Progressive censoring methodology: an appraisal

Evaluating the lifetime performance index of omega distribution based on progressive type-II censored samples

Besides achieving high quality products, statistical techniques are applied in many fields associated with health such as medicine, biology and etc. Adhering to the quality performance of an item to the desired level is a very important issue in various fields. Process capability indices play a vital role in evaluating the performance of an item. In this paper, the larger-the-better process capability index for the three-parameter Omega model based on progressive type-II censoring sample is calculated. On the basis of progressive type-II censoring the statistical inference about process capability index is carried out through the maximum likelihood. Also, the confidence interval is proposed and the hypothesis test for estimating the lifetime performance of products. Gibbs within Metropolis-Hasting samplers procedure is used for performing Markov Chain Monte Carlo (MCMC) technique to achieve Bayes estimation for unknown parameters. Simulation study is calculated to show that Omega distribution's performance is more effective. At the end of this paper, there are two real-life applications, one of them is about high-performance liquid chromatography (HPLC) data of blood samples from organ transplant recipients. The other application is about real-life data of ball bearing data. These applications are used to illustrate the importance of Omega distribution in lifetime data analysis.

Analysis of Adaptive Progressive Type-II Hybrid Censored Dagum Data with Applications

In life testing and reliability studies, obtaining whole data always takes a long time and lots of monetary and human resources. In this case, the experimenters prefer to gather data using censoring schemes that make a balance between the length of the test, the desired sample size, and the cost. Lately, an adaptive progressive type-II hybrid censoring scheme is suggested to enhance the efficiency of the statistical inference. By utilizing this scheme, this paper seeks to investigate classical and Bayesian estimations of the Dagum distribution. The maximum likelihood and Bayesian estimation methods are considered to estimate the distribution parameters and some reliability indices. The Bayesian estimation is developed under the assumption of independent gamma priors and by employing symmetric and asymmetric loss functions. Due to the tough form of the joint posterior distribution, the Markov chain Monte Carlo technique is implemented to gather samples from the full conditional distributions and in turn obtain the Bayes estimates. The approximate confidence intervals and the highest posterior density credible intervals are also obtained. The effectiveness of the various suggested methods is compared through a simulated study. The optimal progressive censoring plans are also shown, and number of optimality criteria are explored. To demonstrate the applicability of the suggested point and interval estimators, two real data sets are also examined. The outcomes of the simulation study and data analysis demonstrated that the proposed scheme is adaptable and very helpful in ending the experiment when the experimenter’s primary concern is the number of failures.

Optimal Test Plan of Step-Stress Model of Alpha Power Weibull Lifetimes under Progressively Type-II Censored Samples

In this study, the estimation of the unknown parameters of an alpha power Weibull (APW) distribution using the concept of an optimal strategy for the step-stress accelerated life testing (SSALT) is investigated from both classical and Bayesian viewpoints. We used progressive type-II censoring and accelerated life testing to reduce testing time and costs, and we used a cumulative exposure model to examine the impact of various stress levels. A log-linear relation between the scale parameter of the APW distribution and the stress model has been proposed. Maximum likelihood estimators for model parameters, as well as approximation and bootstrap confidence intervals (CIs), were calculated. Bayesian estimation of the parameter model was obtained under symmetric and asymmetric loss functions. An optimal test plan was created under typical operating conditions by minimizing the asymptotic variance (AV) of the percentile life. The simulation study is discussed to demonstrate the model’s optimality. In addition, real-world data are evaluated to demonstrate the model’s versatility.

Estimation and prediction for Burr type III distribution based on unified progressive hybrid censoring scheme

The present communication develops the tools for estimation and prediction of the Burr-III distribution under unified progressive hybrid censoring scheme. The maximum likelihood estimates of model parameters are obtained. It is shown that the maximum likelihood estimates exist uniquely. Expectation maximization and stochastic expectation maximization methods are employed to compute the point estimates of unknown parameters. Based on the asymptotic distribution of the maximum likelihood estimators, approximate confidence intervals are proposed. In addition, the bootstrap confidence intervals are constructed. Furthermore, the Bayes estimates are derived with respect to squared error and LINEX loss functions. To compute the approximate Bayes estimates, Metropolis-Hastings algorithm is adopted. The highest posterior density credible intervals are obtained. Further, maximum a posteriori estimates of the model parameters are computed. The Bayesian predictive point, as well as interval estimates, are proposed. A Monte Carlo simulation study is employed in order to evaluate the performance of the proposed statistical procedures. Finally, two real data sets are considered and analysed to illustrate the methodologies established in this paper.

