Abstract
A jointly censored sample is a very useful sampling technique in conducting comparative life tests of the products, its efficiency appears in permitting the selection of two samples from two manufacturing lines at the same time and conducting a life-testing experiment. This article presents estimation information of the joint generalized Pareto distributions parameters using Type-II progressive censoring scheme, which is carried out with binomial removal. The generalized Pareto distribution has many applications in different fields. We outline the problem of parameter estimation using the frequentest maximum likelihood and the Bayesian estimation methods. Furthermore, different interval estimation methods for estimating the four parameters were used: the asymptotic property of the maximum likelihood estimator, the credible confidence intervals, and the Bootstrap confidence intervals. The detailed numerical simulations have been considered to compare the performance of the proposed estimates. In addition, the applicability of the joint generalized Pareto censored model has been performed by applying a real data example.
1. Introduction
In many fields of manufacturing, the products may come from more than one production line under the same processing environment. Hence, selecting two samples from two production lines and conducting a test on a life-testing experiment is essential, therefore a jointly censored sample is quite useful in conducting comparative life tests of the products. Assume the two test samples are independent with sizes m and n, and they are selected from two production lines, then they are located simultaneously on a life-testing experiment. Furthermore, to optimize the cost and the experimental time of the economic life test procedure, suppose that we implement a joint progressive Type-II censoring (JPC) scheme and end the life-testing experiment once r failures occur. In this sequence, we are concerned with developing both point and interval estimation of the unknown parameters of the lifetime density function, and hence estimate the mean lifetimes of units manufactured by the two lines. In this article, we used the JPC scheme which was described by Rasouli and Balakrishnan [1]. The units under consideration are following two parameters generalized Pareto (GP) lifetime distribution. Although much work has been performed on different types of the progressive censoring schemes for one sample, few articles discussed the idea of two sample problems. Inference on the joint Type-II censoring scheme have been discussed earlier in the literature. See for example, Basu [2]; Johnson and Mehrotra [3]; Bhattacharyya and Mehrotra [4]; and Bhattacharyya [5], who have reviewed all issues related to this model. Recently, Balakrishnan and Rasouli [6] developed exact inferential methods based on maximum likelihood estimates (MLEs) and compared their performance with those based on approximate Bayesian and bootstrap methods. In 2010, Rasouli and Balakrishnan generalized the model to be a joint progressive Type-II censoring scheme with two exponential lifetime distributions. Ashour and Eraki [7] used the joint Type-II censoring idea for estimating the parameters of Weibull populations, see also Parsi et al. [8]; Doostparast et al. [9]; Balakrishnan et al. [10]; Mondal and Kundu [11]; Algarni et al. [12]; Shrahili et al. [13]; Alotaibi et al. [14].
GP distribution has many applications and it can model many real life distributions, recently many authors studied GP distribution, for example, one may refer to Martín et al. [15], who discussed baseline methods for the parameter estimation. Huang et al. [16] obtained statistical inference of dynamic conditional GP distribution with weather and air quality factors. Shui et al. [17] discussed outlier-robust truncated maximum likelihood parameter estimators of GP distribution. He et al. [18] introduced risk analysis by GP distribution. Mahgoun et al. [19] discussed GP distribution exploited for ship detection as a model for sea clutter in a Pol-SAR application.
In the present article, we aimed to work on a joint progressive censored data under GP lifetime units. Since not much work had been performed regarding the joint progressive censored samples under GP distribution with binomial removal of the censored units, we will focus our work on making statistical inference for the unknown parameters of GP distribution. Therefore, both frequentest and Bayesian point and interval estimation methods are investigated. Numerical techniques were used to compare the performance of estimation methods to select the most efficient one. Simulation analysis is implemented to obtain the point and interval estimation for the unknown parameters of the GP distribution. Monte Carlo simulation and Gibbs sampling techniques were used to generate samples from the joint GP distribution under the performed scheme, hence the simulation experiments can be obtained easily. Finally real data analysis is achieved to illustrate the purpose of this study.
The rest of the article is organized as follows. In Section 2, model description is given. Point estimation methods are studied in Section 3, while confidence intervals are obtained in Section 4. Data analysis and simulation are used in Section 5 to facilitate comparison between different types of point and interval estimation of GP parameters and a real data set is performed to check the advantage of the new scheme over the old one. Finally, an optimal censoring scheme is suggested in Section 6.
