Abstract
In this article, based on the generalized Type-II progressive hybrid censoring sample from the Burr Type-XII distribution, maximum likelihood and Bayesian inference are discussed. Point and interval estimates of unknown parameters, reliability, and hazard functions are developed. We employed several loss functions, such as squared error, LINEX, and general entropy, as symmetric and asymmetric loss functions and various prior distributions as informative and non-informative priors for Bayesian inference of unknown parameters. Under a generalized Type-II progressive hybrid censoring sample, we also propose a Bayesian one-sample prediction for unobserved failures. We conduct simulation study using the MCMC algorithm for the Bayesian approach based on several prior distributions. Finally, we apply the results of the theoretical research to real data.
1. Introduction
In reliability and survival analysis, the progressive Type-II censoring scheme is the most commonly used scheme. It is preferable to the classical Type-II censoring scheme. Progressive censoring is beneficial in various real-life applications such as industry, life research, and clinical applications. It allows the removal of surviving experimental units until the end of the test. Suppose that a life test is conducted with units, and due to cost and time constraints, it is not desirable to capture all failures. Therefore, only a subset of the unit failures is observed. Such a sample is called a censored sample. Assume that one of the units was broken by accident after the test began, but before all of the units had burned out. If the experiment is still ongoing, this unit must be removed from the life test. The progressive censoring scheme gives a methodology for analyzing this type of data in this case. Some primary referred works are [1–5].
The disadvantages of the Type-II progressive censoring method are that the experiment time can be quite long if the units are highly reliable. As a result, the works of [6, 7] address this issue by suggesting a new type of censoring in which the experiment’s stopping time is , where is fixed before the starting of the life test. This type of censoring is known as progressive hybrid censoring (PHCS). Under PHCS, the total duration of the experiment will not exceed . Several authors have looked into the PHCS, including [8, 9].
However, the disadvantage of PHCS is that it cannot be implemented when so few failures can be detected before . For this reason, Lee et al. [10] proposed a general type of censoring, called generalized Type-II PHCS, in which the lower number of failures is specified. The experiment of lifetime test would save time and cost of failures based on this censoring scheme. Moreover, the estimates of the statistical efficiency of the experiment are improved by more failures. In the following section, the comprehensive notion of generalized Type-II PHCS and its advantages are explained. For recent work on PHCS, see, for example, [11–14].
The contribution in this paper is that we use the generalized Type-II PHCS data from Burr Type-XII distribution to discuss statistical inference for unknown distribution’s parameters and prediction for the removed units in each phase of the generalized Type-II PHCS.
Using the exponential form, the Burr Type-XII distribution has the following density function (PDF), distribution function (CDF), and hazard rate function, respectively, given by
The survival and hazard rate functions are given, respectively, by
The Burr Type-XII distribution has been extensively investigated by numerous researchers due to its applications in various disciplines such as biological, industrial, reliability and life testing, clinical studies, and so on (see [15, 16]). Recently, Nagy et al. in [17] have considered generalized Type-I PHCS with one specific time and two numbers of failures . This is one of the methods’ drawbacks, especially when the units are highly reliable. In this paper, we employed a generalized Type-II PHCS to control the duration of the experiment which involves some additional complications.
The rest of the article is structured as follows. Section 2 gives a summary of the generalized Type-II PHCS. Section 3 calculates maximum likelihood (ML) estimates, while Section 4 calculates Bayesian estimates for unknown parameters and survival and hazard functions using three loss functions. The Bayesian prediction for the removed units in each phaseof the generalized Type-II PHCS is derived in Section 5. Simulation studies are carried out in Section 6 to compare the efficacy of the different inference methodologies. Real data are utilized to demonstrate the theoretical findings in Section 7. Finally, the article is concluded in Section 8.
2. The Model Explanation
Consider a life test in which identical items are put on test. The generalized Type-II PHCS may be described as follows. Let and integer be prefixed such that with is also prefixed integer satisfying . At the time of first failure, of the remaining units are randomly eliminated. Similarly at the time of the second failure, of the remaining units are removed, and so on. This process repeats until the termination time ; at this time, all the remaining units are removed from the experiment. Let and denote the numbers of observed failures up to time and , respectively. Also, let and be the observed values of and , respectively. A schematic representation of the generalized Type-II PHCS can be found in Figure 1.
Under the generalized Type-II PHCS described above, we have one of the following types of observations:(1)If the failure time occurs before , then instead of terminating the test by withdrawing the remaining items after the failure, we continue to observe failures (without any further withdrawals) up to time . Therefore, the observed failure times are .(2)If the failure occurs between and , then the termination time is , and the observed failure times are .(3)If the failure occurs after , then the experiment terminates at and the observed failure times are .
Based on the generalized Type-II PHCS, the likelihood function is given by
Therefore, these cases can be combined and obtained aswherewhere and represent the number of surviving units that are removed at and and are given by
The likelihood function of under the generalized Type-II PHCS can be derived using (1) and (2) in (5), aswhere for simplicity of notation.