On inference and design under progressive type-I interval censoring scheme for inverse Gaussian lifetime model

Purpose

This article considers Inverse Gaussian distribution as the basic lifetime model for the test units. The unknown model parameters are estimated using the method of moments, the method of maximum likelihood and Bayesian methods. As part of maximum likelihood analysis, this article employs an expectation-maximization algorithm to simplify numerical computation. Subsequently, Bayesian estimates are obtained using the Metropolis–Hastings algorithm. This article then presents the design of optimal censoring schemes using a design criterion that deals with the precision of a particular system lifetime quantile. The optimal censoring schemes are obtained after taking into account budget constraints.

Design/methodology/approach

This article first presents classical and Bayesian statistical inference for Progressive Type-I Interval censored data. Subsequently, this article considers the design of optimal Progressive Type-I Interval censoring schemes after incorporating budget constraints.

Findings

A real dataset is analyzed to demonstrate the methods developed in this article. The adequacy of the lifetime model is ensured using a simulation-based goodness-of-fit test. Furthermore, the performance of various estimators is studied using a detailed simulation experiment. It is observed that the maximum likelihood estimator relatively outperforms the method of moment estimator. Furthermore, the posterior median fares better among Bayesian estimators even in the absence of any subjective information. Furthermore, it is observed that the budget constraints have real implications on the optimal design of censoring schemes.

Originality/value

The proposed methodology may be used for analyzing any Progressive Type-I Interval Censored data for any lifetime model. The methodology adopted to obtain the optimal censoring schemes may be particularly useful for reliability engineers in real-life applications.

Type-II progressive censoring with GLM-based random removal mechanism dependent on the experimental conditions

This article presents a novel stochastic removal mechanism under Type-II progressive random censoring in which removal probabilities are allowed to be dependent on the lifetime conditions through Generalized Linear Models (GLM). These conditions potentially include failure distances (the time required to observe the next failure) or other covariate information available in the experiment. The proposed GLM-based random removal mechanism includes a set of tuning parameters that are determined by the researcher according to the possible failure distance category. These parameters allow flexible determination of the removal probabilities leading to necessary experimental cost and time reductions. To establish the proposed mechanism, the Proportional Hazard Rate (PHR) family of distributions is considered. Also, the maximum likelihood estimators of parameters and their asymptotic variances are derived for the Weibull distributed lifetime data. A simple simulation algorithm for generating Type-II progressive censoring samples with GLM-based dependent removal probabilities is also presented. The expected experiment time required to complete the life test under this censoring scheme is also investigated using the Monte Carlo integration method. Several simulation studies are conducted to evaluate and compare the performance of the proposed mechanism. A sensitivity analysis is also considered to study the effect of misspecification of removal mechanism coefficients. Finally, two real data sets are analyzed for illustrative purposes.

Progressive censoring schemes for marshall-olkin pareto distribution with applications: Estimation and prediction

In this paper two prediction methods are used to predict the non-observed (censored) units under progressive Type-II censored samples. The lifetimes of the units follow Marshall-Olkin Pareto distribution. We observe the posterior predictive density of the non-observed units and construct predictive intervals as well. Furthermore, we provide inference on the unknown parameters of the Marshall-Olkin model, so we observe point and interval estimation by using maximum likelihood and Bayesian estimation methods. Bayes estimation methods are obtained under quadratic loss function. EM algorithm is used to obtain numerical values of the Maximum likelihood method and Gibbs and the Monte Carlo Markov chain techniques are utilized for Bayesian calculations. A simulation study is performed to evaluate the performance of the estimators with respect to the mean square errors and the biases. Finally, we find the best prediction method by implementing a real data example under progressive Type-II censoring schemes.

