Accelerated Life Tests under Pareto-IV Lifetime Distribution: Real Data Application and Simulation Study

: In this article, a progressive-stress accelerated life test (ALT) that is based on progressive type-II censoring is studied. The cumulative exposure model is used when the lifetime of test units follows Pareto-IV distribution. Different estimates as the maximum likelihood estimates (MLEs) and Bayes estimates (BEs) for the model parameters are discussed. Bayesian estimates are derived while using the Tierney and Kadane (TK) approximation method and the importance sampling method. The asymptotic and bootstrap conﬁdence intervals (CIs) of the parameters are constructed. A real data set is analyzed in order to clarify the methods proposed through this paper. Two types of the progressive-stress tests, the simple ramp-stress test and multiple ramp-stress test, are compared through the simulation study. Finally, some interesting conclusions are drawn.


Introduction
In most of the classical life testing and reliability experiments, collecting enough number of failure times is not easy, especially when the products are highly reliable with long lifetimes. Under normal conditions the lifetime experiments with a restricted testing time may produce a very few failures, one of the most famous procedures used to accelerate the occurrence of failure is the ALT, in which products are tested at high levels of stress (e.g., humidity, temperature, pressure, voltage, or vibration). The failure time data from ALTs are then analyzed in order to estimate the life characteristics of the products under normal conditions. There are different ways in which the ALT can work, for example, constant, step, and progressive stress ALT. Nelson discussed these different types [1].
In constant-stress ALT, the test is performed completely under a constant level of stress. Several authors have discussed the constant-stress models; see references [2][3][4][5][6][7][8][9] for more details. The second way to apply accelerated stress is step-stress. In this kind of stress, we raise the stress level step by step at pre-specified times to get early failures. Many authors have discussed the step-stress ALTs; see for example Bai et al. [10], Gouno et al. [11], Miller and Nelson [12], Balakrishnan et al. [13], and Mohie El-Din et al. [14][15][16][17]. On the other hand, in progressive-stress ALT, all of the test units are exposed to stress, which is continuously at an increasing rate with time. For more reading about progressive-stress ALT, see Yin and Sheng [18], Mohie El-Din et al. [19], Bai et al. [20], as well as Ronghua and Heliang [21] and Abdel-Hamid and AL-Hussaini [22] who also studied the progressive-stress ALT based on progressive censoring in case of Weibull distribution in [23]. Bayesian prediction intervals under progressively type-II censoring for the half-logistic distribution under progressive-stress model have been discussed by AL-Hussaini et al. [24]. The Bayesian estimates of exponentiated exponential distribution based on type-II progressive hybrid censoring have been discussed by Abdel-Hamid and Abushul [25], depending on the inverse power law and the cumulative exposure model. Abd El-Raheem [26] considered the problem of optimal design of progressive-stress ALT for generalized half normal distribution.
Because of the high cost and long time that the life testing experiments consume, the experimenter needs to end the experiment before observing all failures, so the censoring techniques of data are widely used in order to decrease the test time and cost. Type-I and Type-II conventional censoring are the most two used censoring schemes for life testing. Recently, the most popular censoring type is the progressive type-II censoring scheme (CS). It can be illustrated, as follows: suppose n identical units or devices are put on a life time test with m ≤ n is a pre-specified number of failures. At the occurrence of the first failure t 1:m:n , R 1 surviving units are removed from the test. When the second failure t 2:m:n occurs, R 2 surviving units are also removed from the test and so on until the m-th failure is observed, where the test is terminated and the remaining surviving R m = n − m − (R 1 + ... + R m−1 ) units are removed. For extra studies regarding progressive censoring, see [27,28].
The contribution in this paper is studying statistical inference of progressive-stress ALT for test units whose lifetime follow Pareto-IV distribution in the presence of progressively type-II censored data.
The paper is organized, as follows: in Section 2, the ALT model and test assumptions are presented. In Section 3, the maximum likelihood estimates (MLEs) of the parameters under progressive-stress model are obtained. Using Tierney and Kadane (TK) Approximation and importance sampling method, the Bayes estimates (BEs) for model parameters are discussed in Section 4. The asymptotic and bootstrap confidence intervals for the parameters of the model are constructed in Section 5. In Section 6, a real data set is analyzed in order to illustrate the proposed methods. Section 7 includes the simulation study. Finally, the paper is concluded in Section 8.

