Parameter estimation for a word frequency distribution based on occupancy theory

Compromise Estimators between the generalized Bayes and Bayes estimators with respect to conjugate gamma priors under entropy loss are proposed. The proposed compromise estimators are compared with some suitable generalized Bayes estimators in terms of their frequentist risk performance. Also the RSL approach will be employed to compare the proposed compromise estimators and some admissible generalized Bayes estimators in terms of their Bayes risk performance.     

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.     

Extended linear empirical bayes estimation

In this paper, the linear empirical Bayes estimation method, which is based on approximation of the Bayes estimator by a linear function, is generalized to an extended linear empirical Bayes estimation technique which represents the Bayes estimator by a series of algebraic polynomials. The extended linear empirical Bayes estimators are elaborated in the case of a location or a scale parameter. The theory is illustrated by examples of its application to the normal distribution with a location parameter and the gamma distribution with a scale parameter. The linear and the extended linear empirical Bayes estimators are constructed in these two cases and, then, studied numerically via Monte Carlo simulations. The simulations show that the extended linear empirical Bayes estimators have better convergence rates than the traditional linear empirical Bayes estimators.     

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.     

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.     

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.     

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 estimation under a mixture of the Burr type XII distribution and its reciprocal

The Bayesian estimation for the parameters of the finite mixture of the Burr type XII distribution with its reciprocal are obtained based on the type-I censored data. The Bayes estimators are computed based on squared error and Linex loss functions and using the idea of Markov chain Monte Carlo algorithm. Based on the Monte Carlo simulation, Bayes estimators are compared with their corresponding maximum-likelihood estimators.     

Bayes estimators in the presence of a guess value

The paper deals with the problem of parameter estimation in the presence of a guess value and attempts to justify the use of Bayes estimators as an alternative to ordinary shrinkage estimators. Finally, certain Bayes estimators of exponential parameters are obtained under type II censoring, and these are compared with the corresponding MLEs and ordinary shrinkage estimators using a Monte Carlo study.     

Proper Bayes minimax estimators of the normal mean matrix with common unknown variances

This paper addresses the problem of estimating a matrix of the normal means, where the variances are unknown but common. The approach to this problem is provided by a hierarchical Bayes modeling for which the first stage prior for the means is matrix-variate normal distribution with mean zero matrix and a covariance structure and the second stage prior for the covariance is similar to Jeffreys’ rule. The resulting hierarchical Bayes estimators relative to the quadratic loss function belong to a class of matricial shrinkage estimators. Certain conditions are obtained for admissibility and minimaxity of the hierarchical Bayes estimators.     

Optimal minimax squared error risk estimation of the mean of a multivariate normal distribution

Minimax squared error risk estimators of the mean of a multivariate normal distribution are characterized which have smallest Bayes risk with respect to a spherically symmetric prior distribution for (i) squared error loss, and (ii) zero-one loss depending on whether or not estimates are consistent with the hypothesis that the mean is null. In (i), the optimal estimators are the usual Bayes estimators for prior distributions with special structure. In (ii), preliminary test estimators are optimal. The results are obtained by applying the theory of minimax-Bayes-compromise decision problems.     

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.     

Bayesian inference for the randomly censored Burr-type XII distribution

The article presents the Bayesian inference for the parameters of randomly censored Burr-type XII distribution with proportional hazards. The joint conjugate prior of the proposed model parameters does not exist; we consider two different systems of priors for Bayesian estimation. The explicit forms of the Bayes estimators are not possible; we use Lindley's method to obtain the Bayes estimates. However, it is not possible to obtain the Bayesian credible intervals with Lindley's method; we suggest the Gibbs sampling procedure for this purpose. Numerical experiments are performed to check the properties of the different estimators. The proposed methodology is applied to a real-life data for illustrative purposes. The Bayes estimators are compared with the Maximum likelihood estimators via numerical experiments and real data analysis. The model is validated using posterior predictive simulation in order to ascertain its appropriateness.     

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.     

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.     

The Superiorities of Empirical Bayes Estimation of Variance Components in Random Effects Model

In this article, the Bayes estimators of variance components are derived and the parametric empirical Bayes estimators (PEBE) for the balanced one-way classification random effects model are constructed. The superiorities of the PEBE over the analysis of variance (ANOVA) estimators are investigated under the mean square error (MSE) criterion, some simulation results for the PEBE are obtained. Finally, a remark for the main results is given.     

Bayesian predictive modeling for Inverse Gamma-Pareto composite distribution

Inverse Gamma-Pareto composite distribution is considered as a model for heavy tailed data. The maximum likelihood (ML), smoothed empirical percentile (SM), and Bayes estimators (informative and non-informative) for the parameter θ, which is the boundary point for the supports of the two distributions are derived. A Bayesian predictive density is derived via a gamma prior for θ and the density is used to estimate risk measures. Accuracy of estimators of θ and the risk measures are assessed via simulation studies. It is shown that the informative Bayes estimator is consistently more accurate than ML, Smoothed, and the non-informative Bayes estimators.     

The empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior under Stein's loss function

For the hierarchical Poisson and gamma model, we calculate the Bayes posterior estimator of the parameter of the Poisson distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior by two methods. In numerical simulations, we have illustrated: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we exploit the attendance data on 314 high school juniors from two urban high schools to illustrate our theoretical studies.     

Estimation of p(y<x) in the burr case: a comparative study

This paper provides a simulation study which compares three estimators for R = P(Y<X) when Y and X are two independent but not identically distributed Burr random variables. These estimators are the minimum variance unbiased, the maximum likelihood and Bayes estimators. Moreover, the sensitivity of Bayes estimator to the prior parameters is considered.     

On empirical Bayes estimation of variance components in random effects model

In this paper, Bayes estimators of variance components are derived for the one-way random effects model, and empirical Bayes (EB) estimators are constructed by the kernel estimation method of a multivariate density and its mixed partial derivatives. It is shown that the EB estimators are asymptotically optimal and convergence rates are established. Finally, an example concerning the main results is given.