Statistical Inference of Reliability Parameter for the Skew-Normal Distribution under Progressive Type-II Censored Samples
This study investigates the inference of the reliability parameter for the skew-normal distribution based on progressively Type-II censored samples, where and are two independent random variables. The maximum likelihood estimator and the asymptotic confidence interval for are extracted via an iterative algorithm. Moreover, the Bayes estimator and the corresponding credible interval are computed using Gibbs sampling. Also, the MLE, UMVUE, Bayes estimator, and confidence interval are acquired when the scale parameter is known. We compare the performances of the different estimation methods using Monte Carlo simulations. Finally, the real datasets are analyzed for illustrative purposes.
The reliability parameter, , has received great attention in the literature. The applications of the stress-strength model had rapid growth in many fields, including quality control, engineering statistics, reliability, medicine, psychology, biostatistics, and probabilistic mechanical design. The strength, , and the stress, , are considered random variables. The system fails whenever is less than , and is the reliability of the system performance. Another application arises in the lifetime comparison of the two devices represented by and . The stress-strength studies in the reliability context started with the paper of Birnbaum . Since then, several authors have studied the inference of the by applying different classes of statistical distribution. The estimation of has been studied by Ali and Woo , in which and are Levy random variables. Kotz et al.  provided a comprehensive review of the development of these studies until 2003. On the basis of the complete samples, several papers were published on the inference of , for example, Kundu and Gupta , Huang et al. , Babayi et al.  and Guo and Gui .
Recently, estimation of under progressive censoring samples has been considered by many researchers, for example, the two-parameter Rayleigh distribution discussed by Kohansal and Rezakhah . Extended Chen distribution proposed by Maiti and Kayal , Joukar et al.  studied power Lindley distribution, Kumar and Kumar  investigated inverse Pareto distribution, and Ghanbari et al.  studied Marshall–Olkin distributions. The multicomponent stress-strength model for Weibull distribution is considered by Kohansal et al. .
Ordinarily, the lifetime experiments are costly and time-consuming. There are different types of censoring schemes for reducing the cost and time in life-testing experiments. Type-I and Type-II censoring schemes are the two well-known and extensively applied in the literature. However, none of these schemes allow intermediate removal of the active units from the experiment other than the final termination point. Because of that, the progressive censoring scheme is used, in which some units remove after each failure. This practical scheme was first studied by Cohen . It has become trendy in reliability and life-testing experiments in the last decade, see [15–20].
The Type-II progressive censoring scheme can be described as follows. Let units be placed on a life-testing experiment, and the given scheme is prefixed. of the surviving units are chosen randomly at the time of the first failure and removed from the experiment. Likewise, of the surviving units are withdrawn randomly at the time of the second failure and so on. Last of all, at the time of the th failure, all remaining surviving observations, , are eliminated. Note that Type-II progressive censoring scheme includes complete sampling scheme by choosing and Type-II right censoring scheme by choosing . For more information about progressively censoring, see .
The first problem in data analysis is to find suitable distribution to fit them. The skew-normal density function, which is an extension of the normal one, has acceptable flexibility to model various datasets. Furthermore, having a closed form of the cumulative distribution function that simplifies calculations is another advantage of this distribution. In Section 2, we will provide more details in this regard. Babayi et al.  and Babayi and Khorram  analyzed the strength data based on complete sample data by applying the generalized logistic (GL) and skew-normal distributions, respectively. In this study, as an extension of their work, we study the inference of stress-strength parameter for skew-normal (SN) distribution based on Type-II progressive censored sample data. Also, in this study, we compute the Bayes estimator and the corresponding credible interval using Gibbs sampling. Moreover, we show that the SN model is better than the GL model to fit the strength data reported by Badr and Priest . The rest of this study is as follows. The skew-normal distribution is introduced in Section 2. We derive classical and Bayesian inference for in Section 3 when the scale parameter is unknown. In Section 4, the scale parameter of skew-normal distribution is assumed to be known, and Bayesian estimation, MLE, and UMVUE of are presented. Analysis of the proposed methods is applied to the simulated data in Section 5. Finally, the skew-normal distribution with high flexibility is fitted to real datasets.