Pareto-IV Distribution
Arnold presents the hierarchy of Pareto-IV distribution [29]. It was formed by adding additional parameters (shape, scale, location, and inequality) to the conventional Pareto I distribution. The cumulative distribution function (CDF) of Pareto-IV(µ, γ, θ, α) distribution is given by With a view to present Pareto-IV distribution as a regular family, we consider parameter µ is known and equal to 0, Serfling [30]. Thus, the corresponding probability density function (PDF) of Pareto-IV (0, γ, θ, α) is given by

Test Assumptions
In this subsection, we list the assumptions that are used through this article in the context of progressive-stress ALT: 1. The lifetime of a unit follows Pareto-IV (0, γ, θ, α). 2. The progressive-stress at level i, S i (t) is directly proportional to the time with fixed rate ν i , i.e., The inverse power law is used to model the relationship between the life characteristic θ i and the stress loading S i (t), as follows Suppose that the test is started with n units, S 0 < S 1 (t) < ... < S k (t) be the stress levels used during the test and S 0 be the use-stress. Furthermore, suppose n i units are tested under each progressive-stress level Si(t), i = 1, 2, ..., k. The progressive censoring scheme is carried out as follows: When the time of the first failure t i1:m i :n i is reached, R i1 units are randomly removed from the remaining n i − 1 surviving units. When the time of the second failure t i2:m i :n i is reached, R i2 units from the remaining n i − 2 − R i1 surviving units are randomly withdrawn from the test. The test is terminated at level i when the m i -th failure occurs t im i :m i :n i , at this time all remaining It is clear that the complete samples and type-II censored samples are special cases of this scheme. The observed life times while using the progressive-stress S i (t) are t i1:m i :n i < t i2:m i :n i < ... < t im i :m i :n i , i = 1, 2, ..., k.
Based on the assumption of the of cumulative exposure model, the CDF for the lifetimes that result from progressive-stress S i (t) is , and F i (.) is defined in (2) with scale parameter equals 1. Therefore, where σi = θ 1 ψ b i (b + 1). The PDF of (5) is given by

Maximum Likelihood Estimation
In this section, the MLEs of the parameters γ, α, θ1, and b are obtained. For simplification, let t ij:m i :n i = t ij , which represent the progressive censoring lifetimes under progressive-stress level S i (t), i = 1, 2, ..., k and j = 1, 2, ..., m i . The likelihood function of the proposed model is derived while using the CDF in (5) and the corresponding PDF in (6), as follows: where From (5) and (6) in (7), we obtain Therefore, the log-likelihood function is given by where the likelihood equations of α, γ, and θ 1 and b are, respectively ∂ ∂α The MLEs of the parameters are obtained by solving the system of four nonlinear Equations (10)- (13) in four unknowns γ, α, θ 1 , and b. This can be solved while using the Newton-Raphson iteration method.

Bayesian Estimation
In this section, based on square error (SE), linear exponential (LINEX), and the general entropy (GE) loss functions, the BEs for the parameters α, γ, θ 1 , and b based on progressive type-II censoring are discussed. The prior distributions of the parameters α, γ, θ 1 , and b are assumed to be gamma priors. Thus, Assume that the parameters of the model are independent, then the joint PDF of prior is given by The posterior density function of the parameters γ, α, θ 1 and b can be obtained from (8) and (14), as follows: The BE of the function of parameters U = U(Θ), Θ = (γ, α, θ 1 , b) under the SE loss function is given byÛ Depending on the LINEX loss function, the BE of U = U(Θ) is given bŷ where c = 0 is the shape parameter of the LINEX loss function. Based on GE loss function, the BE of U is given bŷ where q = 0 It is obvious that the integrals in Equations (16)-(18) are complicated. Consequently, the important sampling method and TK method are applied to obtain an approximation for these integrals.

Important Sampling Technique
In this part, we use the important sampling method for estimating the unknown parameters. The posterior distribution presented in (15) can be rewritten as * (γ, α, θ 1 where and Additionally, Algorithm 1 is used to find the BEs of γ, α, θ 1 , b.

Tk-Approximation for Bayesian Estimates
There are a lot of approximation methods to obtain an approximate value for the integration that depend on the posterior distribution as Monte Carlo methods (Kloek and Van Dijk [31] and Zellner and Rossi [32]) and Lindley method [33], which are computationally intensive. For extra studies where the MCMC and Lindley method are used, see, e.g., [34][35][36][37] and [38]. Tierney and Kadane [39] implemented an easily computable integration approximation method that was based on the posterior distribution as the mean and variance of a non-negative parameter or a smooth parameter function that is non-zero within the space of the parameter. The advantage of the Tierney and Kadane (TK) method over Lindley approximation method is that the second method requires the evaluation of the third derivatives of the posterior density or the likelihood function, which can be tiresome and requires great computational precision.
In this subsection, we use the TK approximation method to compute the Bayesian estimates of the parameters that are based on the SE loss function. TK-method is one of the methods to find the approximate value of the ratio of two integrals as given in Equation (16), which can be rewritten, as follows:Û where U(Θ) is any function of parameters (α, γ, θ 1 , b), l(Θ|t) is defined in (9), and ρ(Θ|t) is the logarithm joint prior distribution that is given by To obtain an explicit expression forÛ SE (Θ) using TK approximation, we consider the functions, defined by Now, assume that the following groups of values Hence, we approximateÛ(Θ), as follows: where |Σ| and |Σ * | denote the determinants of negative inverse hessian of δ(α, γ, θ 1 , b) and δ * (α, γ, θ 1 , b), respectively. Moreover, |Σ| and |Σ * | are given by

Interval Estimation
In this part of the paper, the approximate and Bootstrap CIs of the parameters γ, α, θ 1 , and b are obtained.