3. Estimation Using ML Technique
In this section, using the maximum likelihood technique, we compute the ML estimate of the unknown parameters of the Burr Type-XII model with generalized Type-II HPCS. Since the critical points of any function are identical to the critical points of , (8) gives the corresponding log-likelihood function as
To determine the critical points, set the first derivatives of (9) with respect to and equal to zero:
Using numerical technique, we can compute ML estimates of the parameters and , are and respectively, by solving equations (9) and (11). Byusing the invariance property of the ML estimates, we can calculate the ML estimates of the survival and the hazard functions, respectively, as
The conditions necessary for the existence and uniqueness of ML estimates of the parameters and are discussed by Nagy et al. [17].
3.1. Approximate Confidence Intervals
The confidence intervals for any parameters are given bywhere is the right percentile of standard normal distribution and (standard deviation) is the square root of the estimated variance of . Now, the estimated variances and of and are given by the first and the second diagonal elements of the Fisher information matrix of the parameters and . For large , the observed Fisher information matrix of the parameters and is given by
The approximate estimates of and can be constructed by using the delta method (see [18, 19]) aswhere is the transpose of the matrix , with
4. Bayesian Estimations
In this study, we investigate three forms of loss functions for Bayesian estimation. The first is the squared error loss function (SELF), a symmetric function that in parameter estimation assigns equal weight to overestimates and underestimates. The second is the LINEX loss function (LLF), which is asymmetric and provides overestimation and underestimation distinct weights. The generalization of the entropy loss function is the third loss function (GELF). Assume that both and are unknown and are separately distributed as and priors, respectively. The prior density function for joining iswhere are positive constants.
The posterior density function of given by the generalized Type-II PHCS data is obtained as
From (8) and (17),where the normalization constant I is given by
4.1. Bayesian Estimations under SELF
A commonly used loss function is SELF, and the Bayesian estimate relative to SELF is given by
Hence, from (19) and (21), the Bayesian estimates of and under SELF are obtained, respectively, as
4.2. Bayesian Estimations under LLF
Under the assumption that the minimal loss occurs at , LLF can be expressed aswhere . The value which minimizes is given byprovided that the involved expectation is finite. The problem of choosing the value of the parameter has been discussed by Calabria and Pulcini [20].
From (19) and (24), the Bayesian estimator of and based on the LLF is as follows:
4.3. Bayesian Estimations under GELF
Another commonly used asymmetric loss function is GELF which is given by
For , the Bayesian estimate relative to the GELF is given byprovided that the involved expectation is finite. From (19) and (27), one obtains the Bayesian estimator of and using GELF as follows:
Because all of the above integrals in the Bayesian estimate of unknown parameters and cannot be computed analytically, the Markov chain Monte Carlo (MCMC) algorithm is employed to evaluate them. The conditional posterior distributions and of the parameters and can now be computed, respectively, based on the posterior distribution in (19).
The Metropolis–Hastings sampler is employed to create samples of inside the MCMC algorithm because the conditional distribution of in (30) is not a well-known distribution (for further details, see [21–23]). The MCMC algorithm (1) in [17] is used to create and samples from conditional posterior distributions, which will be utilized to approximate Bayes estimates.
Assume that is any function in and ; then, the Bayesian estimates of using the MCMC values are obtained as follows.
Under SELF, the Bayesian estimate of is given by
Based on the LLF,
By using GELF,
The Bayesian confidence interval or credible intervalfor any parameter , if
Since the integration in (34) cannot be solved analytically, the MCMC approximate credible intervals for and using the generated values after sorting it in an ascending order, and , are given as follows:and the lengths of the credible intervals are the absolute difference between the upper and the lower bounds.
5. One-Sample Bayesian Prediction
In this section, the Bayesian point and interval predictions are made for the unobserved or remote units in each phase given the generalized Type-II PHCS. The prediction interval is obtained by using equi-tailed (ET) or two-tailed interval and the highest posterior density (HPD) interval.
Let denote the order statistic out of removed units at stage , (for ). Basak et al. in [24] calculated the conditional density function of , given the observed generalized Type-II PHCS, bywhere
By compensation in (36) by using (1) and (2), the conditional density function of given generalized Type-II PHCS is then given as follows:where and for ; therefore, the Bayesian predictive density function of , given generalized Type-II PHCS, is obtained as
The Bayesian predictive survival function of , given generalized Type-II PHCS, is given as
The Bayesian point predictor of under the squared error loss function is the mean of the predictive density, given by
The equi-tailed (ET) interval Bayesian predictive bounds for can be obtained by solving the following two equations:but the highest posterior density (HPD) interval can be obtained by solving the following two equations:where , , , and denote the upper and lower bounds of interval estimation for ET and HPD methods, respectively.
