Estimation of Parameters for the Gumbel Type-I Distribution under Type-II Censoring Scheme

: This paper aims to decide the best parameter estimation methods for the parameters of the Gumbel type-I distribution under the type-II censorship scheme. For this purpose, classical and Bayesian parameter estimation procedures are considered. The maximum likelihood estimators are used for the classical parameter estimation procedure . The asymptotic distributions of these estimators are also derived. It is not possible to obtain explicit solutions of Bayesian estimators. Therefore, Markov Chain Monte Carlo, and Lindley techniques are taken into account to estimate the unknown parameters. In Bayesian analysis, it is very important to determine an appropriate combination of a prior distribution and a loss function. Therefore, two different prior distributions are used. Also, the Bayesian estimators concerning the parameters of interest under various loss functions are investigated. The Gibbs sampling algorithm is used to construct the Bayesian credible intervals. Then, the efficiencies of the maximum likelihood estimators are compared with Bayesian estimators via an extensive Monte Carlo simulation study. It has been shown that the Bayesian estimators are considerably more efficient than the maximum likelihood estimators. Finally, a real-life example is also presented for application purposes.


Introduction:
The Gumbel distribution (GD) was first proposed by Gumbel in 1941 1 . It is widely used in meteorological phenomena, hydrology, and so on. GD is one of the important distributions in modeling extreme values such as maximum daily flood discharges and snowfalls, rainfalls, and extreme temperatures, see [2][3][4] . This distribution is called extreme value type-I distribution. It has two types. The first type is based on minimum order statistics and the second type is based on maximum order statistics. Here, the first type is discussed.
It is very important to obtain the model parameters of any distribution effectively and precisely. Therefore, the GD has been studied in the literature by numerous researchers. Abbas and Tang 5 obtained the Bayesian parameter estimation methods for the Gumbel type-II distribution. Malinowska and Szynal 6 discussed Bayesian estimators for GD on kth lower record values. Yılmaz et al. 7 compared different parameter estimation methods, including both classical and Bayesian for the two-parameter GD. Saleh 8 studied the unknown parameters of the Gumbel type-I distribution based on the moment and modification moment methods. Reyad and Ahmed 9 obtained E-Bayesian estimators of the GD under the type-II censored scheme. Saad et al. 10 introduced the Gumbel-Pareto distribution.
In survival analysis, the data sets are usually observed as censored samples. Type -II censoring type is well-known and widely used. In the type-II censoring, only the first failures k≤m is observed among m units. In this censoring scheme, it is assumed that a set number of subjects or items are put on a test. The integer k≤m is pre-fixed, and the experiment stops as soon as the k th  failure is observed, see (Kundu and Ragab 11 ). There are estimations of parameters of the different distributions using the type-II censoring in the literature. Altındağ et al. 12 studied maximum likelihood and maximum product spacing estimation methods for Burr-III distribution using type-II censored samples. Nassar et al. 13 discussed E-Bayesian estimation for the simple-stress model of exponential distribution based on the type-II censoring scheme. Okasha et al. 14 examined the properties and parameter estimation of Marshall-Olkin extended inverse Weibull distribution under type-II censoring. Xin et al. 15 obtained the reliability estimation of the three-parameter Burrtype-XII distribution under the type-II censoring scheme.
This paper focuses on the estimation of the unknown parameters of the Gumbel type-I distribution under the type-II censorship scheme with classical and Bayesian parameter estimation. The maximum likelihood (ML) estimation is used in the classical parameter estimation. The asymptotic confidence intervals (ACIs) of the unknown parameters are derived by using the observed Fisher information matrix. Since Bayesian estimators (BEs) cannot be obtained in explicit forms, Lindley (LD) and Markov Chain Monte Carlo (MCMC) methods are used for Bayesian calculations. The selection of a suitable loss function (LF) and prior distribution (PD) is of considerable importance in the Bayesian parameter estimation. Therefore, three different LFs are presented, namely the squared error loss function (SELF), the general entropy loss function (GELF), and the weighted squared error loss function (WSELF). Under these LFs, the normal and gamma PDs are used for the parameters μ and σ, respectively. The Bayesian credible intervals (BCIs) are also constructed based on the Gibbs sampling method. The performances of these estimation methods are compared through an extensive simulation study.
This paper aims to deal with the estimation of the unknown parameters of Gumbel type-I distribution based on the type-II censoring scheme under all these aforementioned estimation methods.
The rest of the paper is organized as follows: In section 2, the brief information about the Gumbel-type-I distribution is given and the ML estimators of the parameters μ and σ are presented. The limiting distributions of the ML estimators are also investigated. In Section 3, BEs of unknown model parameters are obtained by using LD, and MCMC methods. Some numerical comparisons between the ML and Bayes estimators are provided in Section 4. A real-life data set is used to illustrate the computations of the ML and Bayes estimators in Section 5. Concluding remarks are presented at the end of the paper.