Estimation of Reliability Indices for Alpha Power Exponential Distribution Based on Progressively Censored Competing Risks Data

In reliability analysis and life testing studies, the experimenter is frequently interested in studying a specific risk factor in the presence of other factors. In this paper, the estimation of the unknown parameters, reliability and hazard functions of alpha power exponential distribution is considered based on progressively Type-II censored competing risks data. We assume that the latent cause of failures has independent alpha power exponential distributions with different scale and shape parameters. The maximum likelihood method is considered to estimate the model parameters as well as the reliability and hazard rate functions. The approximate and two parametric bootstrap confidence intervals of the different estimators are constructed. Moreover, the Bayesian estimation method of the unknown parameters, reliability and hazard rate functions are obtained based on the squared error loss function using independent gamma priors. To get the Bayesian estimates as well as the highest posterior credible intervals, the Markov Chain Monte Carlo procedure is implemented. A comprehensive simulation experiment is conducted to compare the performance of the proposed procedures. Finally, a real dataset for the relapse of multiple myeloma with transplant-related mortality is analyzed.

Burr XII Distribution for Disease Data Analysis in the Presence of a Partially Observed Failure Mode

Modeling competing failure modes is an important problem in engineering and survival analyses. Competing failure modes are partially observed in many applications and often pose a modeling challenge. This study discusses the inference for partially observed failure modes assuming a Burr XII distribution. In particular, we consider two failure modes, and the failure time data are collected under a hybrid type I censoring scheme. The model parameters are estimated using maximum likelihood and Bayesian methods under a symmetric squared error loss function, whereas the intervals estimation is done with three methods: asymptotic and credible confidence intervals. Besides a simulation study, a real-life data set is taken from individuals who live in an environment with several diseases to present the utility of the work. Additionally, a simulation study is constructed to measure and compare different estimation methods.

Robust Optimum Life-Testing Plans under Progressive Type-I Interval Censoring Schemes with Cost Constraint

This paper considers optimal design problems for the Weibull distribution, which can be used to model symmetrical or asymmetrical data, in the presence of progressive interval censoring in life-testing experiments. Two robust approaches, Bayesian and minimax, are proposed to deal with the dependence of the D-optimality and c-optimality on the unknown model parameters. Meanwhile, the compound design method is applied to ensure a compromise between the precision of estimation of the model parameters and the precision of estimation of the quantiles. Furthermore, to make the design become more practical, the cost constraints are taken into account in constructing the optimal designs. Two algorithms are provided for finding the robust optimal solutions. A simulated example and a real life example are given to illustrate the proposed methods. The sensitivity analysis is also studied. These new design methods can help the engineers to obtain robust optimal designs for the censored life-testing experiments.

Bayesian and Non-Bayesian Inference for Weibull Inverted Exponential Model under Progressive First-Failure Censoring Data

In this article, the estimation of the parameters and the reliability and hazard functions for Weibull inverted exponential (WIE) distribution is considered based on progressive first-failure censoring (PFFC) data. For non-Bayesian inference, maximum likelihood (ML) estimators are acquired; meanwhile, their existence is verified. Via asymptotic normality of ML estimators and delta method, the corresponding confidence intervals (CIs) of the parameters and the reliability and hazard functions are constructed. For Bayesian inference, Lindley’s approximation and Markov chain Monte Carlo (MCMC) techniques are proposed to obain the Bayes estimators and the corresponding credible intervals (CRIs). To this end, both symmetric and asymmetric loss functions are used. A large number of Monte Carlo simulations are implemented to evaluate the efficiency of the developed methods. Eventually, a numerical example is analyzed for illustrative purposes.

Statistical Inference and Optimal Design of Accelerated Life Testing for the Chen Distribution under Progressive Type-II Censoring

This paper discusses statistical inference and optimal design of constant-stress accelerated life testing for the Chen distribution under progressive Type-II censoring. The scale parameter of the life distribution is assumed to be a logarithmic linear function of the stress level. The maximum likelihood estimates of the parameters are obtained. Then, the observed Fisher information matrix is derived and utilized to construct asymptotic confidence intervals. Meanwhile, the parametric bootstrap methods are provided for the interval estimation. In addition, the Bayes estimates under the squared error loss function are obtained by applying the Tierney and Kadane technique and Lindley’s approximation. As for the optimal design, D- and A-optimality criteria are considered to determine the optimal transformed stress level. Finally, the simulation is carried out to demonstrate the proposed estimation techniques and the optimal criteria, and a real data set is discussed.

Classical and Bayesian Inference on Finite Mixture of Exponentiated Kumaraswamy Gompertz and Exponentiated Kumaraswamy Fréchet Distributions under Progressive Type II Censoring with Applications

A finite mixture of exponentiated Kumaraswamy Gompertz and exponentiated Kumaraswamy Fréchet is developed and discussed as a novel probability model. We study some useful structural properties of the proposed model. To estimate the model parameters under the classical method, we use the maximum likelihood estimation using a progressive type II censoring scheme. Under the Bayesian paradigm the estimation is carried out with gamma priors under a progressive type II censored samples with squared error loss function. To demonstrate the efficiency of the proposed model based on progressively type II censoring, a simulation study is carried out. Three actual data sets are used as an example, demonstrating that the suggested model in the new class fits better than the existing finite mixture models available in the literature.

Parameter Estimation in Step Stress Partially Accelerated Life Testing under Different Types of Censored Data

A long testing period is usually required for the life testing of high-reliability products or materials. It is possible to shorten the testing process by using ALTs (accelerated life tests). Due to the fact that ALTs test products in harsher settings than are typical use conditions, the life expectancy of the objects they evaluate is reduced. Censored data in which the specific failure timings of all units assigned to test are not known, or all units assigned to test have not failed, may arise in ALTs for a variety of reasons, including operational failure, device malfunction, expense, and time restrictions. In this paper, we have considered the step stress partially accelerated life test (SSPALT) under two different censoring schemes, namely the type-I progressive hybrid censoring scheme (type-I PHCS) and the type-II progressive censorship scheme (type-II PCS). The failure times of the items are assumed to follow NH distribution, while the tampered random variable (TRV) model is used to explain the effect of stress change. In order to obtain the estimates of the unknown parameters, the maximum likelihood estimation (MLE) approach is adopted. Furthermore, based on the asymptotic theory of MLEs, the approximate confidence intervals (ACIs) are also constructed. The point estimates under two censoring schemes are compared in terms of root mean squared errors (RMSEs) and relative absolute biases (RABs), while ACIs are compared in terms of their lengths and coverage probabilities (CPs). The performance of the estimators has been evaluated and compared under two censoring schemes with various sample sizes through a simulation study. Simulation results show that estimates with type-I PHCS outperform estimates with type-II PCS in terms of RMSEs, RABs, lengths, and CPs. Finally, a real-world numerical example of insulating fluid failure times is presented to show how the approaches will work in reality.

Asymmetric Power Hazard Distribution for COVID-19 Mortality Rate under Adaptive Type-II Progressive Censoring: Theory and Inferences

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.

Optimal Design for a Bivariate Step-Stress Accelerated Life Test with Alpha Power Exponential Distribution Based on Type-I Progressive Censored Samples

We consider an optimization design for the alpha power exponential (APE) distribution as asymmetrical probability distributions under progressive type-I censoring for a step-stress accelerated life test. In this study, two stress variables are taken into account. To save the time and cost of lifetime testing, progressive censoring and accelerated life testing are utilized. The test units’ lifespans are supposed to follow an APE distribution. A cumulative exposure model is used to study the impact of varying stress levels. A log-linear relationship between the APE distribution’s scale parameter and stress is postulated. The maximum likelihood estimators, Bayesian estimators of the model parameters based on the symmetric loss function, approximate confidence intervals (CIs) and credible intervals are provided. Under normal operating conditions, an ideal test plan is designed by minimizing the asymptotic variance of the percentile life.

Inferences for Alpha Power Exponential Distribution Using Adaptive Progressively Type-II Hybrid Censored Data with Applications

One of the most important asymmetrical probability distributions that recently presented as an extension of the conventional exponential distribution is the alpha power exponential distribution. It may be compared to various asymmetrical well-known models, such as Weibull and gamma distributions. As a result, using an adaptive progressive Type-II hybrid censoring scheme, this paper investigates the estimation problems of the alpha power exponential distribution. Maximum likelihood and Bayesian methods are used to estimate unknown parameters, reliability, and hazard rate functions. Under the assumption of independent gamma priors and symmetric loss function, Bayesian estimators are examined. The Bayesian credible intervals and estimated confidence intervals of the relevant values are also calculated. The various estimating approaches are evaluated using a simulation study that considers various sample sizes and censoring schemes. Furthermore, numerous optimality criteria are examined, and the best progressive censoring schemes are offered. Finally, for an explanation, two real data sets from engineering and chemical fields are provided to show the applicability of the asymmetrical alpha power exponential distribution. The Bayesian method for estimating the parameters and reliability indices of the alpha power exponential distribution is recommended based on numerical results, especially when the number of observed data is small.