Approximate Confidence Intervals
Based on the asymptotic distribution of the MLEs of the unknown parameters γ, α, θ 1 , and b, the normal approximation CIs of the parameters are obtained. The asymptotic distribution of the MLEs of (γ, α, θ 1 , b) is given by Miller [40] where η = (κ ij ), i, j = 1, 2, 3, 4 is the variance covariance matrix of the unknown parameters. The 100 (1 − ζ)% two sided CIs for a general parameter ϑ i is given by where and Z q is the 100q-th percentile of a standard normal distribution.

Bootstrap Confidence Intervals
The percentile bootstrap (Boot-p) [41] CIs of the unknown parameters γ, α, θ 1 , and b are obtained. The following steps are used in order to obtain the bootstrap bounds for our distribution parameters. Algorithm 2 is used to find Bootstrap CIs.

Algorithm 2 Bootstrap CIs
1. Find MLE estimates of the parameters α, γ, θ 1 , and b by solving Equations (10)-(13). 2. Use the estimates in step 1 in order to generate a bootstrap sample X * with the same values of the censoring scheme using the algorithm made by [42]. Let us say that the estimates are α * , γ * , θ1 1 * and b * . 3. Repeat steps 1 and 2, 1000 times. 4. Arrange the estimates ascendingly (φ for given x. Accordingly, the upper and lower bounds are given by:

Application
In this section, we illustrate the proposed methods by analyzing reliability experiment carried out previously by Zhu [43]. The data in Table 1 were observed from ramp-voltage experiment of miniature light bulbs. In this ramp-voltage experiment, 62 and 61 light bulbs were experimented under ramp-rate 2.01 V/h, and 2.015 V/h, respectively, with experimental design stress is 2 V.
To determine whether the real data fit the Pareto-IV distribution, we apply Kolmogorov-Smirnov (K-S) test. Using the MLEs of parameters which are presented in Table 3, the values of K-S statistic and corresponding p-values are calculated and shown in Table 2. From results presented in Table 2, we conclude that Pareto-IV distribution is a good fit to the given data according to the p-values. Table 3 includes the MLES and BEs of the model parameters as well as the approximate (App) and Boot CIs for the data set in Table 1. The BEs are obtained while using both TK method and importance sampling method under SE, LINEX, and GE loss functions with c = −2 and q = 2. Furthermore, the values of hyper-parameters used for obtaining BEs are µ 1 = 131.8, µ 2 = 3717.6, µ 3 = 1.3, µ 4 = 2606.2, λ 1 = 0.0027, λ 2 = 0.00016, λ 3 = 0.00027, and λ 4 = 0.00061.

Simulation Studies
The performance of the suggested estimators is studied while using Monte Carlo simulations. The suggested estimators are compared in terms of their relative absolute biases (RABs) and mean square errors (MSEs) for different sample sizes (n i , m i , i = 1, 2, ..., k), and censoring schemes (R ij , j = 1, 2, ..., m i ). Furthermore, the simulation study is carried out for two designs of the progressive-stress ALT, the first one is the simple ramp-stress with two stress levels (k = 2) and their ramp values are ν 1 = 10 and ν 2 = 16. The second one is multiple ramp-stress test contains four stress levels (k = 4) with ramp values ν 1 = 10, ν 2 = 16, ν 3 = 20, and ν 4 = 25. Tables 4-7 contain the results that were obtained from the simulation study. Tables 4 and 5 The simulation study is carried out while using Algorithm 3.

Conclusions
This study addressed statistical inference that was based on progressively type-II censored data from Pareto-IV distribution under progressive-stress accelerated life test. The statistical inference that was covered in this paper includes: estimating the unknown parameters of Pareto-IV distribution by several methods, including the classical method and Bayesian methods. In this context, two methods were used to find Bayes estimates, namely the TK approximation method and importance sampling method. A comprehensive simulation study was conducted to compare different estimation methods; from them, we concluded that Bayes' estimation methods are more accurate than the classical method. Furthermore, TK method is more accurate than importance sampling method according to MSEs and RABs. Regarding to the interval estimation, the normal approximate and bootstrap confidence intervals were obtained. We have deduced from the simulation study that the length of bootstrap CIs is smaller than approximate CIs. Moreover, the coverage probability of bootstrap CIs is greater than the corresponding approximate CIs. We did not rely only on simulation studies, but also analyzed real data from ramp-voltage experiment of miniature light bulbs in order to clarify the different estimation methods.
Author Contributions: All authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Acknowledgments:
The authors are sincerely grateful to the referees and to the Editor-in-Chief for their many constructive comments and careful reading of the paper.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: