On E-Bayesian estimations for the cumulative hazard rate and mean residual life under generalized inverted exponential distribution and type-II censoring

Estimation of reliability and hazard rate is necessary in many applications. To this aim, different methods of estimation have been employed. Each method suffers from its own problems such as complexity of calculations, high risk and so on. Toward this end, this study employed a new method, E-Bayesian, for estimating the parametric functions of the Generalized Inverted Exponential distribution, which is one of the most noticeable distributions in lifetime studies. Relations are derived under a squared error loss function, type-II censoring and a conjugate prior. E-Bayesian estimations are obtained based on different priors of the hyperparameters to investigate the influence of different prior distributions on these estimations. The asymptotic behaviors of E-Bayesian estimations and relations among them have been investigated. Finally, a comparison among the maximum likelihood, Bayes and E-Bayesian estimations in different sample sizes are made, using a real data and the Monte Carlo simulation. Simulations show that the new presented method is more efficient than previous methods and is also easy to operate. Also, some comparisons among the results of Generalized Inverted Exponential distribution, Exponential distribution and Generalized Exponential distribution are provided.KEYWORDS: E-Bayesian estimation, generalized Inverted exponential distribution, type-II censoring, reliability, hazard rate, Monte Carlo simulation     

On a Selection Weibull Distribution

In this article, a selection Weibull distribution is investigated. First, some properties and representations of the model with some plots of the density and hazard rate functions are illustrated. Second, some simple relations of this model with some distributions discussed. In addition, maximum likelihood estimators obtained with numerical methods, and compared by three sub-models with a data set that shows the performance of our model. Finally, a simulation study presented for all parameters.     

Unbiased Weibull Modulus Estimation Using Differential Cumulative Entropy

Recently, Liu (2007 Liu , J. ( 2007 ). Information Theoretic Content and Probability. Ph.D. Thesis. University of Florida, Gainesville, FL . [Google Scholar]) defined a new entropy which measures the distance between a prescribed and an empirical survival function. In this article, we use this measure called Differential Cumulative Entropy (DCE) for Weibull parameters estimation. We show that the DCE method provides biased estimations of the Weibull modulus, but utilizing unbiasing factors derived here we enhance the results. A simulation study shows the higher performance of the new method over commonly used maximum likelihood and linear regression methods in Weibull parameters estimation especially in small sample sizes.     

Estimation of the exponential Pareto II distribution parameters

In this article, we estimate the parameters of exponential Pareto II distribution by two new methods. The first one is based on the principle of maximum entropy (POME) and the second is by Kullback–Leibler divergence of survival function (KLS). Monte Carlo simulated data are used to evaluate these methods and compare them with the maximum likelihood method. Finally, we fit this distribution to a set of real data by estimation procedures.     

Aspects concerning entropy and utility

Expected utility maximization problem is one of the most useful tools in mathematical finance, decision analysis and economics. Motivated by statistical model selection, via the principle of expected utility maximization, Friedman and Sandow (J Mach Learn Res 4:257–291, 2003a) considered the model performance question from the point of view of an investor who evaluates models based on the performance of the optimal strategies that the models suggest. They interpreted their performance measures in information theoretic terms and provided new generalizations of Shannon entropy and Kullback–Leibler relative entropy and called them U-entropy and U-relative entropy. In this article, a utility-based criterion for independence of two random variables is defined. Then, Markov’s inequality for probabilities is extended from the U-entropy viewpoint. Moreover, a lower bound for the U-relative entropy is obtained. Finally, a link between conditional U-entropy and conditional Renyi entropy is derived.     

Weighted Renyi's entropy for lifetime distributions

Recently, Di Crescenzo and Longobardi (2006 Di Crescenzo, A., Longobardi, M. (2006). On weighted residual and past entropies. Sci. Math. Jpn. 64:255–266. [Google Scholar]) have studied “length-biased” shift-dependent information measure and its dynamic versions. On the other hand, Renyi's entropy plays a vital role in the literature of information theory that is a generalization of Shannon's entropy. In this article, the concepts of weighted Renyi's entropy, weighted residual Renyi's entropy, and weighted past Renyi's entropy are introduced and their properties are discussed.     

On generalized order statistics from generalized pareto distribution

The aim of this paper is to estimate parameters of generalized Pareto distribution based on generalized order statistics. Some non-Bayesian methods, such as MLE, bootstrap and unbiased estimators have been obtained to develop point and interval estimations. Bayesian estimations have also been derived under LSE and LINEX loss functions. To compare the performances of the employed methods, numerical results have been computed. To illustrate dependence and association properties of generalized order statistics, correlation coefficient and some informational measures in closed form have been obtained.     

Maximum entropy estimation of income share function from generalized Gini index

Following Sir Anthony and Atkinson who started thinking about the insensitivity of the Gini index to income shares of the lower and the upper income groups, a generalization of the classical Gini index was introduced by Kakwani, Donaldson, Weymark and Yitzhaki which is sensitive to both high and low incomes. In this paper, the maximum entropy method is used to estimate the underlying true income share function based on the limited information of the generalized Gini index about the income shares of a population's percentiles. The income share function is estimated through maximizing both the Shannon entropy and the second-order entropy. In the end, through parametric bootstrap and analyzing a real dataset, the results are compared with the estimator of the share function, which is obtained based on the total information. In contrast to the classic Gini index, the derived share function based on the generalized Gini index provides more accurate approximations for income shares of the lower and the upper percentiles.     

Information Measures for Some Well-Known Families

In this article, results for some well-known families such as Pareto IV, generalized logistic, generalized gamma, double generalized gamma, generalized normal, inverse gamma, and Weibull, and their related families via the links between entropy, variance, Fisher information, and analog of the Fisher information, are derived.