2. Model Description
Suppose denote the ordered lifetimes of units of population 1, and it is assumed that they are independent and identically distributed (i.i.d) from generalized Pareto distribution (GP) with shape and scale parameters and , respectively. Similarly, it is assumed that denote the ordered lifetimes of units of population 2, and it is assumed that they are independent and identically distributed (i.i.d) from generalized Pareto distribution with shape and scale parameters and , respectively.
The generalized Pareto probability density function (pdf) is given by the following equation:and its cumulative density function (CDF) is as follows:where and for , , and for , . For , the GP distribution is one of the several forms of the usual Pareto family of distribution often called the Pareto distribution. For , the support of the distribution is , and the GP distribution has bounded support. For , the GP distribution reduces to the exponential distribution. The special case where corresponds to the uniform distribution . In this work, we considered the case when , with support .
For the joint progressive censored sampling scheme described by Rasouli and Balakrishnan [1], assume are the ordered statistics of the random variables . A JPC scheme between the two samples is described as follows. At the time of the first failure (can be from either X or Y), units are randomly withdrawn from the remaining surviving units. Next, at the time of the second failure (can be from either X or Y), units are randomly withdrawn from the remaining surviving units, and so on. Finally, at the time of the failure (can be again from either X or Y), all remaining surviving units are withdrawn from the life-testing experiment. Let and , where and are the number of units withdrawn at the time of the failure that belong to and sample, respectively, and they are unknown random variables. The data observed in this form will consist of , where with is a fixed integer, , where if the failure takes place from population 1 and , and . The progressive Type-II censoring scheme has the decomposition .
The likelihood function of the joint progressive censored sample under generalized Pareto lifetime (JGP) can be written as follows:whereand and .
In the following section, we provided the point inference for two GP populations under the joint progressive censoring scheme. We obtained the maximum likelihood estimators (MLEs) and Bayes estimators of the unknown parameters; numerical methods will be used to obtain the estimated parameters.
3. Point Estimations
In this section, we will use likelihood inference together with the nonclassical Bayesian estimation method. Numerical methods was used to solve some nonlinear equations since it is impossible to write it in explicit forms. These methods will be used in Section 4 to obtain exact and approximate confidence intervals for the unknown parameters.
3.1. Maximum Likelihood Estimators (MLEs)
The maximum likelihood estimation (MLE) is commonly used inferential statistics, MLE has many nice properties, such as invariance, consistency, and normal approximation properties. It depends basically on maximizing the likelihood function of distribution under consideration. Assume the log-likelihood function of the unknown parameters , and is denoted by , where is a vector of parameters. Now, taking partial derivatives of with respect to the unknown parameters, we obtained the following equation:
Solving this equation by equating it to zero will give and . Numerical methods such as Newton-Raphson was suggested to be used to solve the above system of nonlinear equations. Now, for estimating and , take the partial derivative with respect to and as follows:
Equating the partial derivatives to zero yields, and is given by the following explicit forms:
3.2. Bayes Estimators
In this section, Bayes estimates for the unknown parameters and are observed. In Bayesian method all parameters are considered as random variables with certain distribution called prior distribution. If prior information is not available which is usually the case, we need to select a prior distribution. Since the selection of prior distribution plays an important role in estimation of the parameters, our choice for the prior of , , and are the independent gamma distributions i.e., , , respectively. The reason for choosing this prior density is that Gamma prior has flexible nature as a noninformative prior, specially when the values of hyperparameters are assumed to be zero. Thus, the suggested prior for , , and arerespectively, and , are the hyperparameters of prior distributions. In Bayesian estimation method, we need to also determine the loss function. In this article, we considered quadratic loss function because this loss function is mostly used as symmetrical loss function and is defined as , where is the point estimate of the vector parameter . Under quadratic loss function, Bayes estimators are the posterior mean of the distribution.
The joint prior of , , and is as follows:where , while the joint posterior of , , and is given by the following equation:where is the likelihood function of the GP distribution under PC scheme. Substituting and as defined in equation (1) and (#prior), respectively, the joint posterior is as follows:where and .
The Bayes estimate of any function of , , and , say , under the quadratic loss function is . Since it is difficult to compute this expected value analytically, we will use the Markov Chain Monte Carlo (MCMC) technique, see Lindley [20] and Karandikar [21].
