Generalized Bayes minimax estimators of the mean of multivariate normal distribution with unknown variance

We construct a broad class of generalized Bayes minimax estimators of the mean of a multivariate normal distribution with covariance equal to σ²I_p, with σ² unknown, and under the invariant loss δ(X)−θ²/σ². Examples that illustrate the theory are given. Most notably it is shown that a hierarchical version of the multivariate Student-t prior yields a Bayes minimax estimate.     

Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

Let X∼f(∥x-θ∥²) and let δ_π(X) be the generalized Bayes estimator of θ with respect to a spherically symmetric prior, π(∥θ∥²), for loss ∥δ-θ∥². We show that if π(t) is superharmonic, non-increasing, and has a non-decreasing Laplacian, then the generalized Bayes estimator is minimax and dominates the usual minimax estimator δ₀(X)=X under certain conditions on . The class of priors includes priors of the form for and hence includes the fundamental harmonic prior . The class of sampling distributions includes certain variance mixtures of normals and other functions f(t) of the form e^-αtβ and e^-αt+βφ(t) which are not mixtures of normals. The proofs do not rely on boundness or monotonicity of the function r(t) in the representation of the Bayes estimator as .     

Admissibility and minimaxity of Bayes estimators for a normal mean matrix

In some invariant estimation problems under a group, the Bayes estimator against an invariant prior has equivariance as well. This is useful notably for evaluating the frequentist risk of the Bayes estimator. This paper addresses the problem of estimating a matrix of means in normal distributions relative to quadratic loss. It is shown that a matricial shrinkage Bayes estimator against an orthogonally invariant hierarchical prior is admissible and minimax by means of equivariance. The analytical improvement upon every over-shrinkage equivariant estimator is also considered and this paper justifies the corresponding positive-part estimator preserving the order of the sample singular values.     

Minimax Bayes estimators of a multivariate normal mean

In three or more dimensions it is well known that the usual point estimator for the mean of a multivariate normal distribution is minimax but not admissible with respect to squared Euclidean distance loss. This paper gives sufficient conditions on the prior distribution under which the Bayes estimator has strictly lower risk than the usual estimator. Examples are given for which the posterior density is useful in the formation of confidence sets.     

On minimaxity and admissibility of hierarchical Bayes estimators

This paper obtains conditions for minimaxity of hierarchical Bayes estimators in the estimation of a mean vector of a multivariate normal distribution. Hierarchical prior distributions with three types of second stage priors are treated. Conditions for admissibility and inadmissibility of the hierarchical Bayes estimators are also derived using the arguments in Berger and Strawderman [Choice of hierarchical priors: admissibility in estimation of normal means, Ann. Statist. 24 (1996) 931-951]. Combining these results yields admissible and minimax hierarchical Bayes estimators.     

Proper bayes minimax estimators for a multivariate normal mean with unknown common variance under a convex loss function

Summary Let X ∼ N_p(μ,σ²I_p) and let s/σ² ∼ χ _n ² , independent ofX, where μ and σ² are unknown. This paper considers the estimation of μ (by δ) relative to a convex loss function given by (δ−μ)′[(1−α)I_p/σ²+αQ](δ−μ)/[(1−α)p/σ²+α tr (Q)], whereQ is a knownp×p diagonal matrix and 0≦α≦1. Two classes of minimax estimators are obtained for μ whenp≦3; the first is a new result and the second is a generalization of a result of Strawderman (1973,Ann. Statist.,1, 1189–1194). A proper Bayes estimator is also obtained which is shown to satisfy the conditions of the second class of minimax estimators. The paper concludes by discussing the estimation of μ relative to another convex loss function. This work was supported by the Army, Navy and Air Force under Office of Naval Research Contract No. N00014-80-C-0093. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Admissible estimator of the eigenvalues of the variance-covariance matrix for multivariate normal distributions

An admissible estimator of the eigenvalues of the variance-covariance matrix is given for multivariate normal distributions with respect to the scale-invariant squared error loss.     

