Abstract
This article investigates the estimation of the parameters for power hazard function distribution and some lifetime indices such as reliability function, hazard rate function, and coefficient of variation based on adaptive Type-II progressive censoring. From the perspective of frequentism, we derive the point estimations through the method of maximum likelihood estimation. Besides, delta method is implemented to construct the variances of the reliability characteristics. Markov chain Monte Carlo techniques are proposed to construct the Bayes estimates. To this end, the results of the Bayes estimates are obtained under squared error and linear exponential loss functions. Also, the corresponding credible intervals are constructed. A simulation study is utilized to assay the performance of the proposed methods. Finally, a real data set of COVID-19 mortality rate is analyzed to validate the introduced inference methods.
1. Introduction
To achieve a balance between the total time spent in the experiment and the number of units used in the experiment, the experiment must be subjected to a good control system (censoring scheme) that allows the experimenter to obtain results that enable him to make good statistical inference. At the same time, the working experimental units are saved for future use, as well as the cost and time associated with testing. Type-I (time) and Type-II (failure of units) censoring schemes are the oldest and most common censoring schemes. In these two types of censoring schemes, units cannot be withdrawn from an experiment until the final stage or the number of units fails.
Balakrishnan and Sandhu [1] introduced the progressive Type-II censoring that has more flexibility than the Type-II censoring in allowing units to be withdrawn from the test at different observed failure times. Briefly, it can be described as follows: assume that independent units are placed in the life test and the observed failure times , which are called a progressive sample as , are prefixed. Besides, the scheme is a prefixed censoring plan. At the first failure occurrence , the surviving units are randomly withdrawn from the test; then, at the second failure occurrence , the surviving units are randomly withdrawn from the test and so on until the occurrence of the failure and therefore the remaining surviving units are withdrawn from the test and the test is terminated; see Figure 1. For more details and extensive reviews of the literature on progressive Type II censoring, readers may refer to Balakrishnan [2], Balakrishnan and Sandhu [3], and EL-Sagheer [4].
Kundu and Joarder [5] introduced a Type-II progressive hybrid censoring scheme; the experiment in this type is terminated at a prescribed time with a progressive Type-II right censored. It is clear that this type of censoring has the same disadvantage of the Type-I censoring (time), in which is random and may be a small number. For this reason, Ng et al. [6] developed the adaptive Type-II progressive censoring scheme as a mix of Type-II progressive censoring and Type-I censoring. This type allows to depend on the failure times so that the effective sample size is always , which is fixed in advance. Thus, the life testing experiment can save both the total test time and the cost induced by failure of the units. This censoring scheme can be described as follows: consider identical units under observation in a life testing experiment and suppose the experimenter provides a time , which is an ideal total test time, but we may allow the experiment to run over time . If the progressively censored observed failure occurs before time (i.e., ), the experiment terminates at the time . Otherwise, once the experimental time passes time but the number of observed failures has not reached , we would want to terminate the experiment as soon as possible. Therefore, we should leave as many surviving units on the test as possible. Suppose is the number of failures observed before time , that is, , where and . After the experiment passed time , we set and . This formulation leads us to terminate the experiment as soon as possible if the th failure time is greater than for . One extreme case is when , which means time is not the main consideration for the experimenter; then we will have a usual progressive Type-II censoring scheme with the prefixed progressive censoring scheme . Another extreme case can occur when , which means we always want to end the experiment as soon as possible; then, we will have and , which results in the conventional Type-II censoring scheme. For extensive reviews of the literature on the adaptive Type-II progressive censoring scheme, readers may refer to Cramer and Iliopoulos [7], Mohie El-Din et al. [8], and El-Sagheer et al. [9], which can be briefly described by Figures 2 and 3.
From Figures 1 and 2, we observe that the value of plays an important role in the determination of the values of , , and is a compromise between a shorter experimental time and a higher chance to observe extreme failures.
