Statistical Inference for the Gompertz Distribution Based on Adaptive Type-II Progressive Censoring Scheme

Department of Mathematics and Statistics, Faculty of Science, Taif University, Hawia (888), Taif, Saudi Arabia Department of Mathematics, Faculty of Science, Al-Azhar University, Cairo, Egypt Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Department of Mathematics, Faculty of Science, Fayoum University, Fayoum, Egypt


Introduction
In life testing and survival analysis, there are numerous scenarios, in which units are withdrawn or lost from the experiment before it fails. e data obtained from such an experiment is referred to as censored data. e most important reason for censoring is to reduce the total length of the test, as well as the expense and labor involved with it. A censoring method that can achieve a balance between the total time spent on the experiment, the number of units used in the experiment, and the e ciency of statistical inference based on the experiment's results is also desirable. Because of time constraints and other data gathering constraints, censoring is prevalent in life experiments.
Naturally, there are di erent types of censoring schemes. e most common censoring schemes are Type-I censoring, in which the life testing experiment is terminated at a speci ed time T and in this case, the number of failures (from a sample size n) is a random variable; and Type-II censoring, in which the life testing experiment is terminated upon the rth failure, and in this case, the total time of test is a random variable. One of the disadvantages of the traditional Type-I and Type-II censoring schemes is that they do not have the exibility to allow units to be removed at times other than the experiment's end.
As a result, we investigate a more general censoring scheme known as progressive Type-II censoring, which o ers this feature. In a brief, this is how it works: consider a reliability experiment, in which n units are tested throughout a lifetime experiment. R 1 units of the (n − 1) surviving units are randomly eliminated from the experiment when the rst failure (X 1 ) occurs. Similarly, R 2 units of the (n − 2 − R 1 ) surviving units are randomly removed from the experiment when the second failure (X 2 ) occurs. e test continues until the mth failure (X m ) occurs and at that time the remaining (R m n − m − R 1 − · · · − R m− 1 ) units are removed. Prior to the study, the R i values are set. e progressive Type-II censoring sample is de ned as the ith order observed failure times denoted by X represent the observed values of a progressively Type-II censored sample. e adaptive Type-II progressive censoring, abbreviated by (AT2PC), is a mixture of Type-I censoring and Type-II progressive censoring systems (cf. Ng and Chan [1]). In this censoring scheme, we allow R 1 − R 2 − · · · − R m to depend on the failure times. As a result, the effective sample size is always m, and it is known in advance. e following are some of the benefits of a well-designed AT2PC life testing experiment: (i) reduce the total test time; (ii) reduce the costs associated with unit failure; and (iii) improve the statistical analysis efficiency. e following is a description of the censoring scheme: consider n identical units in a life testing experiment, and presume the experimenter selected an ideal aggregate test time T, while the experiment may run throughout time T.
e experiment stops at time X m: m: n if the m th progressively censored failure happens before time T (i.e., X m: m: n < T). Otherwise, the experiment will be rapidly ended if the experimental time T has passed but the number of observed failures has not surpassed m (i.e., X m: m: n > T). As a result, we attempt to include as many items as possible on the test. e number of failures observed before to time T is denoted by J. Hence, we have where X 0: m: n ≡ 0 and X m+1: m: n ≡ ∞. After the experiment elapsed the time T, we set Use this formula to finish the experiment rapidly if the (J + 1) th failure time is more than T for J + 1 < m. It is worth noting that the value of T influences the values of R i , as well as providing a compromise between a shorter experimental time and a larger chance of observing outlier failures. e usual progressive Type-II censoring scheme with the prefixed progressive censoring scheme (R 1 , . . . , R m ) is produced when T ⟶ ∞. e schematic representation of the adaptive Type-II progressive censoring scheme is shown in Figure 1. Cramer and Iliopoulos [2], Mahmoud et al. [3], Balakrishnan and Cramer [4], Ye et al. [5], El-Sayed et al. [6], Mohie El-Din et al. [7], Almetwally et al. [8], Almetwally et al. [9], Nassr et al. [10], Mohan and Chacko [11], Almongy et al. [12], Haj Ahmad et al. [13], Dutta and Kayal [14,15], and Abo-Kasem et al. [16] provide comprehensive reviews of the literature on the adaptive Type-II progressive censoring scheme. e Gompertz distribution, which is used as a survival model in reliability and survival analysis and plays a significant role in modeling human mortality and fitting actuarial tables, is the underlying distribution in this work. e investigation of statistical methods and characterization of the Gompertz distribution, which was initially described by Gompertz [17], has been contributed by many researchers. e probability density function (PDF), cumulative distribution function (CDF), reliability function, and hazard rate function of the two-parameter Gompertz distribution are provided, respectively, as follows: and where δ > 0 and β > 0 are the scale and shape parameters, respectively. e PDF and hazard rate function of the Gompertz distribution are represented in Figures 2 and 3 withdrawn withdrawn withdrawn withdrawn · · · · · · End Start  e Gompertz distribution is used in a variety of domains, including gerontology, computer science, biology, and marketing science, to describe human mortality and create actuarial tables. Figures 2 and 3 illustrate that the Gompertz distribution has a unimodal PDF and an increasing hazard function, respectively. e features, inferential methods, and a wide range of applications of the Gompertz distribution have been studied by a number of researchers. Willekens [18] looked at the relationships between the Gompertz distribution and other distributions, including the Type-I extreme value and Weibull distributions. Using a progressive Type-II censoring scheme, Chang and Tsai [19] calculated the maximum likelihood (ML) estimates as well as the exact con dence interval for the parameters of the Gompertz distribution. Soliman et al. [20] used progressive rst-failure censoring to compute the ML and Bayesian estimates for the unknown parameters, as well as the hazard rate and reliability functions of the Gompertz distribution. For the unknown parameters, they also calculated approximate and exact con dence intervals. Based on a generalized progressively hybrid censoring scheme, Mohie El-Din et al. [21] derived ML and Bayesian estimates for the parameters of the Gompertz distribution, as well as one-and two-sample Bayesian predictions for future observations from the same population.
is is how the rest of the paper is organized. e ML estimators of the unknown parameters, as well as the associated survival and hazard functions, are derived in Section 2. e approximate con dence intervals for δ and β, as well as the associated survival and hazard rate functions, are also provided. In Section 3, the Gibbs sampling process is utilized to generate a sample from the posterior density function, which is then used to compute Bayesian estimates and create credible intervals. A Monte Carlo simulation study is undertaken in Section 4 to assess the performance of the two estimation methods as well as the con dence intervals. In Section 5, some numerical ndings utilizing a real data set are presented to demonstrate the inferential processes. Finally, in Section 6, we nish the work with some conclusions.

