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.     

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.     

Adaptive nonparametric empirical Bayes estimation via wavelet series: The minimax study

In the present paper, we derive lower bounds for the risk of the nonparametric empirical Bayes estimators. In order to attain the optimal convergence rate, we propose generalization of the linear empirical Bayes estimation method which takes advantage of the flexibility of the wavelet techniques. We present an empirical Bayes estimator as a wavelet series expansion and estimate coefficients by minimizing the prior risk of the estimator. As a result, estimation of wavelet coefficients requires solution of a well-posed low-dimensional sparse system of linear equations. The dimension of the system depends on the size of wavelet support and smoothness of the Bayes estimator. An adaptive choice of the resolution level is carried out using Lepski et al. (1997) method. The method is computationally efficient and provides asymptotically optimal adaptive EB estimators. The theory is supplemented by numerous examples.     

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.     

Bayes and empirical bayes estimation of the probability that z > x + y

Let X, Y and Z be independent random variables with common unknown distribution F. Using the Dirichlet process prior for F and squared erro loss function, the Bayes and empirical Bayes estimators of the parameters λ(F). the probability that Z > X + Y, are derived. The limiting Bayes estimator of λ(F) under some conditions on the parameter of the process is shown to be asymptotically normal. The aysmptotic optimality of the empirical Bayes estimator of λ(F) is established. When X, Y and Z have support on the positive real line, these results are derived for randomly right censored data. This problem relates to testing whether than used discussed by Hollander and Proshcan (1972) and Chen, Hollander and Langberg (1983).     

Constrained Bayes and Empirical Bayes Estimation with Balanced Loss Functions

This article develops constrained Bayes and empirical Bayes estimators under balanced loss functions. In the normal-normal example, estimators of the mean squared errors of the EB and constrained EB estimators are provided which are correct asymptotically up to O(m ^?1), m denoting the number of strata.     

Empirical Bayes sequential estimation of a finite population mean

In this article, we consider the Bayes and empirical Bayes problem of the current population mean of a finite population when the sample data is available from other similar (m-1) finite populations. We investigate a general class of linear estimators and obtain the optimal linear Bayes estimator of the finite population mean under a squared error loss function that considered the cost of sampling. The optimal linear Bayes estimator and the sample size are obtained as a function of the parameters of the prior distribution. The corresponding empirical Bayes estimates are obtained by replacing the unknown hyperparameters with their respective consistent estimates. A Monte Carlo study is conducted to evaluate the performance of the proposed empirical Bayes procedure.     

BAYES EMPIRICAL BAYES APPROACH TO ESTIMATION OF THE FAILURE RATE IN EXPONENTIAL DISTRIBUTION

ABSTRACT

In the empirical Bayes (EB) decision problem consisting of squared error estimation of the failure rate in exponential distribution, a prior Λ is placed on the gamma family of prior distributions to produce Bayes EB estimators which are admissible. A subclass of such estimators is shown to be asymptotically optimal (a.o.). The results of a Monte Carlo study are presented to demonstrate the a.o. property of the Bayes EB 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.     

Bayes methods in the ecological fallacy cntext:estimation of individual correlation from aggregate data

The ecological fallacy is related to Simpson's paradox (1951) where relationships among group means may be counterintuitive and substantially different from relationships within groups, where the groups are usually geographic entities such as census tracts. We consider the problem of estimating the correlation between two jointly normal random variables where only ecological data (group means) are available. Two empirical Bayes estimators and one fully Bayesian estimator are derived and compared with the usual ecological estimator, which is simply the Pearson correlation coefficient of the group sample means. We simulate the bias and mean squared error performance of these estimators, and also give an example employing a dataset where the individual level data are available for model checking. The results indicate superiority of the empirical Bayes estimators in a variety of practical situations where, though we lack individual level data, other relevant prior information is available.     

Nonparametric Bayes estimation of gap-time distribution with recurrent event data

Nonparametric Bayes (NPB) estimation of the gap-time survivor function governing the time to occurrence of a recurrent event in the presence of censoring is considered. In our Bayesian approach, the gap-time distribution, denoted by F, has a Dirichlet process prior with parameter α. We derive NPB and nonparametric empirical Bayes (NPEB) estimators of the survivor function F?=1?F and construct point-wise credible intervals. The resulting Bayes estimator of F? extends that based on single-event right-censored data, and the PL-type estimator is a limiting case of this Bayes estimator. Through simulation studies, we demonstrate that the PL-type estimator has smaller biases but higher root-mean-squared errors (RMSEs) than those of the NPB and the NPEB estimators. Even in the case of a mis-specified prior measure parameter α, the NPB and the NPEB estimators have smaller RMSEs than the PL-type estimator, indicating robustness of the NPB and NPEB estimators. In addition, the NPB and NPEB estimators are smoother (in some sense) than the PL-type estimator.     