Estimation of the Generalized Logarithmic Transformation Exponential Distribution under Progressively Type-II Censored Data with Application to the COVID-19 Mortality Rates

In this paper, classical and Bayesian estimation for the parameters and the reliability function for the generalized logarithmic transformation exponential (GLTE) distribution has been proposed when the life-times are progressively censored. The maximum likelihood estimator of unknown parameters and their corresponding reliability function are obtained under the classical setup. The Bayes estimators are obtained for symmetric (squared error) and asymmetric (LINEX and general entropy) loss functions. This was achieved by considering discrete prior for the scale parameter and conditional gamma prior for the shape parameter. Interval estimation of the unknown parameters and reliability function for classical and Bayesian schemes is also considered. The performances of various derived estimators are recorded using simulation study for different sample sizes and progressive censoring schemes. Finally, the COVID-19 mortality data sets are provided to illustrate the computation of various estimators.

The lifetime analysis of the Weibull model based on Generalized Type-I progressive hybrid censoring schemes

In this study, we estimate the unknown parameters, reliability, and hazard functions using a generalized Type-I progressive hybrid censoring sample from a Weibull distribution. Maximum likelihood (ML) and Bayesian estimates are calculated using a choice of prior distributions and loss functions, including squared error, general entropy, and LINEX. Unobserved failure point and interval Bayesian predictions, as well as a future progressive censored sample, are also developed. Finally, we run some simulation tests for the Bayesian approach and numerical example on real data sets using the MCMC algorithm.

Inference for Inverse Power Lomax Distribution with Progressive First-Failure Censoring

This paper investigates the statistical inference of inverse power Lomax distribution parameters under progressive first-failure censored samples. The maximum likelihood estimates (MLEs) and the asymptotic confidence intervals are derived based on the iterative procedure and asymptotic normality theory of MLEs, respectively. Bayesian estimates of the parameters under squared error loss and generalized entropy loss function are obtained using independent gamma priors. For Bayesian computation, Tierney–Kadane’s approximation method is used. In addition, the highest posterior credible intervals of the parameters are constructed based on the importance sampling procedure. A Monte Carlo simulation study is carried out to compare the behavior of various estimates developed in this paper. Finally, a real data set is analyzed for illustration purposes.

Inference for Step-Stress Partially Accelerated Life Test Model with an Adaptive Type-I Progressively Hybrid Censored Data

Consider estimating data of failure times under step-stress partially accelerated life tests based on adaptive Type-I hybrid censoring. The mathematical model related to the lifetime of the test units is assumed to follow Rayleigh distribution. The point and interval maximum-likelihood estimations are obtained for distribution parameter and tampering coefficient. Also, the work is conducted under a traditional Type-I hybrid censoring plan (scheme). A Monte Carlo simulation algorithm is used to evaluate and compare the performances of the estimators of the tempering coefficient and model parameters under both progressively hybrid censoring plans. The comparison is carried out on the basis of mean squared errors and bias.

Information Geometry of the Exponential Family of Distributions with Progressive Type-II Censoring

In geometry and topology, a family of probability distributions can be analyzed as the points on a manifold, known as statistical manifold, with intrinsic coordinates corresponding to the parameters of the distribution. Consider the exponential family of distributions with progressive Type-II censoring as the manifold of a statistical model, we use the information geometry methods to investigate the geometric quantities such as the tangent space, the Fisher metric tensors, the affine connection and the α-connection of the manifold. As an application of the geometric quantities, the asymptotic expansions of the posterior density function and the posterior Bayesian predictive density function of the manifold are discussed. The results show that the asymptotic expansions are related to the coefficients of the α-connections and metric tensors, and the predictive density function is the estimated density function in an asymptotic sense. The main results are illustrated by considering the Rayleigh distribution.

