General Entropy Loss Research Articles

A new lifetime distribution for a series-parallel system: properties, applications and estimations under progressive type-II censoring

ABSTRACTA compound class of zero truncated Poisson and lifetime distributions is introduced. A specialization is paved to a new three-parameter distribution, called doubly Poisson-exponential distribution, which may represent the lifetime of units connected in a series-parallel system. The new distribution can be obtained by compounding two zero truncated Poisson distributions with an exponential distribution. Among its motivations is that its hazard rate function can take different shapes such as decreasing, increasing and upside-down bathtub depending on the values of its parameters. Several properties of the new distribution are discussed. Based on progressive type-II censoring, six estimation methods [maximum likelihood, moments, least squares, weighted least squares and Bayes (under linear-exponential and general entropy loss functions) estimations] are used to estimate the involved parameters. The performance of these methods is investigated through a simulation study. The Bayes estimates are obtained using Markov chain Monte Carlo algorithm. In addition, confidence intervals, symmetric credible intervals and highest posterior density credible intervals of the parameters are obtained. Finally, an application to a real data set is used to compare the new distribution with other five distributions.

Approximate Bayes Estimators of the Logistic Distribution Parameters Based on Progressive Type-II Censoring Scheme

In this paper we develop approximate Bayes estimators of the parameters, reliability, and hazard rate functions of the Logistic distribution by using Lindley’s approximation, based on progressively type-II censoring samples. Noninformative prior distributions are used for the parameters. Quadratic, linex and general Entropy loss functions are used. The statistical performances of the Bayes estimates relative to quadratic, linex and general entropy loss functions are compared to those of the maximum likelihood based on simulation study.

Bayesian estimation on interval censored Lindley distribution using Lindley’s approximation

Interval censored data commonly arise in engineering and biomedical sciences. The present study deals with Bayesian estimation of interval censored lifetime data while it is assumed that lifetimes follow Lindley distribution. Assuming Jeffrey’s and gamma prior distributions, Bayes estimator of the Lindley parameter has been constructed under symmetric, squared error loss and asymmetric, general entropy loss functions. In addition, Bayes estimators for mean life, reliability and hazard rate have also been constructed. Since posterior distribution can not be reduced to any standard distribution, Lindley’s approximation technique has been utilized for Bayesian computations. The performances of the Bayes estimators has been compared with corresponding maximum likelihood estimators on the basis of simulated samples. Real data sets from engineering and biomedical fields have been analysed for illustration purposes .

Estimation and prediction for the generalized inverted exponential distribution based on progressively first-failure-censored data with application

ABSTRACTIn this paper, the estimation of parameters for a generalized inverted exponential distribution based on the progressively first-failure type-II right-censored sample is studied. An expectation–maximization (EM) algorithm is developed to obtain maximum likelihood estimates of unknown parameters as well as reliability and hazard functions. Using the missing value principle, the Fisher information matrix has been obtained for constructing asymptotic confidence intervals. An exact interval and an exact confidence region for the parameters are also constructed. Bayesian procedures based on Markov Chain Monte Carlo methods have been developed to approximate the posterior distribution of the parameters of interest and in addition to deduce the corresponding credible intervals. The performances of the maximum likelihood and Bayes estimators are compared in terms of their mean-squared errors through the simulation study. Furthermore, Bayes two-sample point and interval predictors are obtained when the future sample is ordinary order statistics. The squared error, linear-exponential and general entropy loss functions have been considered for obtaining the Bayes estimators and predictors. To illustrate the discussed procedures, a set of real data is analyzed.

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

ABSTRACTIn this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.

Estimation using hybrid censored data from a generalized inverted exponential distribution

ABSTRACTWe consider point and interval estimation of the unknown parameters of a generalized inverted exponential distribution in the presence of hybrid censoring. The maximum likelihood estimates are obtained using EM algorithm. We then compute Fisher information matrix using the missing value principle. Bayes estimates are derived under squared error and general entropy loss functions. Furthermore, approximate Bayes estimates are obtained using Tierney and Kadane method as well as using importance sampling approach. Asymptotic and highest posterior density intervals are also constructed. Proposed estimates are compared numerically using Monte Carlo simulations and a real data set is analyzed for illustrative purposes.

