Generalized Bayes minimax estimation of the normal mean matrix with unknown covariance matrix

This paper addresses the problem of estimating the normal mean matrix in the case of unknown covariance matrix. This problem is solved by considering generalized Bayesian hierarchical models. The resulting generalized Bayes estimators with respect to an invariant quadratic loss function are shown to be matricial shrinkage equivariant estimators and the conditions for their minimaxity are given.     

Minimax and admissible minimax estimators of the mean of a multivariate normal distribution for unknown covariance matrix

Let X be a p-variate (p ≥ 3) vector normally distributed with mean μ and covariance Σ, and let A be a p × p random matrix distributed independent of X, according to the Wishart distribution W(n, Σ). For estimating μ, we consider estimators of the form δ = δ(X, A). We obtain families of Bayes, minimax and admissible minimax estimators with respect to the quadratic loss function (δ ? μ)′ Σ^?1(δ ? μ) where Σ is unknown. This paper extends previous results of the author [1], given for the case in which the covariance matrix of the distribution is of the form σ²I, where σ is known.     

Minimax estimation of a multivariate normal mean under polynomial loss

Let X be an observation from a p-variate (p ≥ 3) normal random vector with unknown mean vector θ and known covariance matrix

. The problem of improving upon the usual estimator of θ, δ⁰(X) = X, is considered. An approach is developed which can lead to improved estimators, δ, for loss functions which are polynomials in the coordinates of (δ ? θ). As an example of this approach, the loss L(δ, θ) = |δ ? θ|⁴ is considered, and estimators are developed which are significantly better than δ⁰. When

is the identity matrix, these estimators are of the form $\text{δ(X) = (1 ? (}\text{b}\text{(d + |X|}{}^2\text{)}\text{))X}$.     

Minimax estimation of a multivariate normal mean under arbitrary quadratic loss

Let X be a p-variate (p ≥ 3) vector normally distributed with mean θ and known covariance matrix

. It is desired to estimate θ under the quadratic loss (δ ? θ)^tQ(δ ? θ), where Q is a known positive definite matrix. A broad class of minimax estimators for θ is developed.     

Minimax estimation of means of multivariate normal mixtures

Assume X = (X₁, …, X_p)′ is a normal mixture distribution with density w.r.t. Lebesgue measure, , where Σ is a known positive definite matrix and F is any known c.d.f. on (0, ∞). Estimation of the mean vector under an arbitrary known quadratic loss function _Q(θ, a) = (a − θ)′ Q(a − θ), Q a positive definite matrix, is considered. An unbiased estimator of risk is obatined for an arbitrary estimator, and a sufficient condition for estimators to be minimax is then achieved. The result is applied to modifying all the Stein estimators for the means of independent normal random variables to be minimax estimators for the problem considered here. In particular the results apply to the Stein class of limited translation estimators.     

GAMMA－MINIMAX ESTIMATORS FOR THE MEAN OF A MULTIVARIATE NORMAL DISTRIBUTION WITH PARTIALLY UNKNOWN COVARIANCE MATRIX

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Methods for improvement in estimation of a normal mean matrix

This paper is concerned with the problem of estimating a matrix of means in multivariate normal distributions with an unknown covariance matrix under invariant quadratic loss. It is first shown that the modified Efron-Morris estimator is characterized as a certain empirical Bayes estimator. This estimator modifies the crude Efron-Morris estimator by adding a scalar shrinkage term. It is next shown that the idea of this modification provides a general method for improvement of estimators, which results in the further improvement on several minimax estimators. As a new method for improvement, an adaptive combination of the modified Stein and the James-Stein estimators is also proposed and is shown to be minimax. Through Monte Carlo studies of the risk behaviors, it is numerically shown that the proposed, combined estimator inherits the nice risk properties of both individual estimators and thus it has a very favorable risk behavior in a small sample case. Finally, the application to a two-way layout MANOVA model with interactions is discussed.     

A family of minimax estimators of a multivariate normal mean

Let X be a p-dimensional normal random vector with unknown mean vector θ and covariance σ²I. Let S/σ², independent of X, be chi-square with n degrees of freedom. Relative to the squared error loss, James and Stein (1961) have obtained an estimator which dominates the usual estimator X. Baranchik (1970) has extended James and Stein's results. We obtain a theorem which can provide a different family of minimax estimators containing James-Stein's estimator. Two interesting minimax estimators are presented in this paper.     

Minimax estimation of a variance

The nonparametric problem of estimating a variance based on a sample of sizen from a univariate distribution which has a known bounded range but is otherwise arbitrary is treated. For squared error loss, a certain linear function of the sample variance is seen to be minimax for eachn from 2 through 13, exceptn=4. For squared error loss weighted by the reciprocal of the variance, a constant multiple of the sample variance is minimax for eachn from 2 through 11. The least favorable distribution for these cases gives probability one to the Bernoulli distributions.     

A family of dominating minimax estimators of a multivariate normal mean

Let X have a p-variate normal distribution with mean vector θ and identity covariance matrix I. In the squared error estimation of θ, Baranchik (1970) gives a wide family G of minimax estimators. In this paper, a subfamily C of dominating estimators in G is found such that for each estimator δ₁ in G not in C, there exists an estimator δ₂ in C which which dominates δ₁.     

