In this article, we consider estimation of the parameters of a generalized Pareto distribution and some lifetime indices such as those relating to reliability and hazard rate functions when the failure data are progressive first-failure censored. Both classical and Bayesian techniques are obtained. In the Bayesian framework, the point estimations of unknown parameters under both symmetric and asymmetric loss functions are discussed, after having been estimated using the conjugate gamma and discrete priors for the shape and scale parameters, respectively. In addition, both exact and approximate confidence intervals as well as the exact confidence region for the estimators are constructed. A practical example using a simulated data set is analyzed. Finally, the performance of Bayes estimates is compared with that of maximum likelihood estimates through a Monte Carlo simulation study.

1. Introduction

In life testing and reliability analysis, some units can be lost or withdrawn from the experiment before failure occurs. One of the major reasons for removal of the experimental units is to save the working experimental units for future use, thereby conserving the cost and time associated with testing. This leads us to use the censoring schemes. The type-II censoring can be considered a common type of censored scheme. Many authors have studied the statistical inference for different probability distributions using progressive type-II censoring, including Balakrishnan and Sandhu [1, 2], Cohen [3], Mann [4], Ng [5], Balakrishnan et al. [6], Gibbons and Vance [7], Yuen and Tse [8], Ng et al. [9], Balakrishnan [10], Soliman [11, 12], Madi and Raqab [13], Mahmoud et al. [14], Mahmoud et al. [15], Soliman et al. [16], El-Sagheer [1719], Mahmoud et al. [20], El-Sagheer and Hasaballah [21], El-Sagheer et al. [22], and Soliman et al. [23]. Recently, Zhang and Gui [24] studied the statistical inference for the lifetime performance index of Pareto distribution based on progressive type-II censored sample.

On the other hand, Viveros and Balakrishnan [25] have described a life test in which the experimenter can decide to divide the items being tested into several groups and then run all the items at the same time until occurrence of the first failure in each group. Such a censoring scheme is called first-failure censoring. For more details about statistical inference using first-failure censoring, it is recommended that the reader refers to Wu and Yu [26], Wu et al. [27], Lee et al. [28], and Wu et al. [29]. However, using this censoring scheme does not enable the experimenter to remove experimental units from the test until the first failure is observed. For this reason, Wu and Kuş [30] introduced a life testing scheme, which combines first-failure censoring with a progressive type-II censoring called a progressive first-failure censoring (Pro-F-F-C) scheme. Many previous studies have discussed inference under a Pro-F-F-C scheme for different lifetime distributions, for example, Weibull by Wu and Kuş [30], Burr Type XII by Soliman et al. [31, 32], Gompertz by Soliman et al. [33], Lomax by Mahmoud et al. [34], Compound Rayleigh by Abushal [35], Generalized Inverted Exponential by Ahmed [36], the Mixture of Weibull and Lomax by Mahmoud et al. [37], and exponentiated Frechet by Soliman et al. [38]. Recently, Cai and Gui [39] discussed the classical and Bayesian inference for a Pro-F-F-C left-truncated normal distribution.

Generalized Pareto distribution (GPD) is a significant continuous lifetime distribution. It plays a key role in statistical inference studies and reliability problems. It is also well known for being a distribution that has decreasing failure rate property. The pdf and cdf of a random variable X have a GPD given, respectively, aswhere and are the shape and scale parameters, respectively. The survival and hazard rate functions of GPD at mission time are given by the following expressions:

For more details about GPD, its properties, and applications, see Kremer [40]. In this article, we obtain the Bayes estimates and MLEs for the unknown quantities of the GPD using a Pro-F-F-C scheme. The approximate confidence intervals (ACIs) for and are constructed based on the asymptotic normality of MLEs. In the Bayesian framework, the point estimates of unknown parameters under squared error (SE), linear-exponential (LINEX), and general entropy (GE) showing loss functions are discussed. The process is done using the conjugate gamma prior for the shape parameter and discrete prior for the scale parameter . The exact confidence interval and exact confidence region for the estimators are then derived. To evaluate and compare the performance of these proposed inference procedures, a simulation study with different parameter values is undertaken. Additionally, a numerical example using simulated data set is studied to show the practicality and usefulness of these proposed methods.

