E-Bayesian estimation of the exponentiated distribution family parameter under LINEX loss function

This paper is concerned with using the E-Bayesian method for computing estimates of the exponentiated distribution family parameter. Based on the LINEX loss function, formulas of E-Bayesian estimation for unknown parameter are given, these estimates are derived based on a conjugate prior. Moreover, property of E-Bayesian estimation—the relationship between of E-Bayesian estimations under different prior distributions of the hyper parameters are also provided. A comparison between the new method and the corresponding maximum likelihood techniques is conducted using the Monte Carlo simulation. Finally, combined with the golfers income data practical problem are calculated, the results show that the proposed method is feasible and convenient for application.     

Interval estimation for the Pareto distribution based on the progressive Type II censored sample

For the complete sample and the right Type II censored sample, Chen [Joint confidence region for the parameters of Pareto distribution. Metrika 44 (1996), pp. 191–197] proposed the interval estimation of the parameter θ and the joint confidence region of the two parameters of Pareto distribution. This paper proposed two methods to construct the confidence region of the two parameters of the Pareto distribution for the progressive Type II censored sample. A simulation study comparing the performance of the two methods is done and concludes that Method 1 is superior to Method 2 by obtaining a smaller confidence area. The interval estimation of parameter ν is also given under progressive Type II censoring. In addition, the predictive intervals of the future observation and the ratio of the two future consecutive failure times based on the progressive Type II censored sample are also proposed. Finally, one example is given to illustrate all interval estimations in this paper.     

Efficient estimation in the Pareto distribution

The maximum likelihood estimation (MLE) of the probability density function (pdf) and cumulative distribution function (CDF) are derived for the Pareto distribution. It has been shown that MLEs are more efficient than uniform minimum variance unbiased estimators of pdf and CDF.     

Optimal estimation for the Pareto distribution

This paper proposes an optimal estimation method for the shape parameter, probability density function and upper tail probability of the Pareto distribution. The new method is based on a weighted empirical distribution function. The exact efficiency functions of the estimators relative to the existing estimators are derived. The paper gives L ₁-optimal and L ₂-optimal weights for the new weighted estimator. Monte Carlo simulation results confirm the theoretical conclusions. Both theoretical and simulation results show that the new estimation method is more efficient relative to several existing methods in many situations.     

Bayesian estimation for the Pareto income distribution

The Bayes estimators of the Gini index, the mean income and the proportion of the population living below a prescribed income level are obtained in this paper on the basis of censored income data from a pareto income distribution. The said estimators are obtained under the assumptions of a two-parameter exponential prior distribution and the usual squared error loss function. This work is also extended to the case when the income data are grouped and the exact incomes for the individuals in the population are not available. The method for the assessment of the hyperparameters is also outlined. Finally, the results are generalized for the doubly truncated gamma prior distribution. Now deceased.     

Parameter and reliability estimation for an exponentiated half-logistic distribution under progressive type II censoring

In this article, we deal with a two-parameter exponentiated half-logistic distribution. We consider the estimation of unknown parameters, the associated reliability function and the hazard rate function under progressive Type II censoring. Maximum likelihood estimates (M LEs) are proposed for unknown quantities. Bayes estimates are derived with respect to squared error, linex and entropy loss functions. Approximate explicit expressions for all Bayes estimates are obtained using the Lindley method. We also use importance sampling scheme to compute the Bayes estimates. Markov Chain Monte Carlo samples are further used to produce credible intervals for the unknown parameters. Asymptotic confidence intervals are constructed using the normality property of the MLEs. For comparison purposes, bootstrap-p and bootstrap-t confidence intervals are also constructed. A comprehensive numerical study is performed to compare the proposed estimates. Finally, a real-life data set is analysed to illustrate the proposed methods of estimation.     

On progressively censored inverted exponentiated Rayleigh distribution

In this paper, we discuss a progressively censored inverted exponentiated Rayleigh distribution. Estimation of unknown parameters is considered under progressive censoring using maximum likelihood and Bayesian approaches. Bayes estimators of unknown parameters are derived with respect to different symmetric and asymmetric loss functions using gamma prior distributions. An importance sampling procedure is taken into consideration for deriving these estimates. Further highest posterior density intervals for unknown parameters are constructed and for comparison purposes bootstrap intervals are also obtained. Prediction of future observations is studied in one- and two-sample situations from classical and Bayesian viewpoint. We further establish optimum censoring schemes using Bayesian approach. Finally, we conduct a simulation study to compare the performance of proposed methods and analyse two real data sets for illustration purposes.     

Bayesian estimation for the exponentiated Weibull model under Type II progressive censoring

Based on progressive Type II censored samples, we have derived the maximum likelihood and Bayes estimators for the two shape parameters and the reliability function of the exponentiated Weibull lifetime model. We obtained Bayes estimators using both the symmetric and asymmetric loss functions via squared error loss and linex loss functions. This was done with respect to the conjugate priors for two shape parameters. We used an approximation based on the Lindley (Trabajos de Stadistca 21, 223–237, 1980) method for obtaining Bayes estimates under these loss functions. We made comparisons between these estimators and the maximum likelihood estimators using a Monte Carlo simulation study.     

