by the Monte-Carlo simulation via R-package. The Bayesian estimators were obtained using the gamma priors under the SELF. The main difficulty in the Bayesian procedure was obtaining the posterior distribution. The MH algorithm and the Gibbs sampling were used to simulate deviates from the posterior density. The simulation steps were as follows: Generate 10000 random samples of size n = 50, 100 from the LBWL distribution based on the SS-PALT under the PTIIC. Use the CDF of the LBWL distribution and the uni-root function in the R-package to generate the random number of the LBWL distribution in the numerical algorithm. For different parameters and τ , select two sets of parameters' values as:   • In most situations, the measures of Bayesian estimates were better than those of the ML estimates.

Application to Real Data
In order to further demonstrate the performance of the proposed method, a real data set was used. The R-statistical programming language was used for computation. The dataset was an uncensored dataset consisting of the remission times (in months) of a random observation of 128 bladder cancer patients reported in [27]. The LBWL distribution was fitted to real data using the Kolmogorov-Smirnov goodness of the fit test. The estimated values of parameters were: θ = 4.325,λ = 10.974, and p = 0.9983. The empirical CDF, the histogram of the PDF, PP plots, and QQ plots are displayed in Fig. 1. Based on the real data, the SS-PALT of the LBWL distribution under the PTIIC with binomial removals was considered. The ML and Bayesian estimates of parameters and accelerated factor were calculated. In addition, the ACIs for parameters and accelerated factor of the LBWL distribution at a different significant level for n = 70, β = 8, and p = 0.75 were also calculated.
The history plots, approximate marginal posterior density, and MCMC convergence of θ, λ, and β are represented in Fig. 2.
The ACIs for parameters of the LBWL distribution based on the SS-PALT under the PTIIC data with binomial removals at a different level of significance for n = 70, β = 8, and p = 0.75 are presented in Fig. 3.     In this paper, the Bayesian and maximum likelihood estimation methods for the LBWL distribution are discussed based on SS-PALT using the PTIIC data with binomial removals. The approximate confidence intervals of the ML estimators of the model parameters are assessed based on the Fisher information matrix. In addition, the percentile bootstrap and bootstrapt confidence intervals are determined. Moreover, the effects of sample size n, failure size m, and removal probability p on the accuracy of estimates are studied by simulation studies. The application of the proposed methods to real data is given for illustration purposes.
The simulation results show that, for fixed values of m and p, the performances of both estimation methods improve with n, and for fixed n and p, their performances improve with m, also for fixed n and m, their performances improve with p. Generally, the Bayesian estimates are better than the ML estimates in most situations. The lengths of the percentile bootstrap and bootstrap-t are smaller than those of the corresponding ACIs. For small sample sizes, Bootstrapt CIs are better than the Bootstrap-p CIs in the sense of having smaller widths. However, the differences between the lengths of CIs using both methods decrease when sample sizes increase. In fact, the removal probability p is an important factor on the accuracy of the parameter estimates. When p is large, n-m of the n test units would be dropped out at the early stage of the life test.
As a future work, this study can be extended to explore the situation under type-I progressive censoring. Evaluation of the coverage probabilities can also be computed rather than the lengths of the CIs.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.