Bayes Shrinkage Estimation of the Parameter of Rayleigh Distribution for Progressive Type-II Censored Data

This paper derives Bayes shrinkage estimator of Rayleigh parameter and its associated risk based on conjugate prior under the assumption of general entropy loss function for progressive type-II censored data. Risk function of maximum likelihood estimate, Bayes estimate and Bayes shrinkage estimate have also been derived and compared. A procedure has been suggested to include a guess value in case of the Bayes shrinkage estimation. Risk function of empirical Bayes estimate and empirical Bayes shrinkage estimate have also been derived and compared. In conclusion, an illustrative example is presented to assess how the Rayleigh distribution fits a real data set.

Bayesian estimation for non zero inflated modified power series distribution under linex and generalized entropy loss functions

In this paper, we derive Bayesian estimators of the parameters of modified power series distributions inflated at any of a support point under linex and general entropy loss function. We assume that the prior information can be summarized by a uniform, Beta, two-sided power, Gamma or generalized Pareto distributions. The obtained results are demonstrated on the generalized Poisson and the generalized negative binomial distribution inflated at a given point.

Bayesian parameter learning with an application

This paper deals with prior uncertainty in the parameter learning procedure in Bayesian networks. In most studies in the literature, parameter learning is based on two well-known criteria, i.e., the maximum likelihood and the maximum a posteriori. In presence of prior information, the literature abounds with situations in which a maximum a posteriori estimate is computed as a desired estimate but in those studies, it does not seem that the viewpoint behind its use is according to a loss function-based viewpoint. In this paper, we recall the maximum a posteriori estimator as the Bayes estimator under the zero-one loss function and criticizing the zero-one loss, we suggest the use of the general Entropy loss function as a useful loss when overlearning and underlearning need serious attention. We take prior uncertainty into account and extend the act of parameter learning for the case when prior information is polluted. Addressing a real world problem, we conduct a simulation procedure to study behavior of the proposed estimates. Finally, in order to seek the effect of changing hyperparameters of a chosen prior on the learning procedure, we carry out a sensitivity analysis w.r.t. some chosen hyperparameters.

Posterior analysis of the compound Rayleigh distribution under balanced loss functions for censored data

ABSTRACTThis paper develops Bayesian analysis in the context of progressively Type II censored data from the compound Rayleigh distribution. The maximum likelihood and Bayes estimates along with associated posterior risks are derived for reliability performances under balanced loss functions by assuming continuous priors for parameters of the distribution. A practical example is used to illustrate the estimation methods. A simulation study has been carried out to compare the performance of estimates. The study indicates that Bayesian estimation should be preferred over maximum likelihood estimation. In Bayesian estimation, the balance general entropy loss function can be effectively employed for optimal decision-making.

Bayesian estimation based on generalized order statistics from exponentiated Weibull Poisson model

In this research paper, the estimation of the unknown parameters for the exponentiated Weibull Poisson distribution using the concept of generalized order statistics is investigated from Bayesian approach. The squared error, LINEX and general entropy loss functions are considered for Bayesian computation. Bayes estimates based on Progressively type II censored and the joint density function of ordinary order statistics are considered as special cases of generalized order statistics. Finally simulation study is conducted for illustrative purposes.

Estimating the Entropy of a Weibull Distribution under Generalized Progressive Hybrid Censoring

Recently, progressive hybrid censoring schemes have become quite popular in a life-testing problem and reliability analysis. However, the limitation of the progressive hybrid censoring scheme is that it cannot be applied when few failures occur before time T. Therefore, a generalized progressive hybrid censoring scheme was introduced. In this paper, the estimation of the entropy of a two-parameter Weibull distribution based on the generalized progressively censored sample has been considered. The Bayes estimators for the entropy of the Weibull distribution based on the symmetric and asymmetric loss functions, such as the squared error, linex and general entropy loss functions, are provided. The Bayes estimators cannot be obtained explicitly, and Lindley’s approximation is used to obtain the Bayes estimators. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, a real dataset has been analyzed for illustrative purposes.

The Inverse Weibull Distribution as a Failure Model Under Various Loss Functions and Based on Progressive First-Failure Censored Data

In this article we consider statistical inferences about the unknown parameters of the inverse Weibull distribution based on progressively first-failure censoring using Bayesian procedures. The Bayes estimators are obtained based on both the symmetric and asymmetric (Linex, General Entropy and Precautionary) loss functions. There are no explicit forms for the Bayes estimators; therefore, we propose the Lindley’s approximation method to compute the Bayes estimators. A comparison between these estimators and the maximum likelihood estimator (MLE) is provided by using extensive simulation and two criteria, namely, the bias and the mean squared error. It is concluded that the approximate Bayes estimators outperform the MLEs most of the time. Real life data example is provided to illustrate our proposed estimators.