Stein's idea and minimax admissible estimation of a multivariate normal mean

We consider estimation of a multivariate normal mean vector under sum of squared error loss.We propose a new class of minimax admissible estimator which are generalized Bayes with respect to a prior distribution which is a mixture of a point prior at the origin and a continuous hierarchical type prior. We also study conditions under which these generalized Bayes minimax estimators improve on the James–Stein estimator and on the positive-part James–Stein estimator.     

Trimmed minimax estimator of a covariance matrix

Summary In the problem of estimating the covariance matrix of a multivariate normal population, James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, 361–380, Univ. of California Press) obtained a minimax estimator under a scale invariant loss. In this paper we propose an orthogonally invariant trimmed estimator by solving certain differential inequality involving the eigenvalues of the sample covariance matrix. The estimator obtained, truncates the extreme eigenvalues first and then shrinks the larger and expands the smaller sample eigenvalues. Adaptive version of the trimmed estimator is also discussed. Finally some numerical studies are performed using Monte Carlo simulation method and it is observed that the trimmed estimate shows a substantial improvement over the minimax estimator. The second author's research was supported by NSF Grant Number MCS 82-12968.     

Simultaneous estimation of means of classified normal observations

Simultaneous estimation of normal means is considered for observations which are classified into several groups. In a one-way classification case, it is shown that an adaptive shrinkage estimator dominates a Stein-type estimator which shrinks observations towards individual class averages as Stein's (1966,Festschrift for J. Neyman, (ed. F. N. David), 351–366, Wiley, New York) does, and is minimax even if class sizes are small. Simulation results under quadratic loss show that it is slightly better than Stein's (1966) if between variances are larger than within ones. Further this estimator is shown to improve on Stein's (1966) with respect to the Bayes risk. Our estimator is derived by assuming the means to have a one-way classification structure, consisting of three random terms of grand mean, class mean and residual. This technique can be applied to the case where observations are classified into a two-stage hierarchy.     

Dirichlet invariant processes and applications to nonparametric estimation of symmetric distribution functions

A class of random processes with invariant sample paths, that is, processes which yield (with probability one) probability distributions that are invariant under a given transformation group of interest, are introduced and their properties are studied. These processes, named Dirichlet Invariant processes, are closely related to the Dirichlet processes of Ferguson. These processes can be used as priors for Bayesian analysis of some nonparametric problems. As an application Bayes and Minimax estimates of an arbitrary distribution, symmetric about a known point, are obtained.     

Asymptotic risk behavior of mean vector and variance estimators and the problem of positive normal mean

Asymptotic risk behavior of estimators of the unknow variance and of the unknown mean vector in a multivariate normal distribution is considered for a general loss. It is shown that in both problems this characteristic is related to the risk in an estimation problem of a positive normal mean under quadratic loss function. A curious property of the Brewster-Zidek variance estimator of the normal variance is also noticed.Research supported by NSF Grant DMS 9000999 and by Alexander von Humboldt Foundation Senior Distinguished Scientist Award.University of Münster     

Admissibility of some preliminary test estimators for the mean of normal distribution

Summary Let a random variableX follow ap-variate normal distributionN _p (θ, I _p ) with an unknownp×1 vector θ andp×p identity matrixI _p . The admissibility of a preliminary test estimator using AIC (Akaike's Information Criterion) procedure will be shown ifp=1 and its inadmissibility will be shown ifp≧3 under the loss function based on Kullback-Leibler information measure. Furthermore the two sample case is also considered.     

Ratio methods to the mean estimation with known quantiles

We address the problem of estimating the finite population mean in survey sampling, by exploiting any available auxiliary information in order to increase the precision of classical estimators. The idea is to use any population quantiles of the available auxiliary variables which are known in many real situation from census, administrative files, etc. This is achieved using these known quantities in the construction of the estimators, by modifying the usual ratio estimation methods and afterwards defining a general class of exponentiation ratio estimators. The advantages of the proposed estimators are demonstrated using theoretical asymptotic tools and through a simulation study.     

Improved confidence sets for the mean of a multivariate normal distribution

A new class of confidence sets for the mean of a p-variate normal distribution (p3) is introduced. They are neither spheres nor ellipsoids. We show that we can construct our confidence sets so that their coverage probabilities are equal to the specified confidence coefficient. Some of them are shown to dominate the usual confidence set, a sphere centered at the observations. Numerical results are also given which show how small their volumes are.     

Contracting towards subspaces when estimating the mean of a multivariate normal distribution

The problem of estimating, under unweighted quadratic loss, the mean of a multinormal random vector X with arbitrary covariance matrix V is considered. The results of James and Stein for the case V = I have since been extended by Bock to cover arbitrary V and also to allow for contracting X towards a subspace other than the origin; minimax estimators (other than X) exist if and only if the eigenvalues of V are not “too spread out.” In this paper a slight variation of Bock's estimator is considered. A necessary and sufficient condition for the minimaxity of the present estimator is (1): the eigenvalues of (I ? P) V should not be “too spread out,” where P denotes the projection matrix associated with the subspace towards which X is contracted. The validity of (1) is then examined for a number of patterned covariance matrices (e.g., intraclass covariance, tridiagonal and first order autocovariance) and conditions are given for (1) to hold when contraction is towards the origin or towards the common mean of the components of X. (1) is also examined when X is the usual estimate of the regression vector in multiple linear regression. In several of the cases considered the eigenvalues of V are “too spread out” while those of (I ? P) V are not, so that in these instances the present method can be used to produce a minimax estimate.