Parameter Estimation:
Let Y be a random variable from the Gumbel type-I distribution with the location μ and the scale parameter σ. It's the cumulative distribution function (cdf) and the probability density functions (pdf) are given as: respectively. Suppose that 1 , 2 , … , be independent and identically Gumbel type -I distributed random variables representing the lifetimes of independent units. Based on type-II censoring scheme, a sample of m identical units is put on a life testing experiment and their lifetimes are recorded, and only the first k failure times are considered, i.e. ( (1) < (2) < ⋯ < ( ) ). In this study, ML and Bayesian parameter estimation methods for Eq. 2 under the type-II censorship scheme are discussed.

Maximum Likelihood Estimation:
The likelihood function of Eq. 2 using the type-II censoring scheme is as follows:  However, these equations have no explicit solutions and they have to be found based on iterative solutions. Therefore, the Newton-Raphson method is used in this paper. Let ̂ and ̂ denote the MLEs of σ and μ, respectively. Now, the asymptotic distributions of the ML estimators of the unknown parameters are Here −1 ( , ) is calculated from the inverse of the observed Fisher information matrix ( , ) as given below.

Bayesian Inference:
In this subsection, the BEs of μ and σ parameters of Eq. 2 by using the type -II censored scheming under SELF, WSELF, and GELF are discussed. The loss function is very important in Bayesian parameter estimation, see [16][17][18][19] . One of the well-known loss functions is SELF. It is symmetrical so, it can be used when over and underestimates are equally serious. However, this is not a good criterion in most cases, see Helu and Samawi 16 . In such cases, asymmetric loss functions can be considered. Here, BEs of the parameters under GELF, and WSELF are discussed. These loss functions are asymmetrical. Suppose that the independent PDs of the parameters μ and σ are ( , ) and ( , ) with probability density functions ∈ ℝ and 2 ( ) ∝ −1 − 9 Then, the joint prior density function of μ, and σ is 10 where assume that ( , ) and ( , ) are nonnegative and known. They are also hyperparameters of the PD. Combining Eq. 10 with Eq. 3, the joint posterior distribution of μ and σ is obtained as Then, from Eq. 11, the posteriors of μ and σ are obtained as follows: 13 respectively. The conditional PDs of μ and σ from Eqs. 12, 13 are unknown. Therefore, two different approximations are used, namely LD and MCMC techniques. The details of them are briefly described in the following sections.

Lindley's Approximation:
Let ( , )be a function of μ and σ, then by using Eq. 11, the posterior expectation of an arbitrary function ( , ) is as follows: where ( , ) is the joint prior density function and ( , | ) is the likelihood function. The BE of ( , ) is the solution of Eq. 14. However, the BEs cannot be evaluated analytically. Therefore, the Lindley approximation method is considered to obtain BEs of unknown parameters. This method is introduced by Lindley in 1980 20 . By using Lindley's approximation method, ̂ given in Eq. 14 can be approximated as: and , , = 1,2 are given in Eq. 8. Here, ̂ and ̂ are the ML estimators of μ and σ, respectively. is given in Section 2.
Step4: Repeat steps 2-3 N times and obtain ( 1 , 1 ), … , ( , ). Now, the BEs of the μ and σ based on all these aforementioned loss functions are obtained as follows.

Methodology:
In this Section, the performances of the ML estimators and BEs are investigated via Monte Carlo simulation. The performances of these estimators are compared in terms of absolute biases  Tables 1, 3 respectively. Results of the ACIs, BCIs and the CPs are summarized in Table 2.  The estimators of the Gumbel type-I distribution using MLE and Lindley and MCMC methods are presented in Table 4. Furthermore, the ACIs and BCIs are presented in Table 5.  Although all the estimators are close to each other, there are some differences among them. Based on the simulation results given in Tables 1-3, the most suitable estimators were selected. It has been observed from the simulation results that the estimators obtained by Bayesian methods outperform the ML estimators. Therefore, it is recommended to use Bayesian estimators in these examples.

Conclusions:
Here, the classical and Bayesian estimators are investigated for Gumbel type-I distribution under the type-II censoring scheme based on various censoring rates and sample sizes. In view of classical parameter estimation, the ML estimation method is used. For the Bayesian calculations, LD and MCMC methods are obtained. Then, the Bayesian estimators under different loss functions and different prior distributions are considered. Also, the ACIs are constructed by using the ML estimators. The BCIs are also derived based on the MCMC method. The performances of all the methods discussed in the study are compared with the Monte Carlo simulation study. When the maximum likelihood estimators are compared to Bayesian estimators, it is observed that Bayesian estimators have higher efficiencies than the maximum likelihood estimators. It is also observed that the MCMC method performs slightly better than the LM. Finally, the real-life examination has illustrated the modeling capacity of the Gumbel type-I distribution. For more efficient estimators of the Gumbel type-I distribution, it is recommended to use Bayesian estimation methods with prior-I distribution and the general entropy loss function. Considering both the simulation results and these points, we may suggest that this study can be further extended considering the Gumbel type-I distribution for different censoring schemes with a Bayesian framework under symmetric and asymmetric loss functions.