2. Skew-Normal Distribution
Standard distributions such as the gamma, Weibull, and log-normal distributions are used to analyze the positive real line of skewed data. Azzalini  introduced the skew-normal distribution by following the probability density function (pdf) to skewed data on the real line:where and are the pdf and the cumulative distribution function (cdf) of standard normal distribution, respectively. It is clear that when , (1) corresponds to standard normal distribution. The generalization of (1) presented by Gupta and Gupta  is as follows:where and . Note that (1) is a special case of (2) when .
Now, consider other skew-normal model, suppose that be a continuous random variable with cdf ; then, , is a continuous random variable with baseline distribution . These models are called proportional hazard rate models in reliability theory. By choosing as baseline distribution, the new form of the skew-normal (SN) distribution is resulted in the following cdf and pdf, respectively:andwhere and are the shape and scale parameters, respectively. Plots of the pdf for different values of are shown in Figure 1. If set and , reduces to standard normal distribution function. When is a positive integer number, (3) is the cdf of the first order statistic of random samples of size from the standard normal distribution. It is easy to show that if , then has power normal distribution with cdf, . Hence, the skew-normal model has properties similar to power normal distribution. Kundu and Gupta  surveyed useful applications of the power normal distribution.
3. The MLE of
We want to estimate of in which and , and and are independent random variables. Hence,
Therefore, for estimating of , we first evaluate the MLE of , , and based on progressively Type-II censored data. Let and be two progressively Type-II censored samples from and with censoring schemes and , respectively. So, the likelihood function of the observed sample is given by where
Considering , the log-likelihood function is given by
Hence, the MLEs of and , denotes by and , areand could be computed using the following nonlinear equation via an iterative schemewhereand is the hazard rate function of the standard normal distribution.
Thus, using the invariant property of the ML estimators, the MLE of the is evaluated as
3.1. Asymptotic Distribution
Here, we extract the asymptotic distribution of applying the observed Fisher information matrix, , where and is used to acquire the asymptotic distribution of . The elements of the observed Fisher information matrix can be derived as follows:and using , the last element of the matrix will be as
The confidence interval (CI) of based on the asymptotic distribution of is determined. In the regularity conditions, the distribution of vector converges to the normal with , and .
Theorem 1. As and , thenwhere is the symmetric matrix as
Proof. The proof is resulted by applying the asymptotic normality of the MLE.
Theorem 2. As and , thenwhere
Proof. Here, , whereProof, using Theorem 1 and the delta method, is achieved as follows:□
By Theorem 2, the asymptotic CI of is given bywhere is th percentile of standard normal distribution.
3.2. Bayesian Estimation of
We develop the Bayesian inference of based on the Bayesian estimation of the parameters , , and . A starting point of any Bayesian analysis is the choice of the prior. We have assumed the priors follow gamma distribution because, firstly, the supports of , , and are positive. Secondly, gamma priors are conjugate ones. The advantage of using conjugate priors is that the posterior belongs to the same class as the prior, so updating the prior reduces updating its parameters. As a rule, conjugate priors lead to straightforward mathematical calculations. Therefore, we assume , , and follow the conjugate independent gamma priors. Therefore,where , and .
The joint density of the data, , , and , is
Therefore, the joint posterior density of , , and given data can be written as
Since (24) cannot be determined in a closed form, therefore, for inference of , the MCMC methods are used. The posterior PDFs of , , and are given in the following way:where stands for Gamma distribution.
Since the posterior PDF of is unknown, the normal distribution is a good proposal based on Figure 3. Hence, we apply the normal proposal distribution to simulate using the Metropolis–Hasting method. Therefore, the Gibbs sampling algorithm is the following.(1)Step 1: take the initial values for .(2)Step 2: set .(3)Step 3: simulate from .(4)Step 4: simulate from .(5)Step 5: simulate fromwith the as a proposal distribution using the Metropolis–Hastings method.(6)Step 6: determine using (5).(7)Step 7: set .(8)Step 8: times repeat steps 3–7.
Figure 4 illustrates the convergence of the Metropolis–Hasting algorithm based on the normal proposal. In addition, the cumulative average graph (center) shows that the algorithm converges after 1000 iteration, and the autocorrelation graph demonstrates no correlation.
The posterior mean and variance of can be evaluated as
By choosing the shortest length from , the HPD credible interval is determined in which is the th-order statistics from sample size .
4. Estimation of with Known Scale Parameter
The estimation of is considered when the common scale parameter is known and equals .
4.1. The MLE of
Considering (12), the MLE of will bewhere and .
Lemma 3. Let be progressively Type-II censored samples from with censoring schemes . By definingthen distribution of , , is exponential with mean .