We will use Gibbs sampling method to generate a sample from the posterior density function and compute Bayes estimates. Gibbs Sampling is a Markov chain Monte Carlo (MCMC) algorithm which is used to obtain a sequence of observations that are approximately following a certain probability distribution. In another words, it is a specific case of the Metropolis–Hastings algorithm. Gibbs Sampling is applicable when the joint distribution is not known explicitly or is difficult to sample from directly, but the conditional distribution of each variable is known and is easier to sample from. For more details, one may refer to Andrew et al. [22]; Muhammed and Almetwally [23]; and El-Sherpieny et al. [24]. For the purpose of generating a sample from the posterior distribution, we assumed that the pdf of prior densities are as described in equation (2). The full conditional posterior densities of , and and the data is given by the following equation:
To apply Gibbs technique, we need the following algorithm:(1)Start with initial values (2)Use M-H algorithm to generate posterior sample for , , and from equation (8).(3)Repeat Step 2 times and obtain ,(4)After obtaining the posterior sample, the Bayes estimates of , , and with respect to quadratic loss function are as follows:where is the burn-in-period of Markov Chain.
4. Interval Estimation
Normal approximation method for constructing confidence intervals is an efficient method, it has an advantage and performs well when the sample size is large enough, otherwise this method will not be useful. If this is the case, bootstrap methods may provide more accurate approximate confidence intervals. In this section, four different approximate confidence interval methods are proposed. Our goal is to select the best interval with respect to the interval lengths, i.e., the interval with the shortest length is the best.
4.1. Asymptotic Confidence Interval
When the sample size is large enough, the normal approximation of the MLE can be used to construct asymptotic confidence intervals for the parameters , , and . The asymptotic normality of MLE can be stated as , where , is a vector of parameters, denotes convergence in distribution, and is the Fisher information matrix, i.e.,
The expected values of the second order partial derivatives can be evaluated using integration techniques. Therefore, the approximate CIs for , , and arerespectively, where , and are the entries in the main diagonal of fisher matrix and is the lower percentile of standard normal distribution.
4.2. Bootstrap Confidence Interval
Since the asymptotic confidence intervals do not perform very well for small sample size, an alternative approach to the traditional one is used, namely, the bootstrap method. Parametric and nonparametric bootstrap methods are presented in Davison and Hinkley [25] and Efron and Tibshirani [26]. In this section, we used two parametric bootstrap methods: (a) percentile bootstrap and (b) t-bootstrap (see Hall [27] and Efron [28, 29]).
4.2.1. Percentile Bootstrap Confidence Interval
The confidence intervals based on percentile bootstrap are performed by using the following algorithm:(1)Step (1): Given the original data set , and or 1 depending on whether the failure is from population one or two. Estimate , , and using the maximum likelihood estimation (say ).(2)Step (2): Generate a bootstrap sample from joint Weibull distribution with parameters obtained in Step (1).(3)Step (3): With respect to the bootstrap sample estimation is .(4)Step (4): Repeat Step 2 and 3 M-times to obtain different bootstrap samples.(5)Step (5): Arrange the different bootstrap samples in an ascending order as , where and .
A two-sided 100(1-)% percentile bootstrap confidence intervals for the unknown parameters , , and are given by the following equation:
4.2.2. Bootstrap-t Confidence Intervals
For this method, use the following algorithm:(1)Given the original data set , and or 1 depending on whether the failure is from population one or two. Estimate , , and using the maximum likelihood estimation (say ).(2)Generate a bootstrap sample from joint Weibull distribution with parameters (3)The bootstrap sample estimation is (4)Compute the t-statistics , and , where , and are the asymptotic variance of , , and , respectively(5)Repeat Steps 2 to 4 times .(6)Arrange the T values obtained in Step 5 in ascending order .
Two-sided t-bootstrap confidence intervals for the unknown parameters , , and are given by the following equation:
4.3. Credible Intervals
Using MCMC techniques in Section 3.2, Bayes credible intervals of the parameters , , and can be obtained as follows:(1)Arrange , and , ascending order as follow , , and .(2)Two-sided credible intervals for the unknown parameters , , and are given by the following equation:
5. Data Analysis and Simulations
In this section, comparisons are proposed regarding the different methods of point and interval estimation that were used in the previous sections. These comparisons need numerical analysis methods and simulation, Monte Carlo simulation was carried out. We analyzed a real data set for illustrative purposes; also, a simulation study was carried out to compare the performances of the different estimators, using different parameter values and different sampling schemes.