Mugdadi [10] proposed the two-parameter power hazard function distribution, denoted by PHFD as an alternative to the Weibull, Rayleigh, and exponential distributions and studied its different properties. This distribution has increasing hazard rate function (HRF) for , and it has decreasing HRF for . The probability density function (PDF), cumulative distribution function (CDF), reliability function RF, and HRF are given, respectively, bywhere and are the scale and shape parameters, respectively. The PHFD has been shown to be useful for modeling and analyzing the life time data in medical and biological sciences, engineering, and so on. This distribution is a very flexible model that approaches different models when its parameters are changed. It contains the following special models: PHFD refers to Rayleigh when and , PHFD reduces to Weibull () when , and PHFD is exponential distribution with mean when .
The coefficient of variation is used in many science areas such as medical sciences, chemistry, biology, economics, finance, engineering, and reliability theory to assess homogeneity of bone test samples, dose-response studies, wildlife studies, and other fields; see Subrahmanya Nairy and Rao [11], Creticos et al. [12], Versaci et al. [13], and El-Sagheer et al. [9]. Given a set of observations from PHFD , the sample coefficient of variation iswhere and are the first and the second moments of the PHFD , given bywhere is the gamma function . Thus,where
Inferences for the PHFD have been studied by several authors including Kınacı [14] discussed the stress-strength reliability model for PHFD, El-Sagheer [15] investigated the problem of point and interval estimations of the parameters for PHFD based on progressive Type-II censoring scheme, Mugdadi and Min [16] discussed Bayes estimation of PHFD based on a complete and Type-II censored samples, and Khan [17] established the recurrence relations for single and product moments of generalized order statistics from the PHFD. Recently, El-Sagheer et al. [18] constructed the Bayesian and non-Bayesian approaches for the lifetime performance index with the progressive Type-II censored sample from the PHFD. In this article, we use the adaptive Type-II progressive censored (AT2PC) scheme to study the problem of estimating the shape and scale parameters, RF, HRF, and for two-parameter PHFD.
The structure of the paper is organized as follows. Section 2 deals with the maximum likelihood estimate and asymptotic confidence intervals. Bayesian estimates using Markov chain Monte Carlo technique are provided in Section 3. In Section 4, a simulation study is conducted to compare the performance of these estimation methods. A real data set of COVID-19 is presented to illustrate the application of the proposed inferences in Section 5. Finally, a brief conclusion is given in Section 6.
2. Maximum Likelihood Inference
Suppose that is a AT2PC-order statistics from the PHFD with progressive censored scheme . According to Ng et al. [6], the log-likelihood function without normalized constant can be written aswhere is used instead of . After differentiating with respect to and , respectively, and equating each of them to zero, the likelihood equations can be written as follows:
From (10), the MLE of can be illustrated as
Since (10) and (12) do not have closed-form solutions, Newton–Raphson iteration method is widely used to obtain the desired MLEs in such situations. For more details, see El-Sagheer [4]. Once MLEs of and are obtained, the MLEs of , , and for a given mission time can be obtained after replacing and by and according to the invariant property of the MLEs as
2.1. Approximate Confidence Intervals
According to Cohen [19], the Fisher information matrix (FIM), which is defined by the negative expectation of the partial second derivative of the log-likelihood function, is needed to construct the approximate confidence intervals (ACIs) for parameters .
The elements of the FIM are
Since MLE has asymptotic normality property under certain regularity conditions, the estimator has asymptotic distribution , and the inverse matrix of is
Thus, the ACIs of iswhere . is the percentile of the standard normal distribution with right-tail probability .