Estimation in Maxwell distribution with randomly censored data

In many practical situations, complete data are not available in lifetime studies. Many of the available observations are right censored giving survival information up to a noted time and not the exact failure times. This constitutes randomly censored data. In this paper, we consider Maxwell distribution as a survival time model. The censoring time is also assumed to follow a Maxwell distribution with a different parameter. Maximum likelihood estimators and confidence intervals for the parameters are derived with randomly censored data. Bayes estimators are also developed with inverted gamma priors and generalized entropy loss function. A Monte Carlo simulation study is performed to compare the developed estimation procedures. A real data example is given at the end of the study.

Bayesian inference for exponentiated Pareto model with application to bladder cancer remission time

Maximum likelihood and Bayes estimators of the unknown parameters and the expected experiment times of the exponentiated Pareto model have been obtained for progressive type-II censored data with binomial removal scheme. Markov Chain Monte Carlo (MCMC) method is used to compute the Bayes estimates of the parameters of interest. The generalized entropy loss function and squared error loss function have been considered for obtaining the Bayes estimators. Comparisons are made between Bayesian and maximum likelihood (ML) estimators via Monte Carlo simulation. The proposed methodology is illustrated through real data.

Classical and Bayesian Inference for an Extension of the Exponential Distribution under Progressive Type-II Censored Data with Binomial Removals

Maximum likelihood and Bayes estimators of the unknown parameters of an extension of the exponential (EE) distribution have been obtained for Progressive Type-II Censored data with Binomial removals. Markov Chain Monte Carlo (MCMC) method is used to compute the Bayes estimates of the parameters of interest. The General Entropy Loss Function (GELF) and Squared Error Loss Function (SELF) have been considered for obtaining the Bayes estimators. Comparisons are made between Bayesian and Maximum likelihood estimators (MLEs) via Monte Carlo simulation. An example is discussed to illustrate its applicability.

Bayes estimation of change point in the count data model: a Particular case of Discrete Burr Type III Distribution

A sequence of independent count data X1, X2..., Xm, Xm+1,...., Xnwhere observations from a particular case of discrete Burr family-type III distribution with distribution function F1(t) at time t later it was found that there was change in the system at some point of time m and it is a reflected in the sequence Xmby change in distribution function F2(t) at time t. The Bayes estimates of change point and parameters of Particular case of Bur Type III Distribution are derived under Linex and General Entropy loss functions.

Estimation of the inverse Weibull distribution based on progressively censored data: Comparative study

In this article we consider statistical inferences about the unknown parameters of the Inverse Weibull distribution based on progressively type-II censoring using classical and Bayesian procedures. For classical procedures we propose using the maximum likelihood; the least squares methods and the approximate maximum likelihood estimators. The Bayes estimators are obtained based on both the symmetric and asymmetric (Linex, General Entropy and Precautionary) loss functions. There are no explicit forms for the Bayes estimators, therefore, we propose Lindley׳s approximation method to compute the Bayes estimators. A comparison between these estimators is provided by using extensive simulation and three criteria, namely, Bias, mean squared error and Pitman nearness (PN) probability. It is concluded that the approximate Bayes estimators outperform the classical estimators most of the time. Real life data example is provided to illustrate our proposed estimators.

Double Stage Shrinkage Estimation of the Reliability Function of the Proportional Hazard Family of Distribution Function under Different Loss Functions Using Progressive Type II Censored Sample

In this paper we propose two double stage shrinkage estimators of the reliability function of the proportional hazard family of distribution functions, using progressive type II censored sample. The risk functions and relative risk functions of the suggested estimators are derived under symmetric and asymmetric loss functions, viz., the squared error loss function and the general entropy loss function. The results show that the proposed estimators have better performance than a classical estimator in terms relative risk.

Bayesian Inference Based on the Interval Grouped Data from the Weibull Model with Application

In this paper, based on the interval grouped data, Bayesian approach is used to obtain estimator for the Weibull scale parameter and some lifetime such as the reliability and hazard functions. The estimation procedures have been developed and modified under squared error loss function (SELF) and general entropy loss function (GELF). The estimators are derived using the inverted gamma conjugate prior. Credible intervals and high posterior density (HPD) credible intervals are also obtained. Prediction for the future number of failures in the corresponding intervals is presented. Finally, real life example is applied to illustrate the performance of the estimation procedures.