6. Simulation Study
For passing through and understanding statistical inferential approaches, and combination of theoretical and applied skills, sothe analytical technique was discussed in previous sections, and this section will focus on the numerical technique through a simulation study. We simulated random samples from the Burr Type-XII distribution in equation (2) and evaluated the coverage probabilities of the resulting CIs. We used parameter values and , for sample of size , with different values of , and . For the point estimate, we computed the ML and Bayesian estimates of , , , and . For Bayesian estimates, we used SELF, LLF (with ), and GELF (with ) using informative () and non-informative priors (); for each estimate, the mean square error (MSE) and estimated expected bias (EB) have been determined. We also construct the average confidence (ACL) and the coverage probabilities (CP) of the and asymptotic confidence intervals and Bayesian credible intervals for , , , and using simulations. We used three different progressive Type-II censored sampling schemes, namely,(1)Scheme 1: if .(2)Scheme 2: if .(3)Scheme 3: if and .
For computing the Bayes estimates, we generate the MCMC algorithm suggested in Section 4 with observations. The ML estimates for unknown parameters and have been used as initial values for running the MCMC algorithm. The generated sequences’ starting values may be different from converged sequences, so the first values are removed here to avoid the effects of the initial values. To check the convergence of MCMC samples and determine the B-burn-in, we provide the key diagnostic test, trace plots, and posterior density plots for different parameters and censoring schemes. Figures 2–5 show the trace plots of iterations for posterior densities of , , , and using the three above schemes at , , and with and . All censoring schemes are plotted in the same way, and it has been found that the trace plots of all censoring schemes converge very well. Furthermore, the approximate marginal posterior density with histograms for , , , and is shown in Figures 6–9, respectively.
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The trace plots, as seen in these diagrams, exhibit delicate mixing of the chains and converge to their distributions. Furthermore, in all situations, histograms and density plots are approximately symmetrical about their means. As a result, the MCMC-generated sample can be used to develop Bayesian estimates and approximate credible intervals, as well as to estimate parameters and their functions. The values of and of the ML and Bayesian estimates for , , , and are presented in Tables 1–4, respectively. Tables 5–8 show the ACL of and confidence intervals (CIs), as well as the associated coverage portability (CP).
7. Numerical Example
For this example, we use the real data in ([25], p.105) which is consideredas breakdown times in minutes of an insulating fluid between electrodes at voltage 34 kV.Zimmer et al. in [26] indicated that the Burr Type-XII distribution fits these data and used it to obtain the estimates of the Burr Type-XII parameters. Table 9 shows the 19 breakdown times of an insulating fluid between electrodes. We will use these data to consider the following progressively censored schemes.
For generating generalized Type-II progressive hybrid censored samples, suppose , ; then, we would have the following progressive data: 0.19, 0.78, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 31.75, 32.52, and 33.91, with 0.96, 12.06, 36.71, and 72.89 being randomly selected and removed from the given data. If we consider different values of and , then we have different kinds of generalized Type-II PHCS, namely,(1)Scheme I: suppose , ; since , then the experiment would have terminated at , with , and we would have the following data: 0.19, 0.78, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 31.75, 32.52, 33.91, and 36.71.(2)Scheme II: suppose , ; since , then the experiment would have terminated at , with , and we would have the following data: 0.19, 0.78, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, 8.27, 31.75, 32.52, and 33.91.(3)Scheme III: suppose , ; since , then the experiment would have terminated at , with , and we would have the following data: 0.19, 0.78, 1.31, 2.78, 3.16, 4.15, 4.67, 4.85, 6.50, 7.35, 8.01, and 8.27, with 0.96, 12.06, 36.71, and 72.89 being removed from the data and 31.75, 32.52, and 33.91 being removed after .
Table 10 shows the ML and Bayesian estimates for the unknown parameters and and survival and hazard functions based on the generalized Type-II PHCS and two different selections and . Table 11 also includes the asymptotic CIs of and , as well as the credible intervals. Table 12 shows the point predictors and Bayesian prediction boundaries of for three different censoring schemes with and .
8. Conclusions
When the observed sample is a generalized Type-II PHCS sample, Bayesian and ML estimates of the unknown parameters, as well as the survival and hazard functions of the Burr Type-XII distribution, are produced. Squared error, LINEX, and general entropy loss functions based on and distributions are considered in the Bayesian method. The parameters, as well as the survival and hazard functions, are given asymptotic and credible CIs of and . In the generalized Type-II PHCS, Bayesian point and interval prediction of non-observed failures were also established. The following conclusions can be drawn from the numerical results:(1)In most cases, the Bayesian estimates with are better than the MLEs.(2)The results of the estimates from ML that can be seen in Tables 1–4 are similar to the Bayesian estimators using . Thus, when we have no prior knowledge about the unknown parameters, it is often easier to use the ML instead of the Bayesian estimators because the computation of the Bayesian estimators is more complicated.(3)In most cases, the MSE increases as decreases.(4)The ACL of the CIs increases as and decrease.(5)The credible intervals perform well compared to the asymptotic CIs.(6)In all cases, the ACL of CIs is larger than that of CIs.(7)The HPD prediction intervals appear to be more accurate than the ET prediction intervals.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The authors acknowledge the financial support from the Researchers Supporting Project (RSP-2022/323), King Saud University, Riyadh, Saudi Arabia.