5.1. Simulation Study
In this subsection, the Monte Carlo simulation study was utilized to analyze the point and confidence interval estimation for the parameters of GP distribution based on the JPC scheme with binomial removal. The simulation results are summarized in Tables 1–3 and some concluding remarks of simulation results are pointed out. To evaluate the performance of the estimation procedures described in this article, we performed an extensive simulation study. Since the JPC samples scheme with binomial removal of the GP distribution is given in terms of the cdf and pdf of GP distribution, it is easy to generate a random sample from this model.where and . Therefore, we need to determine some values of actual parameters for this model as follows:
Case 1. .
Case 2. .
Case 3. .
Also, we need to suggest different samples sizes, hence when the samples size are and , we selected different failure sizes for this sample as and . When the sample sizes are and , we selected different failure sizes as and . Also, when the sample sizes are and , we selected different failure sizes as and . The probability of binomial removal for JPC is supposed to have two values as and 0.5.
After generating the data for and , we combined these variables to obtain the from two different samples with the same probabilities. Furthermore, denote the order statistics, where . Then, generate R-removal of censored form binomial with probability. Generate progressive censored sample. For more information about generating progressive censored samples, see Balakrishnan [30] and Balakrishnan and Cramer [31]. For the MLE method, we used Newton–Raphson algorithm and for the Bayesian estimation method, we used MCMC by MH algorithm. Confidence intervals of Bayesian estimation method is performed using credible intervals algorithm of MCMC results when 10000 loop is performed.
In a confidence interval, we get the length of the interval (L.CI) when . The Biases and MSEs are used to compare MLE and Bayesian estimation methods. The results are obtained in Tables 1–3 after 10000 loops.
The following concluding remakes are noticed based on these Tables:(1)As sample size increases with fixing all other values of model, the bias, MSE, and L.CI associated with the parameter of the GP distribution based on JPC estimates decrease for all methods of estimation.(2)As the number of failed units increases with fixing all other values of the model, the bias, MSE, and L.CI associated with the parameter of the GP distribution based on JPC estimates decrease for all methods of estimation.(3)The Bayesian estimation method is the best estimation method to estimate the parameters of the GP distribution based on JPC, since it has the smallest bias and MSE values and shortest L.CI.(4)The bootstrap confidence interval is the shortest interval length for estimation of the GP parameters based on JPC.(5)When comparing the asymptotic confidence intervals with the credible confidence intervals, we can realize that the latter have shorter interval lengths.
5.2. Application of Data
Abu-Zinadah [32] used this data to inference the jointly Type-II censored samples from two Pareto distributions. The data are as follow: X = 0.152, 0.548, 0.759, 0.778, 0.916, 0.976, 1.017, 1.433, 1.558, 1.822, 1.888, 2.395, 3.066, 3.901, 5.489, 5.809, 17.886, 21.829, 43.239, and 90.793. Y = 0.006, 0.383, 0.489, 0.925, 1.25, 1.337, 1.448, 1.976, 2.426, 5.484, 8.611, 9.430, 16.120, 37.360, 41.090, 49.276, 152.313, 442.915, 12510.900, and 63621.000.
The empirical and fitted distribution functions, CDF, and PP-plots are presented in Figures 1 and 2. The graphical tools such as trace plots and auto-correlation function (ACF) plots are used to check the convergence of MCMC. Figures 3 and 4 show the trace and ACF plots for , , and of a chain of different number of iterations. The ACF plots for , , and show that the chains have a low auto-correlation. Also, they indicate a rapid convergence of the MCMC subject to the normal distribution. For more information about convergence of MCMC one may refer to Freitas et al. [33]. From Table 4, it is clear that Bayesian estimation performs better than MLE for different number of failures and different binomial probabilities, this is because it has less mean squared error (SE). Figure 5 shows the Contour plots of log-likelihood function with different parameter values, the MLE results of model with and are unique and attain their maximum points.
6. Conclusions
In this article, we considered point and interval estimation for two joint populations with generalized Pareto lifetimes under progressive Type-II censoring schemes. Classical and nonclassical estimation methods were proposed and numerical methods were implemented to evaluate the performance of the different methods of estimation, it was shown through a real data example that Bayesian methods were superior to the classical method (MLE). While comparing the confidence intervals it was realized that Bootstrap confidence interval has the shortest interval lengths compared to asymptotic and credible confidence intervals.
Data Availability
The data used to support this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Project no. GRANT2550). The authors, therefore, acknowledge technical and financial support of the Deanship of Scientific Research at KFU.