Reducing bias of the maximum-likelihood estimation for the truncated Pareto distribution

The Pareto distribution is an important distribution in statistics, which has been widely used in finance, physics, hydrology, geology, astronomy, and so on. Even though the parameter estimation for the Pareto distribution has been well established in the literature, the estimation problem for the truncated Pareto distribution becomes complex. This article investigates the bias and mean-squared error of the maximum-likelihood estimation for the truncated Pareto distribution, and some useful results are obtained.     

On detecting outliers in the Pareto distribution

In this paper, we introduce two new statistics for detecting outliers in the Pareto distribution. These new statistics are the extension of the statistics for detecting outliers in exponential and gamma distributions. In fact, we compare the power of our test statistics with the other statistics and select the best test statistic for detecting outliers in the Pareto distribution. Finally, numerical examples of different insurance claims are used to see the performance of the test.     

Bias-corrected maximum likelihood estimation of the parameters of the generalized Pareto distribution

ABSTRACT

We derive analytic expressions for the biases, to O(n^?1), of the maximum likelihood estimators of the parameters of the generalized Pareto distribution. Using these expressions to bias-correct the estimators in a selective manner is found to be extremely effective in terms of bias reduction, and can also result in a small reduction in relative mean squared error (MSE). In terms of remaining relative bias, the analytic bias-corrected estimators are somewhat less effective than their counterparts obtained by using a parametric bootstrap bias correction. However, the analytic correction out-performs the bootstrap correction in terms of remaining %MSE. It also performs credibly relative to other recently proposed estimators for this distribution. Taking into account the relative computational costs, this leads us to recommend the selective use of the analytic bias adjustment for most practical situations.     

The E-Bayesian and hierarchical Bayesian estimations of Pareto distribution parameter under different loss functions

In this paper, we study the E-Bayesian and hierarchical Bayesian estimations of the parameter derived from Pareto distribution under different loss functions. The definition of the E-Bayesian estimation of the parameter is provided. Moreover, for Pareto distribution, under the condition of the scale parameter is known, based on the different loss functions, formulas of the E-Bayesian estimation and hierarchical Bayesian estimations for the shape parameter are given, respectively, properties of the E-Bayesian estimation – (i) the relationship between of E-Bayesian estimations under different loss functions are provided, (ii) the relationship between of E-Bayesian and hierarchical Bayesian estimations under the same loss function are also provided, and using the Monte Carlo method simulation example is given. Finally, combined with the golfers income data practical problem are calculated, the results show that the proposed method is feasible and convenient for application.     

On the mixtures of Weibull and Pareto (IV) distribution: An alternative to Pareto distribution

Finite mixture models have provided a reasonable tool to model various types of observed phenomena, specially those which are random in nature. In this article, a finite mixture of Weibull and Pareto (IV) distribution is considered and studied. Some structural properties of the resulting model are discussed including estimation of the model parameters via expectation maximization (EM) algorithm. A real-life data application exhibits the fact that in certain situations, this mixture model might be a better alternative than the rival popular models.     

A matching prior for extreme quantile estimation of the generalized Pareto distribution

Extreme quantile estimation plays an important role in risk management and environmental statistics among other applications. A popular method is the peaks-over-threshold (POT) model that approximate the distribution of excesses over a high threshold through generalized Pareto distribution (GPD). Motivated by a practical financial risk management problem, we look for an appropriate prior choice for Bayesian estimation of the GPD parameters that results in better quantile estimation. Specifically, we propose a noninformative matching prior for the parameters of a GPD so that a specific quantile of the Bayesian predictive distribution matches the true quantile in the sense of Datta et al. (2000).     

Estimation and prediction for an inverted exponentiated Rayleigh distribution under hybrid censoring

In this paper we consider estimation of unknown parameters of an inverted exponentiated Rayleigh distribution when it is known that data are hybrid Type I censored. The maximum likelihood and Bayes estimates are derived. In sequel interval estimates are also constructed. We further consider one- and two-sample prediction of future observations and also obtain prediction intervals. The performance of proposed methods of estimation and prediction is studied using simulations and an illustrative example is discussed in support of the suggested methods.     

Estimation for an inverted exponentiated Rayleigh distribution under type II progressive censoring

In this paper, we consider estimation of unknown parameters of an inverted exponentiated Rayleigh distribution under type II progressive censored samples. Estimation of reliability and hazard functions is also considered. Maximum likelihood estimators are obtained using the Expectation–Maximization (EM) algorithm. Further, we obtain expected Fisher information matrix using the missing value principle. Bayes estimators are derived under squared error and linex loss functions. We have used Lindley, and Tiernery and Kadane methods to compute these estimates. In addition, Bayes estimators are computed using importance sampling scheme as well. Samples generated from this scheme are further utilized for constructing highest posterior density intervals for unknown parameters. For comparison purposes asymptotic intervals are also obtained. A numerical comparison is made between proposed estimators using simulations and observations are given. A real-life data set is analyzed for illustrative purposes.