2.2. Delta Method
In order to construct the ACIs of , , and , we need to find the variances of them. So, Greene [20] implemented the delta method for this purpose. In this method, for analytically computing the variance, a linear approximation of the functions of the MLEs is created, and then the variance of the simpler linear function that can be used for large sample inference is computed. Let be the first partial derivative of . Then, the approximate estimates of variances can be written aswhere is the transpose matrix of . Thus, the ACIs for can be given by
Moreover, Meeker and Escobar [21] suggested using the natural approximation for the log-transformed MLE to prevent the negative lower bound of the ACIs. Thus, two-sided normal ACIs for , , , or are given by
3. Bayesian Inference
To characterize the problems more rationally and reasonably, we must take into consideration both the information from observed sample data and the prior information; this is the main idea of Bayesian inference. In this section, Bayesian inference procedures using Markov chain Monte Carlo (MCMC) technique are proposed to estimate the parameters and as well as , , and under both squared error (SE) and linear exponential (LINEX) loss functions. Also, the corresponding CRIs are constructed under MCMC technique. We consider here that the parameters and follow the gamma prior distributions with PDFswhere and reflect the knowledge of prior about and they are assumed to be known and nonnegative hyperparameters. In addition, the parameters and as well as the corresponding priors are considered here to be independent. Further, the joint prior of the parameters and can be written as
Via Bayes’ theorem across using the likelihood function with the joint prior , the joint posterior density of and can be obtained as
Under SE loss function, the Bayesian estimator of any function of and , say , is given by
Under LINEX loss function, we havewhere . It is clear that the ratio of two integrals in (25) and (26) cannot be obtained in a closed form. Thus, it is necessary to use suitable numerical methods to approximate these integrals. So, in this case, we apply MCMC technique to obtain the Bayes estimates of , , , and and corresponding credible intervals (CRIs).
3.1. Markov Chain Monte Carlo Technique
MCMC technique is one of the most general technique for an estimation, which is provided here to compute the Bayes estimates and the corresponding CRIs for , , , and . It is known that there are several procedures of MCMC technique available in which samples are generated from the conditional posterior densities. One of the simplest MCMC procedures is the Gibbs sampling procedure, which was proposed by Geman and Geman [22]. Another procedure is considered the Metropolis–Hastings (M-H) algorithm, which was proposed by Metropolis et al. [23] and later extended by Hastings [24]. A more general procedure of MCMC procedures which we will use is considered the M-H within Gibbs sampling. Gibbs sampler is required to decompose the joint posterior distribution into full conditional distributions for each parameter and then sample from them. From (26), the posterior conditional density function of given can be written as
Similarly,
Moreover, the conditional posterior density of given in (27) is gamma density with shape parameter and scale parameter . Thus, by implementing any gamma generating routine, samples of can be simply generated. In addition, the conditional posterior density of cannot be reduced analytically to well-known distribution. So, according to Tierney [25], M-H algorithm within Gibbs sampling with normal proposal distribution is used to conduct the MCMC methodology. The hybrid M-H algorithm and Gibbs sampler works as follows:(1)Start with an , and set .(2)Generate from gamma distribution .(3)Using M-H algorithm, generate from with the proposal distribution, where is from a variance-covariance matrix.(a)Generate from .(b)Evaluate the acceptance probability(c)Generate a from a uniform distribution.(d)If , accept the proposal and set ; else, set .(4)Compute , , and as(5)Set .(6)Reiterate Steps (3–5) times.(7)Under SE and LINEX loss functions, the Bayes estimate of (where , , , , , , and ) can be obtained by where is the burn-in period and , , , , and .(8)To compute the CRIs of , order as . Then, the symmetric CRIs of is
4. Simulation Study
Monte Carlo simulations were implemented utilizing 1000 AT2PC samples for each simulation to compare the estimators of parameters and as well as some lifetime parameters , , and discussed in the preceding sections. By using Mathematica ver. 12, all computations were conducted. We conducted a simulation study using different combinations of , , and and different censored scheme . We used the algorithm proposed by Ng et al. [6] to simulate an AT2PC sample from the PHFD with parameters . The true values of , , and at time are evaluated to be 0.9222, 0.506, and 0.4279, respectively. The performance of estimators is evaluated in terms of mean square error (MSE), which is computed as , where , , , , , , and for the point estimates, also average lengths (ALs) and coverage probability (CPs), which are computed as the number of CIs that covered the true values divided by 1000, for interval estimates. Bayes estimates and the CRIs are computed based on MCMC samples and discard the first values as “burn-in.” In addition, we assume the informative gamma priors for and , that is, when the hyperparameters are and . Moreover, CRIs were computed for each simulated sample. In our study, we consider two different values of , 5 and the following censoring schemes (CS):(i)I: , for (ii)II: , for if odd; , for if even(iii)III: , for
The results of MSE, ALs, and CPs of estimates are shown in Tables 1–5.
