A note on james-stein and bayes empiricl bayes estimators

In an empirical Bayes decision problem, a simple class of estimators is constructed that dominate the James-Stein

estimator, A prior distribution A is placed on a restricted (normal) class G of priors to produce a Bayes empirical Bayes estimator, The Bayes empirical Bayes estimator is smooth, admissible, and asymptotically optimal. For certain A rate of convergence to minimum Bayes risk is 0(n^-1)uniformly on G. The results of a Monte Carlo study are presented to demonstrate the favorable risk bebhavior of the Bayes estimator In comparison with other competitors including the James-Stein estimator.     

On asymptotic optimality of bayes empirical bayes estimators

In an empirical Bayes decision problem, a prior distribution ? is placed on a one-dimensfonal family G of priors G_w, wεΩ, to produce a Bayes empirical Bayes estimator, The asymptotic optimaiity of the Bayes estimator is established when the support of ? is Ω and the marginal distributions H_w have monotone likelihood ratio and continuous Kullback-Leibler information number.     

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.     

On empirical bayes sequential estimation

We study the empirical Bayes approach to the sequential estimation problem. An empirical Bayes sequential decision procedure, which consists of a stopping rule and a terminal decision rule, is constructed for use in the component. Asymptotic behaviors of the empirical Bayes risk and the empirical Bayes stopping times are investigated as the number of components increase.     

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.     

Empiric bayes estimation of binomial probability

An empirical Bayes estimator of a binomial parameter, based on orthogonal polynomials on (0,1), is introduced. The resulting estimator of the prior density is asymptotically optimal. The method allows one to combine Bayes and empiric Bayes methods with smoothing in a natural way.     

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.     

Empirical bayes estimation of binomial parameter with symmetric priors

This paper deals with the problem of estimating the binomial parameter via the nonparametric empirical Bayes approach. This estimation problem has the feature that estimators which are asymptotically optimal in the usual empirical Bayes sense do not exist (Robbins (1958, 1964)), However, as pointed out by Liang (1934) and Gupta and Liang (1988), it is possible to construct asymptotically optimal empirical Bayes estimators if the unknown prior is symmetric about the point 1/2, In this paper, assuming symmetric priors a monotone empirical Bayes estimator is constructed by using the isotonic regression method. This estimator is asymptotically optimal in the usual empirical Bayes sense. The corresponding rate of convergence is investigated and shown to be of order n^-1, where n is the number of past observations at hand.     

Minimax linear empirical bayes estimation of the binomial parameter

The minimax linear Empirical Bayes estimators for a binomial parameter are obtained, assuming some information about the moments of the prior. The form of these estimates is used to propose a criterion which may be helpful in determining whether Empirical Bayes estimation is Indicated for a given problem.     

Hierarchical bayes quality measurement plan

Quality Measurement Plan (QMP) as developed by Hoadley (1981) is a statistical method for analyzing discrete quality audit data which consist of the expected number of defects given the standard quality. The QMP is based on an empirical Bayes (EB) model of the audit sampling process. Despite its wide publicity, Hoadley's method has often been described as heuristic. In this paper we offer an hierarchical Bayes (HB) alternative to Hoadley's EB model, and overcome much of the criticism against this model. Gibbs sampling is used to implement the HB model proposed in this paper. Also, the convergence of the Gibbs sampler is monitored via the algorithm of Gelman and Rubin (1992).     

An empirical bayes approach to a variables acceptance sampling plan problem

An empirical Bayes approach to a variables acceptance sampling plan problem is presented and an empirical Bayes rule is developed which is shown to be asymptotically optimal under general conditions. The problem considered is one in which the ratio of the costs of accepting defective items and rejecting non-defective items is specified. Sampling costs are not considered and the size of the sample taken from each lot is fixed and constant. The empirical Bayes estimation of the Bayes rule is shown to require the estimation of a conditional probability. An estimator for conditional probabilities of the form needed is derived and shown to have good asymptotic properties.     

Empirical bayes estimation of the log odds ratio in 2×2 contingency tables

We consider a sequence of contingency tables whose cell probabilities may vary randomly. The distribution of cell probabilities is modelled by a Dirichlet distribution. Bayes and empirical Bayes estimates of the log odds ratio are obtained. Emphasis is placed on estimating the risks associated with the Bayes, empirical Bayes and maximum lilkelihood estimates of the log odds ratio.     

The umvue and bayes estimate of reliability of mixed failure time distribution

We study the reliability estimates of the non-standard mixture of degenerate (degenerated at zero) and exponential distributions. The Uniformly Minimum Variance Unbiased Estimator (UMVUE) and Bayes estimator of the reliability for some selective prior when the mixing proportion is known and unknown are derived. The Bayes risk is computed for each Bayes estimator of the reliability. A simulated study is carried out to assess the performance of the estimators alongwith the true and Maximum Likelihood Estimate (MLE) of the reliability. An example from Vannman (1991) is also discussed at the end of the paper.     

Bayes and empirical bayes estimation of survival function under progressive censoring

The purpose of this note is to derive the Bayes and the empirical Bayes estimators of an unknown survival function F under progressively censored data with respect to the squared error loss function and a Dirichlet process prior using the fact that the posterior distribution of F given the data is a mixture of Dirichlet processes, and the assumption that the survival and the censor in0- distributions are continuous.     

Nonparametric empirical bayes estimation of certain funtionals

For estimating functionals of the form ∫∫φ(x,y)dF(x) dF(y), nonparametric empirical Bayes estimators are developed which are competitors of the classical U-statistics. Asymptotic optimality of the proposed estimators is proved     

A formal bayes multiple shrinkage estimator

A new minimax multiple shrinkage estimator is constructed. This estimator which can adaptively shrink towards many subspace targets, is formal Bayes with respect to a mixture of harmonic priors. Unbiased estimates of risk and simulation results suggest that the risk properties of this estimator are very similar to those of the multiple shrinkage Stein estimator proposed by George (1986a). A special case is seen to be admissible.     

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.     

Estimation in the generalized linear empirical bayes model using the extended quasi-likelihood

A generalized linear empirical Bayes model is developed for empirical Bayes analysis of several means in natural exponential families. A unified approach is presented for all natural exponential families with quadratic variance functions (the Normal, Poisson, Binomial, Gamma, and two others.) The hyperparameters are estimated using the extended quasi-likelihood of Nelder and Pregibon (1987), which is easily implemented via the GLIM package. The accuracy of these estimates is developed by asymptotic approximation of the variance. Two data examples are illustrated.     

Nonparametric bayes estimates of estimable parameters with a dirichlet invarant process and inveriant u-statistics

Tne Bayes estimates of estimable parameters of arbitrary degree in the one sample case are obtained against a Dirichlet invariant. process prior and the squared error loss. We also oive the limits of Bayes estimates, which are related to the in- a- variant U-statistics. For a fixed distribution, the limits of Bayes estimates have the asymptotic normal distribution under certain conditjons.