Proof. See the work of Balakrishnan and Aggarwala .
Theorem 4. Suppose that and be two progressively Type-II censored samples from and with censoring schemes and , respectively. Then, CI for iswhere and are the lower and upper th percentile points of the distribution with and degrees of freedom.
Proof. From Lemma 3, it is evident thatand is the chi-square random variable with df, and likewise, is the chi-square random variable with df. Thus, by usingThe proof is completed.
From Theorem 4, the PDF of is resulted as
4.2. The UMVUE of
The UMVUE of is obtained using the following lemma. Because of the convenience of proof, we ignore it.
Lemma 5. If have i.i.d Exp , then conditional density function of given is
Theorem 6. The UMVUE of iswhere is a CS statistic for .
Proof. From Lemma 3, and , so by definingIt is clear that is the unbiased estimator of . Then, using Lehmann and Scheffe theorem and the fact that is a CS statistic , the UMVUE is given bywhereand and . Equation (37) using Lemma 5 and for is obtained in the following way:Considering the binomial expansion , for , is simplified asBy the same method, one can show that, for ,□
4.3. The Bayesian Estimation of
Since we assumed that the parameters are a priori independent with gamma density, the posterior density of and are independent and , receptively. Therefore, the posterior distribution of will bewhere ,and .
By assuming the quadratic loss function, the Bayesian estimation will be the posterior mean which could be computed by considering the following well-known equation:where and are beta and hypergeometric functions, respectively.
Therefore, the Bayesian estimation of iswhere . The variance of the Bayesian estimator could be achieved by usingwhere . As the posterior is not tractable, we can generate a sample from the posterior using an indirect sampling algorithm, such as the accept-reject method for constructing the HPD intervals.
5. Simulation Study
For evaluating the proposed methods, we implement the methods on artificial datasets. For inference of , two cases are considered: (i) unknown and (ii) known .
In the first case, the performances of the MLE and the Bayes estimation in terms of biases and mean squares errors (MSE) are compared. Furthermore, the asymptotic confidence intervals (CIs) and the HPD credible intervals are compared with average confidence lengths and coverage percentages. We use MATLAB software for all the computations. For comparing the performance of the MLE and Bayes estimation, we do the following steps (Table 1).(1)Step 1: we consider different parameter values, , , and , and we assign and for all schemes.(2)Step 2: we consider three progressively censored schemes in Table 1.(3)Step 3: for generating two independent progressive Type-II censored samples from and from based on the algorithm of Balakrishnan and Sandhu .(4)Step 4: to evaluate the Bayes estimation and HPD credible intervals, noninformative prior (P1) and two informative gamma priors (P2 and P3) are assumed as(5)Step 5: for extracting MLE of , the estimation of is computed from (10). The iterative process stops when the difference between the two consecutive iterates becomes less than . The 95% asymptotic confidence intervals are obtained by applying Theorem 2.(6)Step 6: in the algorithm of Gibbs sampling, we take T = 10000 and assume of as burn-in-period experimentally for eliciting the Bayes estimators and HPD credible intervals. Therefore, the results will not be affected by the initial values of the parameters .(7)Step 7: the average biases and MSE for the MLE and Bayesian estimations are reported in Table 2 for 1000 replications.(8)Step 8: average lengths and coverage percentages of the 95% CIs and HPD credible intervals for are reported in Table 3 for various sets of parameters.
As seen in Table 2, both estimators are quite competitive. The results of Table 2 indicate that the MLE outperforms the Bayes estimation in terms of bias and MSE. Also, Table 2 demonstrates that almost all the Bayes estimators are not affected by the values of hyperparameters. From Table 3, we observe that the average lengths of credible intervals are less than the average lengths of the CIs. Furthermore, the coverage percentage of the MLE is less than the Bayes estimator.
We study the second case in which the scale parameter is known. In this case, the MLE and the 95% exact CI of are constructed using (28) and (30), respectively. Also, the UMVUE and Bayesian estimation of can be computed using (35) and (45), respectively. All results are presented in Table 4, demonstrating the Bayesian estimator outperforms others and has the smallest MSE of other estimators. As expected, the coverage probabilities for the 95% exact CI of are close to 95%. Also, the results show that the scheme has the smallest bias and MSE of other schemes.