Maximum Likelihood Estimation
e ML estimators of δ, β, s(t), and h(t) are discussed in this section. Also we calculated the approximate con dence intervals δ, β, s(t), and h(t).
Let X 1: m: n < ··· < X J: m: n <T < X J+1: m: n < ··· <X m: m: n be an adaptive Type-II progressive censored sample from a continuous population with CDF F(x) and PDF f(x) along with a censoring scheme R (R 1 , ...,R j , 0, ..., 0,R * m ), where T is pre xed and R * m n − m − J i 1 R i . en, the joint density function of X 1: m: n < ··· < X J: m: n < T< X J+1: m: n < ··· < X m: m: n is given as (Ng et al. [22]) where From (3), (4), and (7), the likelihood function of δ and β is given by the following equation: and then, the log-likelihood function may be written as follows: e likelihood equations are constructed by differentiating (10) with regard to δ and β and equating to zero: Based on a progressive Type-II censored sample, Ghitany et al. [23] formulated the necessary and sufficient condition for the existence and uniqueness of the ML estimators of the parameters of the Gompertz distribution. e following theorem introduces the necessary and sufficient condition for the existence and uniqueness of the ML estimators of the parameters of the Gompertz distribution based on an adaptive progressive Type-II censored sample by substituting m into eorem 1, obtained by Ghitany et al. [23]. Theorem 1. Let X 1: m: n < · · · < X J: m: n < T < X J+1: m: n < · · · < X m: m: n be an adaptive Type-II progressive censored sample with a censoring schemeR � (R 1 , . . . , R j , 0, . . . , 0, R * m ). en, the maximum likelihood estimates δ ML and β ML of the parameters δ and β of the Gompertz distribution exist, and they are unique with and δ ML as the solution of the nonlinear equation, if and only if e ML estimators of the reliability function s(t) and hazard rate function h(t) may be produced utilizing the invariance property of the ML estimator by substituting δ ML and β ML in (5) and (6), respectively, and given by the following equation: and 2.1. Approximate Confidence Intervals. From the log-likelihood function in (10), we have and Taking the expectation of minus equations (18), (19), and (20) yields the Fisher information matrix I(δ, β). However, because the expectation is difficult to get, so under some regularity criteria, (δ, β) is approximately bivariately normal distributed with mean (δ, β) and covariance matrix is the inverse of the observed Fisher information matrix given by the following equation: us, the 100(1 − c)% approximate confidence intervals for δ and β are, respectively, where var(δ) and var(β) are the first and second elements on the main diagonal of the covariance matrix I − 1 0 (δ, β) and z c/2 is the percentile of the standard normal distribution with right-tail probability r/2.

