Estimating risk reduction in stein estimation

It is shown that the unbiased estimator of the risk reduction in Stein estimation is unsatisfactory from a mean-squared-error point of view. A truncated form of the unbiased estimator and various empirical Bayes estimators of the risk reduction are shown to perform much better than the unbiased estimator. A simple practical estimator is proposed whose performance is a compromise between that of the truncated and empirical Bayes estimators.     

Bayes/frequentist compromise decision rules for Gaussian sampling

Bayesian methods have the potential to confer substantial advantages over frequentist when the assumed prior is approximately correct, but otherwise can perform poorly. Therefore, estimators and other inferences that strike a compromise between Bayes and frequentist optimality are attractive. To evaluate potential trade-offs, we study Bayes vs. frequentist risk under Gaussian sampling for families of point estimators and interval estimators. Bayes/frequentist compromises for interval estimation are more challenging than for point estimation, since performance involves an interplay between coverage and length. Each family allows ‘purchasing’ improved frequentist performance by allowing a small increase in Bayes risk over the Bayes rule. Any degree of increase can be specified, thus enabling greater or lesser trade-offs between Bayes and frequentist risk.     

Bayesian estimation of mean and square of mean of normal distribution using linex loss function

In this paper, the Bayes estimators for mean and square of mean ol a normal distribution with mean μ and vaiiance σ r2 (known), relative to LINEX loss function are obtained Comparisons in terms of risk functions and Bayes risks of those under LINEX loss and squared error loss functions with their respective alternative estimators viz, UMVUE and Bayes estimators relative to squared error loss function, are made. It is found that Bayes estimators relative to LINEX loss function dominate the alternative estimators m terms of risk function snd Bayes risk. It is also found that if t2 is unknown the Bayes estimators are still preferable over alternative estimators.     

ESTIMATION OF EXPONENTIATED WEIBULL SHAPE PARAMETERS UNDER LINEX LOSS FUNCTION

ABSTRACT

The paper deals with Bayes estimation of the exponentiated Weibull shape parameters under linex loss function when independent non-informative type of priors are available for the parameters. Generalized maximum likelihood estimators have also been obtained. Performances of the proposed Bayes estimator, generalized maximum likelihood estimators, posterior mean (i.e., Bayes estimator under squared error loss function) and maximum likelihood estimators have been studied on the basis of their risks under linex loss function. The comparison is based on a simulation study because the expressions for risk functions of these estimators cannot be obtained in nice closed forms.     

Bayes admissible estimation of the means in Poisson decomposable graphical models

We investigate the Bayes estimation of the means in Poisson decomposable graphical models. Some classes of Bayes estimators are provided which improve on the maximum likelihood estimator under the normalized squared error loss. Both proper and improper priors are included in the proposed classes of priors. Concerning the generalized Bayes estimators with respect to the improper priors, we address their admissibility.     

Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring

We consider the problem of estimating unknown parameters, reliability function and hazard function of a two parameter bathtub-shaped distribution on the basis of progressive type-II censored sample. The maximum likelihood estimators and Bayes estimators are derived for two unknown parameters, reliability function and hazard function. The Bayes estimators are obtained against squared error, LINEX and entropy loss functions. Also, using the Lindley approximation method we have obtained approximate Bayes estimators against these loss functions. Some numerical comparisons are made among various proposed estimators in terms of their mean square error values and some specific recommendations are given. Finally, two data sets are analyzed to illustrate the proposed methods.     

Estimation of three-parameter exponentiated-Weibull distribution under type-II censoring

The present article obtains the point estimators of the exponentiated-Weibull parameters when all the three parameters of the distribution are unknown. Maximum likelihood estimator generalized maximum likelihood estimator and Bayes estimators are proposed for three-parameter exponentiated-Weibull distribution when available sample is type-II censored. Independent non-informative types of priors are considered for the unknown parameters to develop generalized maximum likelihood estimator and Bayes estimators. Although the proposed estimators cannot be expressed in nice closed forms, these can be easily obtained through the use of appropriate numerical techniques. The performances of these estimators are studied on the basis of their risks, computed separately under LINEX loss and squared error loss functions through Monte-Carlo simulation technique. An example is also considered to illustrate the estimators.     

Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring

In this paper, the statistical inference of the unknown parameters of a two-parameter inverse Weibull (IW) distribution based on the progressive type-II censored sample has been considered. The maximum likelihood estimators (MLEs) cannot be obtained in explicit forms, hence the approximate MLEs are proposed, which are in explicit forms. The Bayes and generalized Bayes estimators for the IW parameters and the reliability function based on the squared error and Linex loss functions are provided. The Bayes and generalized Bayes estimators cannot be obtained explicitly, hence Lindley's approximation is used to obtain the Bayes and generalized Bayes estimators. Furthermore, the highest posterior density credible intervals of the unknown parameters based on Gibbs sampling technique are computed, and using an optimality criterion the optimal censoring scheme has been suggested. Simulation experiments are performed to see the effectiveness of the different estimators. Finally, two data sets have been analysed for illustrative purposes.     

Empirical bayes estimation in contingency tables

In the estimation of cell probabilities from a two–way contingency table, suppose that a priori the classification variables are believed independent. New empirical Bayes and Bayes estimators are proposed which shrink the observed proportions towards classical estimates under the model of independence. The estimators, based on a Dirichlet mixture class of priors, compare favorably to an estimator of Laird (1978) that is based on a normal prior on terms of a log–linear model. The methods are generalized to three–way tables.     

Multivariate limited translation empirical Bayes estimators

The paper develops multivariate limited translation empirical Bayes estimators of the normal mean vector which serve as a compromise between the empirical Bayes and the maximum likelihood estimators. These compromise estimators perform better than the regular empirical Bayes estimators, in a frequentist sense, when there is wide departure of an individual observation from the grand average.     

Nonparametrio bayes and empirical bayes estimation of the residual survival function at age t

Nonparametric Bayes and empirical Bayes estimations of the

survival function of a unit of age t (> 0) using Dirichlet

process prior are presented. The proposed empirical Bayes

estimators are found to be “asymptotically optimal” in the sense of Robbins (1955). The performances of the proposed

empirical Bayes estimators are compared with those of certain

rival estimators in terms of relative savings loss, The exact

expressions for Bayes risks are also provided in certain cases.     

On estmation of poisson lovits

Simultaneous estimation of poisson logits is considered in a 2xp contingency table under entropy loss. Classical estimators, which are corrected versions of the maximum likeihood estimators, are obtained as generalized Bayes and empirical Bayes estimators. Finally, improved estimators are obtained which domicate the generalized Bayes estimators.     

Bayesian estimation for randomly censored generalized exponential distribution under asymmetric loss functions

This paper deals with the Bayesian estimation of generalized exponential distribution in the proportional hazards model of random censorship under asymmetric loss functions. It is well known for the two-parameter lifetime distributions that the continuous conjugate priors for parameters do not exist; we assume independent gamma priors for the scale and the shape parameters. It is observed that the closed-form expressions for the Bayes estimators cannot be obtained; we propose Tierney–Kadane's approximation and Gibbs sampling to approximate the Bayes estimates. Monte Carlo simulation is carried out to observe the behavior of the proposed methods and one real data analysis is performed for illustration. Bayesian methods are compared with maximum likelihood and it is observed that the Bayes estimators perform better than the maximum-likelihood estimators in some cases.     

Estimation with the generalized exponential distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches have been considered for the two-parameter generalized exponential distribution based on record values with the number of trials following the record values (inter-record times). The maximum likelihood estimates are obtained under the inverse sampling and the random sampling schemes. It is shown that the maximum likelihood estimator of the shape parameter converges in mean square to the true value when the scale parameter is known. The Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The confidence intervals for the parameters are constructed based on asymptotic and Bayesian methods. The Bayes and the maximum likelihood estimators are compared in terms of the estimated risk by the Monte Carlo simulations. The comparison of the estimators based on the record values and the record values with their corresponding inter-record times are performed by using Monte Carlo simulations.     