A subclass of bayes linear estimators that are minimax

In the linear regression model with ellipsoidal parameter constraints, the problem of estimating the unknown parameter vector is studied. A well-described subclass of Bayes linear estimators is proposed in the paper. It is shown that for each member of this subclass, a generalized quadratic risk function exists so that the estimator is minimax. Moreover, some of the proposed Bayes linear estimators are admissible with respect to all possible generalized quadratic risks. Also, a necessary and sufficient condition is given to ensure that the considered Bayes linear estimator improves the least squares estimator over the whole ellipsoid whatever generalized risk function is chosen.     

On Bayes estimators with uniform priors on spheres and their comparative performance with maximum likelihood estimators for estimating bounded multivariate normal means

For independently distributed observables: X_i∼N(θ_i,σ²),i=1,…,p, we consider estimating the vector θ=(θ₁,…,θ_p)^′ with loss ‖d−θ‖² under the constraint , with known τ₁,…,τ_p,σ²,m. In comparing the risk performance of Bayesian estimators δ_α associated with uniform priors on spheres of radius α centered at (τ₁,…,τ_p) with that of the maximum likelihood estimator , we make use of Stein’s unbiased estimate of risk technique, Karlin’s sign change arguments, and a conditional risk analysis to obtain for a fixed (m,p) necessary and sufficient conditions on α for δ_α to dominate . Large sample determinations of these conditions are provided. Both cases where all such δ_α’s and cases where no such δ_α’s dominate are elicited. We establish, as a particular case, that the boundary uniform Bayes estimator δ_m dominates if and only if m≤k(p) with , improving on the previously known sufficient condition of Marchand and Perron (2001) [3] for which . Finally, we improve upon a universal dominance condition due to Marchand and Perron, by establishing that all Bayesian estimators δ_π with π spherically symmetric and supported on the parameter space dominate whenever m≤c₁(p) with .     

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.     

Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distributions

The problem of estimating a mean vector of scale mixtures of multivariate normal distributions with the quadratic loss function is considered. For a certain class of these distributions, which includes at least multivariate-t distributions, admissible minimax estimators are given.     

Admissibility and minimaxity of generalized Bayes estimators for spherically symmetric family

We give a sufficient condition for admissibility of generalized Bayes estimators of the location vector of spherically symmetric distribution under squared error loss. Compared to the known results for the multivariate normal case, our sufficient condition is very tight and is close to being a necessary condition. In particular, we establish the admissibility of generalized Bayes estimators with respect to the harmonic prior and priors with slightly heavier tail than the harmonic prior. We use the theory of regularly varying functions to construct a sequence of smooth proper priors approaching an improper prior fast enough for establishing the admissibility. We also discuss conditions of minimaxity of the generalized Bayes estimator with respect to the harmonic prior.     

The Stein phenomenon for monotone incomplete multivariate normal data

We establish the Stein phenomenon in the context of two-step, monotone incomplete data drawn from , a (p+q)-dimensional multivariate normal population with mean and covariance matrix . On the basis of data consisting of n observations on all p+q characteristics and an additional N−n observations on the last q characteristics, where all observations are mutually independent, denote by the maximum likelihood estimator of . We establish criteria which imply that shrinkage estimators of James-Stein type have lower risk than under Euclidean quadratic loss. Further, we show that the corresponding positive-part estimators have lower risk than their unrestricted counterparts, thereby rendering the latter estimators inadmissible. We derive results for the case in which is block-diagonal, the loss function is quadratic and non-spherical, and the shrinkage estimator is constructed by means of a nondecreasing, differentiable function of a quadratic form in . For the problem of shrinking to a vector whose components have a common value constructed from the data, we derive improved shrinkage estimators and again determine conditions under which the positive-part analogs have lower risk than their unrestricted counterparts.     

