Bayesian testing of an exponential point null hypothesis

The problem of testing a point null hypothesis involving an exponential mean is The problem of testing a point null hypothesis involving an exponential mean is usual interpretation of P-values as evidence against precise hypotheses is faulty. As in Berger and Delampady (1986) and Berger and Sellke (1987), lower bounds on Bayesian measures of evidence over wide classes of priors are found emphasizing the conflict between posterior probabilities and P-values. A hierarchical Bayes approach is also considered as an alternative to computing lower bounds and “automatic” Bayesian significance tests which further illustrates the point that P-values are highly misleading measures of evidence for tests of point null hypotheses.     

Asymptotic optimality of a two-stage procedure in Bayes sequential estimation

The problem of sequential estimation of the mean with quadratic loss and fixed cost per observation is considered within the Bayesian framework. Instead of fully sequential sampling, a two-stage sampling technique is introduced to solve the problem. The proposed two-stage procedure is robust in the sense that it does not depend on the distribution of outcome variables and the prior. It is shown to be asymptotically not worse than the optimal fixed-sample-size procedures for the arbitrary distributions, and to be asymptotically Bayes for the distributions of one-parameter exponential family.     

On robust empirical bayes analysis of means and variances from stratified samples

Ghosh and Lahiri (1987a,b) considered simultaneous estimation of several strata means and variances where each stratum contains a finite number of elements, under the assumption that the posterior expectation of any stratum mean is a linear function of the sample observations - the so called“posterior linearity” property. In this paper we extend their result by retaining the “posterior linearity“ property of each stratum mean but allowing the superpopulation model whose mean as well as the variance-covariance structure changes from stratum to stratum. The performance of the proposed empirical Bayes estimators are found to be satisfactory both in terms of “asymptotic optimality” (Robbins (1955)) and “relative savings loss” (Efron and Morris (1973)).     

Empirical bayes estimation of the mean in a multivariate normal distribution

We consider the problem of estimating the mean vector of a multivariate normal distribution under a variety of assumed structures among the parameters of the sampling and prior distributions. We adopt a pragmatic approach. We adopt distributional familites, assess hyperparmeters, and adopt patterned mean and coveariance structures when it is relatively simple to do so; alternatively, we use the sample data to estimate hyperparameters of prior distributions when assessment is a formidable task; such as the task of assessing parameters of multidimensional problems. James-Stein-like estimators are found to result. In some cases, we've been abl to show that the estimators proposed uniformly dominate the MLE's when measured with respect to quadratic loss functions.     

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.     

On the admissibility of c[Xbar] + d with respect to the linex loss function

Let X₁, …,X_n be a random sample from a normal distribution with mean θ and variance σ² The problem is to estimate θ with loss function L(θ,e) = v(e-θ) where v(x) = b(exp(ax)-ax-l) and where a, b are constants with b>0, a¦0. Zellner (1986), showed that [Xbar] ? σ²a/2n dominates [Xbar] and hence [Xbar] is inadmissible. The question of what values of c and d render c[Xbar]+ d admissible is studied here.     

Bayesian approaches to the detection of outliers in poisson samples

This paper is concerned with the detection of upper outliers in a Poisson sample.The approach is Bayesian throughout. It is supposed that a small number of observations are contaminated, that is they are generated from a Poisson sample with mean inflated by a factor §.Bayes factors for the cases when (i) § is known, (ii) it is given a proper conjugate prior or (iii) it is completely unknown are discussed. It is suggested, in contrast to classical approaches, that transforming the data to normality does not simplify the problem.     

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.     

Bayesian shrinkage estimators for a measure of dispersion of an inverse gaussian distribution

The Bayesian shrinkage estimation for a measure of dispersion with known mean is studied for the inverse Gaussian distribution. An optimum choice of the shrinkage factor and the properties of the proposed Bayesian shrinkage estimators are being studied. It is shown that these estimators have smaller risk than the usual estimator of the reciprocal measure of dispersion.     

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.     

Estimating Damages in a Class Action Litigation

In a class action litigation, actual damages are not known exactly and must be estimated. Various estimators are proposed and assessed by using a model that identifies possible sources of error. Estimators that have been used in practice are shown to be seriously biased. An empirical Bayes estimator and an empirical minimal mean squared error estimator are found to be more satisfactory methods for estimating damages.     

Empirical and Hierarchical Bayesian Estimation in Finite Population Sampling under Structural Measurement Error Models

Abstract.  This paper considers simultaneous estimation of means from several strata. A model-based approach is taken, where the covariates in the superpopulation model are subject to measurement errors. Empirical Bayes (EB) and Hierarchical Bayes estimators of the strata means are developed and asymptotic optimality of EB estimators is proved. Their performances are examined and compared with that of the sample mean in a simulation study as well as in data analysis.     

A Bivariate Geometric Distribution with Applications to Reliability

This article studies a bivariate geometric distribution (BGD) as a plausible reliability model. Maximum likelihood and Bayes estimators of parameters and various reliability characteristics are obtained. Approximations to the mean, variance, and Bayes risk of these estimators have been derived using Taylor's expansion. A Monte-Carlo simulation study has been performed to compare these estimators. At the end, the theory is illustrated with a real data set example of accidents.     