The rest of the paper is arranged as follows. Section 2 deals with the classical method of estimation. Bayes estimators relative to different loss functions are considered in Section 3. In Section 4, the ACIs, exact confidence intervals, and exact confidence regions for the parameters are discussed. In Section 5, the proposed procedures obtained in the previous sections are investigated using simulated data. A simulation study is conducted to compare the proposed procedures in Section 6. Finally, a conclusion is provided in Section 7.

2. Maximum Likelihood Estimation

Let , , be a Pro-F-F-C order statistics from the GPD with the progressive censoring scheme . According to Wu and Kuş [30], the joint probability density function can be written as

From (1), (2), and (5), the likelihood function is given by

Thus, the log-likelihood function is

By equating each result of the first-order derivatives of log-likelihood function with respect to and , to zero, we obtain

Hence,and the solution of

Since there is no closed form of the solution to the above equations, the Newton–Raphson method (NRM) is widely used to obtain the desired MLEs in such situations. Once MLEs of and are obtained, the MLEs of and for given can be obtained by the invariant property of the MLEs as

3. Bayesian Estimation

Bayes estimation is quite different from the MLE method because it takes into consideration both the information from observed sample data and the prior information. Bayes’ theorem is completely dependent on the parameter estimation through calculation of the posterior distribution. As calculating the posterior distribution is conditional on the data, this requires explicit specification of the prior distribution model parameters. Furthermore, in order to gain the best estimate of the unknown parameter, it is necessary to determine the appropriate loss functions.

The next step is to take into account different loss functions. First, we consider the square error (SE) loss function which is widely used in the literature. Because of the symmetry nature of this function, it gives equal weight to overestimation as well as underestimation. Under SE, the Bayesian estimate (BE) of any function of parameters, say , is the unconditional posterior mean which is given as

However, in many situations, the parameter may be overestimated or show serious consequences of underestimation, or vice versa. In such cases, an asymmetric loss function, which associates greater importance to overestimation or underestimation, can be taken into consideration for parameters estimation. A beneficial asymmetric loss function is the LINEX loss as follows:where is a shape parameter whose sign refers to the direction and its magnitude represents the degree of symmetry. Moreover, for figure close to zero, the LINEX loss more or less becomes a SE loss. Thus, the BE of under this loss function is given by

Next, we consider the GE loss function as follows:where is a shape parameter which represents departure from symmetry. Subsequently, based on the GE loss functions, the BE of is obtained as

It is remarked that for , the BE of concurs with the BE under SE loss function.

3.1. Posterior Analysis

In this subsection, we consider that the parameter a discrete prior and has a conjugate gamma prior. Suppose that , , thenwhere and . Further, has

Then, the posterior distribution of takes the form as follows:where

The joint posterior of and using (6), (19), and (20) iswhere

By using the Bayes theorem for discrete variables, the marginal posterior probability of iswhere and are given in (24); the marginal posterior probability of is

3.2. BE under SE Loss

In this subsection, we obtain the BE of , , , and under SE loss function. By using (14), (21), and (25), the BEs , , , and are given by

3.3. BE under LINEX Loss

Based on (16), (21), and (25), the BEs , , , and are

3.4. BE under GE Loss

From (18), (21), and (25), the BEs , , , and are, respectively,

To perform the calculations in these subsections, the values of and must be found in (20). We use the prior expectation of conditional on . Thus, from (3) and (20), we get

4. Interval Estimation

This section deals with ACIs, exact CIs, and exact confidence regions for the parameters and of GPD based on Pro-F-F-C.

4.1. Asymptotic Confidence Intervals

The asymptotic normality of the MLEs can be used to construct ACIs for parameters and by using Fisher information matrix (FIM). The FIM can be written as wherewhere . The asymptotic variance-covariance matrix of the parameters and can be obtained by inverting the observed FIM as follows:with


The ACIs for and becomewhere is upper percentile of standard normal variate .

4.2. Exact Confidence Intervals

Let denote a Pro-F-F-C sample from GPD with parameters and , and let

It is remarked that is a progressively censored sample of exponential distribution (ED) with mean 1. Let us assume the following:

According to Thomas and Wilson [41], the generalized spacings are iid as standard ED; hence,has , andhas . To construct the confidence intervals for and , we consider pivotal quantities:

It can be easily shown that where , , and . Also, and are independent. To construct an exact confidence interval for and exact joint confidence region for and , we need to analyze the following two lemmas.

Lemma 1. For any positive real numbers , is a strictly increasing function of , where .