Approximate Confidence Intervals Using the Delta
Method. Greene [24] proposed the delta technique as a general way for computing approximate confidence intervals for ML estimator functions. See Agresti [25] for a description of the delta approach, which takes a function that is too complex to compute the variance analytically, generates a linear approximation of it, and then computes the variance of the simpler linear function that can be utilized for large sample inference. e approximate confidence intervals for s(t) and h(t) are calculated using the delta approach in this subsection. Let where en, the approximate estimates of var(s(t)) and var(h(t)) are given, respectively, by the following equation: where V T i is the transpose of the vector V i , i � 1, 2. So, the 100(1 − c)% approximate confidence interval of s(t) and h(t) are and

Bayesian Estimation
e Bayesian estimation of δ, β, s(t), and h(t) is developed in this section. As Arnold and Press [26] have highlighted, there is no mechanism for selecting adequate priors for Bayesian estimation. Under the assumption that both parameters δ and β are unknown and have independent gamma priors, Bayesian estimation is utilized here, with PDFs provided by the following equation: respectively, where a, b, c, d > 0 and Γ(.) denotes the complete gamma function. It is worth mentioning that the gamma prior's class is flexible because these distributions may be used to model a wide range of prior information. Furthermore, by setting hyperparameters to zero, the improper priors of δ and β may be derived as special cases of independent gamma priors. Several researchers have employed gamma priors, such as Maiti and Kayal [27] and Dey et al. [28]. e joint prior density of δ and β is given as follows: e joint posterior density of δ and β is generated by using the likelihood function in (9) and the joint prior in (29), which is given by the following equation: erefore, the Bayesian estimator of some function of δ and β say g(δ, β), with respect to the squared error loss function, will be the posterior expectation of g(δ, β), i.e., Unfortunately, in most cases, the integrals in (31) cannot be derived in explicit form. Even if this integration can be done explicitly, the corresponding credible interval may be impossible to create, and numerical approaches may fail. In this case, we propose using the Markov chain Monte Carlo (MCMC) method to approximate (31).

3.1.
e Metropolis-Hastings Algorithm within Gibbs Sampling. We use MCMC technique in this subsection, which approximates the generation of random variables from a posterior distribution ϕ(δ, β), then computes the Bayesian estimation of δ, β, s(t), and h(t), as well as the corresponding credible intervals. ere are many different MCMC schemes to choose from, and it might be difficult to decide which one to use. Within Gibbs sampling, we primarily focus on a special form of the MCMC known as Metropolis-Hastings, which was developed by Metropolis et al. [29] and then extended by Hastings [30].
From (30), the posterior conditional density function of δ given β can be obtained as follows: Similarly, the posterior conditional density function of β given δ can be obtained as follows:

Mathematical Problems in Engineering
Both δ and β posterior density functions in equations (32) and (33) cannot be reduced analytically to well-known distributions. As a result, standard methods cannot be used to sample directly, although the plots indicate that they are similar to the normal distribution. To generate random numbers from this distribution, we use Metropolis-Hastings sampling with a normal proposal distribution. e following approach is proposed for generating δ and β from posterior density functions and obtaining Bayesian estimates of δ and β, as well as the corresponding credible intervals.