A New Approach to the Estimation of Inter-Variable Correlation

The use of different measures of similarity between observed vectors for the purposes of classifying or clustering them has been expanding dramatically in recent years. One result of this expansion has been the use of many new similarity measures, designed for the purpose of satisfying various criteria. A noteworthy application involves estimating the relationships between genes using microarray experimental data. We consider the class of ‘correlation-type’ similarity measures. The use of these new measures of similarity suggest that the whole problem needs to be formulated in statistical terms to clarify their relative benefits. Pursuant to this need, we define, for each given observed vector, a baseline representing the ‘true’ value common to each of the component observations. These ‘true’ values are taken to be parameters. We define the ‘true correlation’ between each two observed vectors as the average (over the distribution of the observations for given baseline parameters) of Pearson's correlation with sample means replaced by the corresponding baseline parameters. Estimators of this true correlation are assessed using their mean squared error (MSE). Proper Bayes estimators of this true correlation, being based on the predictive posterior distribution of the data, are both difficult to calculate/analyze and highly non robust. By constrast, empirical Bayes estimators are: (i) close to their Bayesian counterparts; (ii) easy to analyze; and (iii) strongly robust. For these reasons, we employ empirical Bayes estimators of correlation in place of their Bayesian counterparts. We show how to construct two different kinds of simultaneous Bayes correlation estimators: the first assumes no apriori correlation between baseline parameters; the second assumes a common unknown correlation between them. Estimators of the latter type frequently have significantly smaller MSE than those of the former type which, in turn, frequently have significantly smaller MSE than their Pearson estimator counterparts. For purposes of illustrating our results, we examine the problem of inferring the relationships between gene expression level vectors, in the context of observing microarray experimental data.     

Bayes estimation of a langevin mean direction

A Langevin distribution with two parameters (mean direction and concentration parameter) has been extensively used for modeling and analyzing problems related to directional data. In this article, we examine the estimation problem for the mean direction. Bayes estimators are derived with respect to a conjugate as well as the Jeffreys’ priors. Further in case of unknown concentration parameter, other priors are also chosen. An extensive analysis of risk behavior of Bayes estimators is carried out with the help of simulations.     

The Slippage Problem for the Circular Normal Distribution

The slippage problem occurs when an unspecified observation in a given random sample is from a distribution other than that for all the remaining observations. This paper studies the problem in terms of the 'slip' in the mean direction of a circular normal distribution. The slippage problem is first treated as a multiple decision problem with a prior which is invariant under the permutations of the hypotheses. The probabilities of accepting the various hypotheses for the Bayes rule with respect to this prior are explicitly obtained. The likelihood ratio tests for this slippage problem, for the cases when the mean directions are both known and unknown, are shown to be easily computable. The tests are illustrated through two well-known datasets. The performances of a range of tests are compared using extensive simulation.     

Parametric and semiparametric hypotheses in the linear model

The independent additive errors linear model consists of a structure for the mean and a separate structure for the error distribution. The error structure may be parametric or it may be semiparametric. Under alternative values of the mean structure, the best fitting additive errors model has an error distribution which can be represented as the convolution of the actual error distribution and the marginal distribution of a misspecification term. The model misspecification term results from the covariates' distribution. Conditions are developed to distinguish when the semiparametric model yields sharper inference than the parametric model and vice versa. The main conditions concern the actual error distribution and the covariates' distribution. The theoretical results explain a paradoxical finding in semiparametric Bayesian modelling, where the posterior distribution under a semiparametric model is found to be more concentrated than is the posterior distribution under a corresponding parametric model. The paradox is illustrated on a set of allometric data. The Canadian Journal of Statistics 39: 165–180; 2011 ©2011 Statistical Society of Canada     

Empirical bayes approach to bayes sequential estimation: the poisson and bernoulli cases

The problem considered is the Bayes sequential estimation of the mean with quadratic loss and fixed cost per observation. Assume the prior distribution is not completely known. Some empirical Bayes procedures are proposed in the Poisson and Bernoulli cases, and they are shown to be asymptotically non-deficient in the sense of Woodroofe (1981).     

Une condition nécessaire d'admissibilité et ses conséquences sur les estimateurs à rétrécisseur de la moyenne d'un vecteur normal

Considering exponential families of distributions, we estimate parameters which are not the natural parameters. We prove that the admissible estimators of these parameters are limits of Bayes estimators and can be expressed through a given functional form. An important particular case of this model pertains to the estimation of the mean of a multidimensional normal distribution when the variance is known up to a multiplicative factor. We deduce from the main result a necessry condition for the admissibility of matricial shrinkage estimators.     

Bayesian least squares estimates of univariate regression functions

The regression function R(?) to be estimated is assumed to have an expansion in terms of specified functions, orthogonalized vich respect to values of the explanatory variable. Relative precisions of OBSERVATION are assumed known. The estimate is the posterior linear mean of R(?) given the data. The investigator plots graphs of appropriate functions as an aid in eliciting his prior means and precisions for the coefficients in the expansion. The method is illustrated by an example using simulated data, an example in which effects of various dosages of Vitamin D are estimated, and an example in which a utility function is estimated.