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 the bounded regret of empirical bayes estimators

Let (θ₁,x₁),…,(θ_n,x_n) be independent and identically distributed random vectors with E(xθ) = θ and Var(x|θ) = a + bθ + cθ². Let t_i be the linear Bayes estimator of θ_i and θ~_i be the linear empirical Bayes estimator of θ_i as proposed in Robbins (1983). When Ex and Var x are unknown to the statistician. The regret of using θ~_i instead of t_i because of ignorance of the mean and the variance is r_i = E(θ_i ? θ_i)² ?E(t_i ?θ_i)². Under appropriate conditions cumulative regret R_n = r₁+…r_n is shown to have a finite limit even when n tends to infinity. The limit can be explicitly computed in terms of a,b,c and the first four moments of x.     

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.     

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.     

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.     

Linear. empirical bayes estimation in the case of the wishart distribution

We consider independent pairs (X₁,∑₁), (X₂,∑₂),…,(X_n∑_n), where each Si is distributed according to some unknown density function g(∑) and, given ∑_i = ∑, X has a conditional density function g(x|∑) of the Wishart type. In each pair, the first component is observable but the second is not. After the (n + l)-th observation X_n+i is obtained, the objective is to estimate ∑ _n+i corresponding to X_n+i. This estimator is called an empirical Bayes (EB) estimator of ∑. We construct a linear EB estimator of ∑ and examine its precision.     

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.     

Bootstrap adjustments for empirical bayes interval estimates of small-area proportions

Empirical Bayes approaches have often been applied to the problem of estimating small-area parameters. As a compromise between synthetic and direct survey estimators, an estimator based on an empirical Bayes procedure is not subject to the large bias that is sometimes associated with a synthetic estimator, nor is it as variable as a direct survey estimator. Although the point estimates perform very well, naïve empirical Bayes confidence intervals tend to be too short to attain the desired coverage probability, since they fail to incorporate the uncertainty which results from having to estimate the prior distribution. Several alternative methodologies for interval estimation which correct for the deficiencies associated with the naïve approach have been suggested. Laird and Louis (1987) proposed three types of bootstrap for correcting naïve empirical Bayes confidence intervals. Calling the methodology of Laird and Louis (1987) an unconditional bias-corrected naïve approach, Carlin and Gelfand (1991) suggested a modification to the Type III parametric bootstrap which corrects for bias in the naïve intervals by conditioning on the data. Here we empirically evaluate the Type II and Type III bootstrap proposed by Laird and Louis, as well as the modification suggested by Carlin and Gelfand (1991), with the objective of examining coverage properties of empirical Bayes confidence intervals for small-area proportions.     

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.     

Empirical bayes interval estimates involving uniform distributions

Bayesian and empirical Bayesian decision rules are exhibited for the interval estimation of the parameter 0 of a Uniform (0,θ) distribution. The estimate ?,δ>resulting in the interval [?,?+δ]suffers loss given by L(?,δ>,θ)=1-[?≦e≦?+δ]+c₁((?-θ)²+(?+δ?θ)²))+c₂δ. The solution is presented for prior distributions G which have bounded support, no point masses,∫θ^?mdG(θ)<∞ and for some integer m. An example is presented involving a particular parametric form for G and rates of risk convergence in the empirical Bayes problem for this example are calculated.     

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.     

An empirical bayes estimate of multinomial probabilities

This paper deals with the prblem of estimating simultaneously the parameters (Cell probabilities) of m ≤ 2 independent multinomial distributions, with respect to a quadratic loss functions. An empirical Bayes estimator is proposed which is shown to have smaller risk than the maximum likelihood estimator for sufficiently large values of mq, where q is a measure of the average diversity of the given multinomial populations. Some numerical results are given on the performance of the proposed estimator.     

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.     

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).     

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.     

Nonparametric linear bayes estimation of survival curves from incomplete observations

A linear Bayes estimator of a survival curve is derived.The estimator has a relatively simple interpretation as a Kaplan-Meier estimator based on an augemented data base - prior information plus sampling information.It is Bayes if the prior is a Dirichlet process, and otherwise an approximation to the Bayes rule against any prior.     

Empirical bayes rules for selecting good populations

A problem of selecting populations better than a control is considered. When the populations are uniformly distributed, empirical Bayes rules are derived for a linear loss function for both the known control parameter and the unknown control parameter cases. When the priors are assumed to have bounded supports, empirical Bayes rules for selecting good populations are derived for distributions with truncation parameters (i.e. the form of the pdf is f(x|θ)= p_i(x)c_i(θ)I_{(0, θ)}(x)). Monte Carlo studies are carried out which determine the minimum sample sizes needed to make the relative errors less than ε for given ε-values.     

Empirical bayes tests in a positive exponential family with partial information on priors

We investigate an empirical Bayes testing problem in a positive exponential family having pdf f{x/θ)=c(θ)u(x) exp(?x/θ), x>0, θ>0. It is assumed that θ is in some known compact interval [C₁, C₂]. The value C₁ is used in the construction of the proposed empirical Bayes test δ^* _n. The asymptotic optimality and rate of convergence of its associated Bayes risk is studied. It is shown that under the assumption that θ is in [C₁, C₂] δ^* _n is asymptotically optimal at a rate of convergence of order O(n^?1/n n). Also, δ^* _n is robust in the sense that δ^* _n still possesses the asymptotic optimality even the assumption that "C₁≦θ≦C₂ may not hold.     

Minimaxity of empirical bayes estimators of the means of independent normal variables with unequal variances

The problem of simultaneous estimation of normal means is considered when variances are unequal and the loss is sum of squared errors. Minimaxity or non-minimaxity of empirical Bayes estimators is investigated when the common prior distribution is given by normal one with mean 0. Minimaxity results for the case when the loss is a weighted sum of squared errors is also given. Monte Carlo simulation results are given to compare the risk behavior of the empirical Bayes estimator with those of other minimax ones.