A unified and generalized set of shrinkage bounds on minimax Stein estimates

Consider the problem of estimating the mean vector θ of a random variable X in , with a spherically symmetric density f(x−θ²), under loss δ−θ². We give an increasing sequence of bounds on the shrinkage constant of Stein-type estimators depending on properties of f(t) that unify and extend several classical bounds from the literature. The basic way to view the conditions on f(t) is that the distribution of X arises as the projection of a spherically symmetric vector (X,U) in . A second way is that f(t) satisfies (−1)^jf^(j)(t)≥0 for 0≤j≤ℓ and that (−1)^ℓf^(ℓ)(t) is non-increasing where k=2(ℓ+1). The case ℓ=0 (k=2) corresponds to unimodality, while the case ℓ=k=∞ corresponds to complete monotonicity of f(t) (or equivalently that f(x−θ²) is a scale mixture of normals). The bounds on the minimax shrinkage constant in this paper agree with the classical bounds in the literature for the case of spherical symmetry, spherical symmetry and unimodality, and scale mixtures of normals. However, they extend these bounds to an increasing sequence (in k or ℓ) of minimax bounds.     

Fiducial inference on the largest mean of a multivariate normal distribution

Inference on the largest mean of a multivariate normal distribution is a surprisingly difficult and unexplored topic. Difficulties arise when two or more of the means are simultaneously the largest mean. Our proposed solution is based on an extension of R.A. Fisher’s fiducial inference methods termed generalized fiducial inference. We use a model selection technique along with the generalized fiducial distribution to allow for equal largest means and alleviate the overestimation that commonly occurs. Our proposed confidence intervals for the largest mean have asymptotically correct frequentist coverage and simulation results suggest that they possess promising small sample empirical properties. In addition to the theoretical calculations and simulations we also applied this approach to the air quality index of the four largest cities in the northeastern United States (Baltimore, Boston, New York, and Philadelphia).     

Third-order power comparisons for a class of tests for multivariate linear hypothesis under general distributions

The purpose of this paper is, in multivariate linear regression model (Part I) and GMANOVA model (Part II), to investigate the effect of nonnormality upon the nonnull distributions of some multivariate test statistics under normality. It is shown that whatever the underlying distributions, the difference of local powers up to order N⁻¹ after either Bartlett’s type adjustment or Cornish-Fisher’s type size adjustment under nonnormality coincides with that in Anderson [An Introduction to Multivariate Statistical Analysis, 2nd ed. and 3rd ed., Wiley, New York, 1984, 2003] under normality. The derivation of asymptotic expansions is based on the differential operator associated with the multivariate linear regression model under general distributions. The performance of higher-order results in finite samples, including monotone Bartlett’s type adjustment and monotone Cornish-Fisher’s type size adjustment, is examined using simulation studies.     

Combining the data from two normal populations to estimate the mean of one when their means difference is bounded

In this paper we address the problem of estimating θ₁ when , are observed and |θ₁−θ₂|?c for a known constant c. Clearly Y₂ contains information about θ₁. We show how the so-called weighted likelihood function may be used to generate a class of estimators that exploit that information. We discuss how the weights in the weighted likelihood may be selected to successfully trade bias for precision and thus use the information effectively. In particular, we consider adaptively weighted likelihood estimators where the weights are selected using the data. One approach selects such weights in accord with Akaike's entropy maximization criterion. We describe several estimators obtained in this way. However, the maximum likelihood estimator is investigated as a competitor to these estimators along with a Bayes estimator, a class of robust Bayes estimators and (when c is sufficiently small), a minimax estimator. Moreover we will assess their properties both numerically and theoretically. Finally, we will see how all of these estimators may be viewed as adaptively weighted likelihood estimators. In fact, an over-riding theme of the paper is that the adaptively weighted likelihood method provides a powerful extension of its classical counterpart.     

Sequences of elliptical distributions and mixtures of normal distributions

Two conditions are shown under which elliptical distributions are scale mixtures of normal distributions with respect to probability distributions. The issue of finding the mixing distribution function is also considered. As a unified theoretical framework, it is also shown that any scale mixture of normal distributions is always a term of a sequence of elliptical distributions, increasing in dimension, and that all the terms of this sequence are also scale mixtures of normal distributions sharing the same mixing distribution function. Some examples are shown as applications of these concepts, showing the way of finding the mixing distribution function.     

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.