Empirical Bayes Estimation of Small Area Means under a Nested Error Linear Regression Model with Measurement Errors in the Covariates

Abstract.  Previously, small area estimation under a nested error linear regression model was studied with area level covariates subject to measurement error. However, the information on observed covariates was not used in finding the Bayes predictor of a small area mean. In this paper, we first derive the fully efficient Bayes predictor by utilizing all the available data. We then estimate the regression and variance component parameters in the model to get an empirical Bayes (EB) predictor and show that the EB predictor is asymptotically optimal. In addition, we employ the jackknife method to obtain an estimator of mean squared prediction error (MSPE) of the EB predictor. Finally, we report the results of a simulation study on the performance of our EB predictor and associated jackknife MSPE estimators. Our results show that the proposed EB predictor can lead to significant gain in efficiency over the previously proposed EB predictor.     

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.     

A linear regression with unobserved dependent variables

Let (?,X) be a random vector such that E(X|?) = ? and Var(x|?) a + b? + c?² for some known constants a, b and c. Assume X₁,…,X_n are independent observations which have the same distribution as X. Let t(X) be the linear regression of ? on X. The linear empirical Bayes estimator is used to approximate the linear regression function. It is shown that under appropriate conditions, the linear empirical Bayes estimator approximates the linear regression well in the sense of mean squared error.     

Empirical bayes testing of a distribution function with d1richlet process priors

This paper considers the generalised empirical Bayes two-action (testing) and multiple action problems concerning a distribution function The Dirichiet process priors p of Ferguson have been used as the prior distributions on the space of distribution functions on the real line. The two-action component problem is considered in detail and when p is unknown partially empirical Bayes procedures {6 } which are asymptotically optimal with rates 0{jT1/2) and OCCmCn+l))     

The Statistical Properties of the Equity Estimator

In this article we consider the Equity estimator proposed by Krishnamurthi and Rangaswamy. We show that this estimator is inconsistent and does not necessarily improve on the mean squared error (MSE) of the least squares (LS) estimator. We perform a Monte Carlo experiment based on the price-promotion model used in marketing research, with marketing data, comparing the MSE of the Equity estimator to that of two empirical Bayes estimators and the LS estimator. We find that the empirical Bayes estimators have substantially smaller MSE than the Equity estimator in almost every case.     

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.     

Constrained Bayes and empirical Bayes estimation under random effects normal ANOVA model with balanced loss function

The paper develops constrained Bayes and empirical Bayes estimators in the random effects ANOVA model under balanced loss functions. In the balanced normal–normal model, estimators of the Bayes risks of the constrained Bayes and constrained empirical Bayes estimators are provided which are correct asymptotically up to O(m^-1)$\mathrm{O}(m^{-1})$, that is the remainder term is o(m^-1)$\mathrm{o}(m^{-1})$, m$m$ denoting the number of strata.     

EMPIRICAL BAYES ESTIMATION WITH NON-IDENTICAL COMPONENTS. CONTINUOUS CASE.

In this paper a variant of the standard empirical Bayes estimation problem is considered where the component problems in the sequence are not identical in that the conditional distribution of the observations may vary with the component problems. Let {(Θ_n, X_n)} be a sequence of independent random vectors where Θ_n? G and, given Θ_n=Θ_n, X_n -P_Θ,m(n) with {m(n)} a sequence of positive integers where m(n)≤m? < ∞ for all n. With P_Θ,m in a continuous exponential family of distributions, asymptotically optimal empirical Bayes procedures are exhibited which depend on uniform approximations of certain functions on compact sets by polynomials in e^Θ. Such approximations have been explicitly calculated in the case of normal and gamma families. In the case of normal families, asymptotically optimal linear empirical Bayes estimators in the class of all linear estimators are derived and shown to have rate O(n^-1/2).     

Robust empirical Bayes tests for continuous distributions

In this paper, we study the empirical Bayes two-action problem under linear loss function. Upper bounds on the regret of empirical Bayes testing rules are investigated. Previous results on this problem construct empirical Bayes tests using kernel type estimators of nonparametric functionals. Further, they have assumed specific forms, such as the continuous one-parameter exponential family for {F_θ:θΩ}, for the family of distributions of the observations. In this paper, we present a new general approach of establishing upper bounds (in terms of rate of convergence) of empirical Bayes tests for this problem. Our results are given for any family of continuous distributions and apply to empirical Bayes tests based on any type of nonparametric method of functional estimation. We show that our bounds are very sharp in the sense that they reduce to existing optimal or nearly optimal rates of convergence when applied to specific families of distributions.