Applying Transformer Insulation Using Weibull Extended Distribution Based on Progressive Censoring Scheme

In this paper, the Weibull extension distribution parameters are estimated under a progressive type-II censoring scheme with random removal. The parameters of the model are estimated using the maximum likelihood method, maximum product spacing, and Bayesian estimation methods. In classical estimation (maximum likelihood method and maximum product spacing), we did use the Newton–Raphson algorithm. The Bayesian estimation is done using the Metropolis–Hastings algorithm based on the square error loss function. The proposed estimation methods are compared using Monte Carlo simulations under a progressive type-II censoring scheme. An empirical study using a real data set of transformer insulation and a simulation study is performed to validate the introduced methods of inference. Based on the result of our study, it can be concluded that the Bayesian method outperforms the maximum likelihood and maximum product-spacing methods for estimating the Weibull extension parameters under a progressive type-II censoring scheme in both simulation and empirical studies.

Approximated optimal design for a bivariate step-stress accelerated life test with generalized exponential distribution under type-I progressive censoring

Purpose

In this paper, the author proposed an optimization design for a step-stress accelerated life test (SSALT) with two stress variables for the generalized exponential (GE) distribution under progressive type-I censoring.

Design/methodology/approach

In this paper, two stress variables were considered. Progressive censoring and accelerated life testing were used to reduce the time and cost of testing. It was assumed that the lifetimes of the test units followed a GE distribution. The effects of changing stress were considered as a cumulative exposure model. A log-linear relationship between the scale parameter of the GE distribution and the stress was proposed. The maximum likelihood estimators and approximate and bootstrap confidence intervals (CIs) for the model parameters were obtained. An optimum test plan was developed using minimization of the asymptotic variance (AV) of the percentile life under the usual operating condition.

Findings

According to the simulation results, the bootstrap CIs of the model parameters gave more accurate results than approximate CIs through the length of CIs. The sensitivity analysis was performed to illustrate the effect of initial estimates on optimal values that has been studied. Simulation results also indicated that the optimal times were not too sensitive to the initial values of parameters; thus, the proposed design was robust.

Originality/value

In most studies, only one accelerating stress variable is used. Sometimes accelerating one stress variable does not yield enough failure data. Thus, two stress variables may be needed for additional acceleration. In this paper, two stress variables are considered. The inclusion of two stress variables in a test design will lead to a better understanding of the effect of two simultaneously operating stress variables. Also, the author assumes that the failure time of the test units follows a GE distribution. It is observed that the GE distribution can be used quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions. Also, most studies in this field have focused on the derivation of optimum test plans. In this paper, the author examined the estimation of model parameters and the optimization of the test design. In this paper, the asymptotic and bootstrap CIs for the model parameters are calculated. In addition, a sensitivity analysis is performed to examine the effect of the changes in the pre-estimated parameters on the optimal hold times. For determining the optimal test plan, due to nonlinearity and complexity of the objective function, the particle swarm optimization (PSO) algorithm is developed to calculate the optimal hold times. In this method, the research speed is very fast and optimization ability is more.

Estimation for Entropy and Parameters of Generalized Bilal Distribution under Adaptive Type II Progressive Hybrid Censoring Scheme

Entropy measures the uncertainty associated with a random variable. It has important applications in cybernetics, probability theory, astrophysics, life sciences and other fields. Recently, many authors focused on the estimation of entropy with different life distributions. However, the estimation of entropy for the generalized Bilal (GB) distribution has not yet been involved. In this paper, we consider the estimation of the entropy and the parameters with GB distribution based on adaptive Type-II progressive hybrid censored data. Maximum likelihood estimation of the entropy and the parameters are obtained using the Newton–Raphson iteration method. Bayesian estimations under different loss functions are provided with the help of Lindley’s approximation. The approximate confidence interval and the Bayesian credible interval of the parameters and entropy are obtained by using the delta and Markov chain Monte Carlo (MCMC) methods, respectively. Monte Carlo simulation studies are carried out to observe the performances of the different point and interval estimations. Finally, a real data set has been analyzed for illustrative purposes.