Bootstrap Confidence Interval
When the effective sample size m is large, the approximate and Bayesian confidence intervals are adequate. When m is small, the bootstrap resampling approach is preferable for constructing the confidence interval. e procedure for obtaining the confidence interval for δ, β, s(t), and h(t) using the bootstrap resampling approach is described as follows: Step 1: based on n, m, R, and T, compute δ ML and β ML .
Step 2: use δ ML and β ML and the same values of n, m, R, and T to generate a bootstrap resample. and

Monte Carlo Simulation Study
In this section, Monte Carlo simulation is used to compare the different estimation methods presented in the previous sections. We employed a variety of sample sizes and different sets of R i 's, as shown in Table 1. We next simulated adaptive Type-II progressively censored samples from the Gompertz distribution with parameters δ � 0.8 and β � 1.1 when T � 0.5 using the algorithm reported by Ng et al. [23]. We computed the ML and Bayesian estimates for the two unknown parameters as well as the corresponding reliability and hazard rate function (when t � 0.3) in each case. We also       (0.3) and h(0.3).    We used a noninformative gamma prior (NI-Bayesian: a b c d 0) and an informative gamma prior (I-Bayesian: a 10, b 8, c 9, d 7) to compute Bayesian estimates and credible intervals based on 11000 MCMC samples, discarding the rst 1000 values as burn-in. We repeated the process 1000 times and computed the average, bias, and mean squared error (MSE) of the ML and Bayesian estimates for δ and β, as shown in Table 2 and Figure 4, and for s(0.3) and h(0.3), as shown in Table 3. Also, we calculated the length and the coverage probability (Cov) for the 95% approximate and Bayesian and Bootstrap estimation intervals for δ and β, as shown in Table 4 and Figure 5, and for s(0.3) and h(0.3), as shown in Table 5.

Illustrative Example
In this section, some numerical results based on real data sets are presented to demonstrate inferential procedures.
Real data set: Lyu [31] has represented 86 times between failures, as shown in Table 6.
is leads to j 13. us, the resulting adaptive Type-II progressive censored sample is shown in Table 7.
We assume the noninformative prior, where (a, b, c, d) (0, 0, 0, 0), because we have no prior information about the unknown parameters. In Table 8, the ML and noninformative prior Bayesian estimates for the parameters δ, β, s(0.3), and h(0.3), as well as the lower and upper  bounds of 95% confidence intervals for them, using the asymptotic distributions and noninformative prior Bayesian methods are presented. e MCMC trace plots and the marginal posterior density with histograms of δ and β are shown in Figure 6.

Concluding Remarks
In this work, we looked at several estimating approaches using an adaptive Type-II progressive censored sample from the Gompertz distribution. For the unknown parameters, we calculated the ML and Bayesian estimators, as well as the corresponding reliability and hazard rate functions. For the unknown parameters and their corresponding reliability and hazard rate functions, we constructed approximate, Bayesian (using MCMC approach), and Bootstrap confidence intervals. Furthermore, we conducted simulation studies using a variety of sample sizes and censoring schemes to compare and evaluate the suggested estimate methods' effectiveness. Finally, we used a numerical example based on real data to show the computations of the methods proposed in this study.
From Tables 2-5 and 8 and Figures 4-6, we can notice that (1) e results obtained using the Bayesian method are better than those obtained using the ML method. (2) e results of the Bayesian method with informative prior are better than those of the Bayesian technique with noninformative prior. (3) e Bayesian method with noninformative prior produces results that are quite close to the results using the ML method. Because the Bayesian method is computationally more expensive, it is always preferable to use the ML method rather than the Bayesian method with noninformative prior. (4) e average length of all confidence intervals and the mean squared error of all estimations decrease as the number of data increases.
(5) e marginal posterior density plots and histograms are approximately symmetric around their means, and the MCMC trace plots of δ and β converge. As a result, the unknown parameters may be estimated using the MCMC generated sample.

Data Availability
All data generated or analyzed during this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this paper.