5. Application of COVID-19 Data
In this section, for illustrative purposes, a real-life data for the coronavirus is presented to inspect the inference procedures discussed in the previous sections. We consider the set of real-life data, which is reported by Almongy et al. [26]. This data represents a COVID-19 mortality rate data that belongs to Italy of 59 days, which is recorded from 27 February to April 2020. The data are as follows:
For the goodness of fit test, we compute the Kolmogorov–Smirnov (K–S) distances between the empirical distribution and the fitted distribution functions. The K–S is 0.09254 and the associated p-value is 0.73124. Therefore, according to the p-value, we can say that the PHFD fits quite well to the above data. Empirically, and plots are shown in Figure 4, which clearly show that the PHFD fits the data very well.
(a)
(b)
(c)
(d)
Under the previous data, we use the algorithm proposed by Ng et al. [6] to generate an AT2PC sample with , , and {3, 0, 2, 0, 1, 0, 2, 0, 0, 3, 0, 3, 0, 1, 2, 0, 0, 3, 0, 0, 2, 1, 0, 3, 0, 1, 0, 2, 0, 0}. Thus, the resulting AT2PC sample is as follows:
Based on the previous sample of AT2PC, the MLEs and ACIs for , , , , and are determined to be as in Table 6. Moreover, to compute the Bayesian estimates, the prior distributions of the parameters need to be specified. Since we have no prior information, we assume that noninformative gamma priors for and , that is, when the hyperparameters are and , . Under MCMC technique, the posterior analysis was done across combining M-H algorithm within Gibbs sampler. To conduct the MCMC algorithm which was described in Section 3.1, the initial values of the parameters and were taken to be their MLEs. In addition, 12000 MCMC samples were generated. To avoid the effect of the initial values (starting point), we expunge the first 2000 samples as “burn-in.” Table 6 shows the Bayesian estimates as well as CRIs for , , , , and .
6. Conclusion
The purpose of this paper is to develop different methods to estimate the unknown quantities , , , , and of the PHFD using AT2PC scheme, which is introduced by Ng et al. [6]. The MLEs as well as ACIs using asymptotic distributions are obtained. Furthermore, to obtain the CIs of the reliability characteristics and coefficient of variation, we used delta method. It is clear that, after studying the Bayesian estimates, the posterior distribution equations of the unknown quantities are complicated and so hard to reduce analytically to well-known forms. For this reason, we have applied MCMC technique to compute the Bayes estimators. The Bayes estimates have been computed under both SE and LINEX loss functions. For illustrative purpose, real data set of COVID-19 mortality rates is considered. To check and compare the performance of the proposed methods, a simulation study was implemented with different sample sizes and different CSs . According to the results, we note the following:(1)As expected, from Tables 1–5, as sample sizes increases, the MSEs and ALs decrease.(2)From Tables 1–5, it is clear that the MSEs and ALs for all estimates at time are smaller than those at time .(3)For fixed sample sizes and observed failures, the first scheme is the best scheme in the sense of having smaller MSEs and ALs.(4)Bayes’ estimates have the smallest MSEs and ALs for all estimates. Hence, Bayes’ estimates perform better than the MLEs.(5)Bayes’ estimate under LINEX with provides better estimates for all estimates because of having the smallest MSEs.(6)Bayes’ estimates under LINEX for the choice perform better than their estimates for the choice in the sense of having smaller MSEs.(7)Bayes’ estimates under SE perform better than their estimates under LINEX with in the sense of having smaller MSEs.(8)From Tables 1–5, we observe that as the time increases, the MSEs and ALs associated with all estimates increase.(9)Both MLE and Bayesian methods have very close estimates and their ACIs have quite high CPs (around 0.95). Also, the Bayesian CRIs have the highest CPs.
Data Availability
The datasets generated during the current study are available from the corresponding author on reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.