The relative performances of improved ridge estimators and an empirical bayes estimator: some monte carlo results

The relative 'performances of improved ridge estimators and an empirical Bayes estimator are studied by means of Monte Carlo simulations. The empirical Bayes method is seen to perform consistently better in terms of smaller MSE and more accurate empirical coverage than any of the estimators considered here. A bootstrap method is proposed to obtain more reliable estimates of the MSE of ridge esimators. Some theorems on the bootstrap for the ridge estimators are also given and they are used to provide an analytical understanding of the proposed bootstrap procedure. Empirical coverages of the ridge estimators based on the proposed procedure are generally closer to the nominal coverage when compared to their earlier counterparts. In general, except for a few cases, these coverages are still less accurate than the empirical coverages of the empirical Bayes estimator.     

Bayesian Estimation for Poisson-exponential Model under Progressive Type-II Censoring Data with Binomial Removal and Its Application to Ovarian Cancer Data

In this article, we propose Maximum likelihood estimators (MLEs) and Bayes estimators of parameters of Poisson-exponential distribution (PED) under General entropy loss function (GELF) and Squared error loss function (SELF) for Progressive type-II censored data with binomial removals (PT-II CBRs). The MLEs and corresponding Bayes estimators are compared in terms of their risks based on simulated samples from PED. The proposed methodology is illustrated on a real dataset of ovarian cancer.     

Empirical Bayes estimation for the mean of the selected normal population when there are additional data

ABSTRACT

This paper is concerned with the problem of estimation for the mean of the selected population from two normal populations with unknown means and common known variance in a Bayesian framework. The empirical Bayes estimator, when there are available additional observations, is derived and its bias and risk function are computed. The expected bias and risk of the empirical Bayes estimator and the intuitive estimator are compared. It is shown that the empirical Bayes estimator is asymptotically optimal and especially dominates the intuitive estimator in terms of Bayes risk, with respect to any normal prior. Also, the Bayesian correlation between the mean of the selected population (random parameter) and some interested estimators are obtained and compared.     

On the Bayesian Estimation for the Uniform Scale Parameter via a Functional Equation

Bayes uniform model under the squared error loss function is shown to be completely identifiable by the form of the Bayes estimates of the scale parameter. This results in solving a specific functional equation. A complete characterization of differentiable Bayes estimators (BE) and generalized Bayes estimators (GBE) is given as well as relations between degrees of smoothness of the estimators and the priors. Characterizations of strong (generalized Bayes) Bayes sequence (SBS or SGBS) are also investigated. A SBS is a sequence of estimators (one for each sample size) where all its components are BE generated by the same prior measure. A complete solution is given for polynomial Bayesian estimation.     

Bayesian inference for the parameters of Weibull distribution under progressive Type-I interval censored data with beta-binomial removals

This article considers the problem of estimating the parameters of Weibull distribution under progressive Type-I interval censoring scheme with beta-binomial removals. Classical as well as the Bayesian procedures for the estimation of unknown model parameters have been developed. The Bayes estimators are obtained under SELF and GELF using MCMC technique. The performance of the estimators, has been discussed in terms of their MSEs. Further, expression for the expected number of total failures has been obtained. A real dataset of the survival times for patients with plasma cell myeloma is used to illustrate the suitability of the proposed methodology.     

Empirical bayes shrinkage estimation of reliability in the exponential distribution

In this paper we propose two empirical Bayes shrinkage estimators for the reliability of the exponential distribution and study their properties. Under the uniform prior distribution and the inverted gamma prior distribution these estimators are developed and compared with a preliminary test estimator and with a shrinkage testimator in terms of mean squared error. The proposed empirical Bayes shrinkage estimator under the inverted gamma prior distribution is shown to be preferable to the preliminary test estimator and the shrinkage testimator when the prior value of mean life is clsoe to the true mean life.