Lemma 2. For a given set of observations , the function is a strictly increasing function of when . Furthermore,(I)For , there is a unique solution for the given equation , where .(II)Let . For , there is a unique solution for the given equation wherefor and . Using the same arguments and notations in Wu et al. [42], Lemma 1 and Lemma 2 can be proved.

4.3. Exact Confidence Interval for

Suppose that , , denote a Pro-F-F-C sample from GPD , with censoring scheme . For any , confidence interval for is as follows. We know that by Lemma1 and Lemma 2 strictly increases in when , where(1)For , there is a unique solution for the given equation , where .(2)Let . For , there is a unique solution for the equation .

Hence, for , from (44), we obtain

Thus, a confidence interval for iswhere and is the solution for for the equation:

4.4. Exact Confidence Region for and

By the same way, from (45), it is clear thatwhere . For , we have

Then, we obtain

This is equivalent to

5. Numerical Computations

Consider a Pro-F-F-C sample generated from GPD showing and . The data consist of 120 observations, grouped into sets, with 4 items within each group . The Pro-F-F-C sample of size 10 out of 30 groups with the corresponding censoring scheme is given in Table 1. The MLEs of and using NRM are computed, and then both and are calculated at .

To compute the BEs, we first estimate two values of using a nonparametric procedure , . Using the available data, we obtained and . These two priors are substituted into (33), where and are obtained numerically for each given , and , , using the NRM. Table 2 displays the values of , and for each given and . The results of MLE and BE for , , , and are presented in Table 3. By using (45), the ACIs of and are and . For , we need the percentiles and to construct the CI for . According to (44), the exact confidence interval of is calculated as . For the given , , , and , the joint confidence region for and is

After the following integration,

We obtain the confidence area at , by 74.214 1. Similarly, the confidence areas for some values of are presented in Table 4. Figure 1 shows the 95% confidence region for and .

6. Simulation Study

To compare the proposed BEs with the MLEs, a simulation study is performed using various combinations of , , and and different censored schemes of (different values). A Pro-F-F-C sample from GPD with the parameters is generated. The true values of and at time and 0.5 are evaluated to be and . The performance of the resulting estimators of , , , and has been considered in terms of the mean squared error (MSE), which are computed, for , , , , , and as . These results were obtained using Mathematica ver. 13. Considering two different group sizes and the following censoring schemes,Scheme I: for Scheme II: for if odd, and for if evenScheme III: for

The results of MSE of estimates are reported in Tables 5 and 6.

7. Conclusion

The main aim of this article is to develop different methods to estimate the unknown quantities of the GPD based on a Pro-F-F-C scheme, which was introduced by Wu and Kuş [30]. We applied the classical and the Bayesian inferential procedures for the unknown parameters and reliability measures. The ACIs have been derived based on the asymptotic normality of MLEs. Under the Bayesian approach, we obtained the BEs based on the SE, LINEX, and GE loss functions. Furthermore, we assumed the conjugate gamma prior for the shape parameter and discrete prior for the scale parameter. The exact confidence interval and exact confidence region for the estimators have been constructed based on pivotal quantities. A numerical example using a simulated data set has been studied to show the practicality of these proposed procedures. The performance of the different estimation methods is realized via a simulation study which is revealed in the following:(1)The BEs based on SE, LINEX, and GE loss functions perform better than the MLEs, in terms of MSEs(2)The BEs based on LINEX and GE loss functions when and 2 and and 2 perform better than BEs based on SE, in terms of MSEs(3)The BEs based on the SE loss function perform better than BEs based on LINEX and GE loss functions when and and and , in terms of MSEs(4)From Tables 5 and 6, for a fixed scheme, the MSE values of all estimates, a model’s parameters, and the reliability measures decrease as increases which is consistent with the statistical theory that the larger the sample size, the more accurate the estimate(5)It can be seen from Tables 5 and 6 that the three CS methods vary in terms of preference and sometimes CS I is the best while at other times the CS II or III is the best in the sense of having smaller MSEs(6)The MSEs for and estimates based on the Pro-F-F-C scheme with increase in those for P-type-II-C with while the MSEs for and estimates based on the Pro-F-F-C scheme with decrease in those for P-type-II-C with

Data Availability

The data used are theoretically generated from the equations in the manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


This research was supported by Taif University Researchers Supporting Project (number TURSP-2020/318), Taif University, Taif, Saudi Arabia.