6. Data Analysis
In this section, two real strength data reported by Badar and Priest  are analyzed (Table 5). The data represent the strength measured in GPA (GigaPascal, GPA = KN/mm2, Kilonewten/squared millimeter) for single carbon fibers and impregnated 1000 carbon fiber tows. Single fibers were tested under tension at gauge lengths of 10 mm (dataset I) and 20 mm (dataset II). The fibers were randomly selected; therefore, the observations are iid. Data in Table 5 were reported in the ascending order. Babayi and Khorram  used Type-II generalized Logistic distribution (GL) to analyze these data sets. We try to show the SN distribution models for fitting these datasets are better than GL models. For this purpose, we apply the maximized value of the log-likelihood function (log L), the Kolmogorov–Smirnov (K–S) test statistic, the corresponding value, and the Akaike information criterion , where is the number of parameters in the model. The results are presented in Table 6. According to the results, for data set II, AIC and log-likelihood display that the GL model is better than the SN model. Still, the K–S statistic and p value indicate that the SN model is better than the GL model. For dataset I, all of the log-likelihood, AIC, K–S statistics, and p value indicate that the SN model is better than the GL model. Figure 5 shows the empirical and theoretical cdf graphs. Also, Figure 6 displays the histograms with estimated models for the two data sets. We can observe that the SN distribution fits very well for these datasets.
Based on the four different progressive censoring schemes, we compute the MLE and corresponding %95 asymptotic confidence intervals of from (12) and Theorem 2, respectively. Also, we extract the Bayes estimates and HPD credible intervals using the Gibbs sampling algorithm. According to Congdon  and Kundu and Gupta , we set noninformative prior (P1) to compute the Bayesian estimation. All the results are presented in Table 7. In the first censoring scheme in Table 7, we consider inference of based on completed datasets. Comparing the results show that both the ML and Bayes estimators perform well, and there is no significant difference between them. Also, we observe that the length of HPD credible intervals is less than the length of the asymptotic confidence intervals.
In this study, we have studied the inference of reliability parameter for skew-normal distribution in the presence of progressive Type-II censored samples. First, we have considered the scale parameter to be unknown. We have extracted MLE of and corresponding asymptotic confidence interval using an iterative procedure. Also, we have derived Bayes estimation and HPD credible interval of applying Gibbs sampling technique. In continuation of our discussion, assuming that the scale parameter is known, we have obtained MLE, UMVUE, and Bayes estimators and the exact confidence interval of . Furthermore, a simulation study is implemented to compare the performances of different proposed methods. It has been observed that the average lengths of credible intervals are less than the average lengths of the confidence intervals and the coverage percentage of the MLE is less than the Bayes estimator. Finally, the real dataset analysis is presented to illustrate the applicability of the model. In future work, the inference of stress-strength based on ranked set sampling or record values for skew-normal distribution can be made. Also, the neutrosophic inference of stress-strength for skew-normal distribution can be extended [32, 33].
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
S. Kotz, Y. Lumelskii, and M. Pensky, The Stress-Strength Model and its Generalization: Theory and Applications, World Scientific, Singapore, 2003.
A. Chaturvedi, N. Kumar, and K. Kumar, “Statistical inference for thereliability functions of a family of lifetime distributions based on progressive type II right censoring,” Statistica, vol. 78, no. 1, pp. 81–101, 2018.View at: Google Scholar
A. J. Fernández, C. J. Pérez-González, M. Aslam, and C. H. Jun, “Design of progressively censored group sampling plans for Weibull distributions: an optimization,” European Journal of Operational Research, vol. 211, no. 3, pp. 154–160, 2010.View at: Google Scholar
N. Balakrishnan and R. Aggarwala, Progressive Censoring: Theory, Methods and Applications, Birkhauser, Boston, MA, USA, 2000.
M. G. Badar and A. M. Priest, “Statistical aspects of fiber and bundle strength in hybrid composites,” Progress in Science and Engineering Composites, ICCM-IV, Tokyo, Japan, pp. 1129–1136, 1982.View at: Google Scholar
A. A. Azzalini, “Class of distributions which include the normal,” Scandinavian Journal of Statistics, vol. 12, pp. 171–178, 1985.View at: Google Scholar
E. L. Lehmann and H. Scheffe, “Completeness, similar regions and unbiased estimation,” Sankhya, vol. 10, pp. 305–340, 1950.View at: Google Scholar
P. Congdon, Bayesian Statistical Modeling, Wiley, New York, USA, 2001.