Estimation of a covariance matrix $\sum$ is a notoriously difficult problem; the standard unbiased estimator can be substantially suboptimal. We approach the problem from a noninformative prior Bayesian perspective, developing the reference noninformative prior for a covariance matrix and obtaining expressions for the resulting Bayes estimators. These expressions involve the computation of high-dimensional posterior expectations, which is done using a recent Markov chain simulation tool, the hit-and-run sampler. Frequentist risk comparisons with previously suggested estimators are also given, and determination of the accuracy of the estimators is addressed.

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Estimation of a Covariance Matrix Using the Reference Prior

Two stage conditionally unbiased estimators of the selected mean

In this review of estimation problems in restricted parameter spaces, we focus through a series of illustrations on a number of methods that have proven to be successful. These methods relate to the decision-theoretic aspects of admissibility and minimaxity, as well as to the determination of dominating estimators of inadmissible procedures obtained for instance from the criteria of unbiasedness, maximum likelihood, or minimum risk equivariance. Finally, we accompany the presentation of these methods with various related historical developments.

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Estimation in restricted parameter spaces: a review

: In the problem of estimating the covariance matrix of a multivariate normal population James and Stein obtained a minimax estimator by considering the best invariant estimator with respect to the triangular group. In this paper its authors propose an orthogonally invariant estimator obtained by averaging the minimax estimator with respect to the invariant measure on the orthogonal group. Explicit forms of the proposed estimator are given for dimensions 2 and 3. Risk is evaluated for various population covariance matrices and it shows a substantial improvement over the minimax estimator for a wide range of population covariance matrices.

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An orthogonally invariant minimax estimator of the covariance matrix of a multivariate normal population

Let Sp × p have a Wishart distribution with parameter matrix Σ and n degrees of freedom. We consider here the problem of estimating the precision matrix Σ−1 under the loss functions L1(σ) tr (σ) - log |σ| and L2(σ) = tr (σ). James-Stein-type estimators have been derived for an arbitrary p. We also obtain an orthogonal invariant and a diagonal invariant minimax estimator under both loss functions. A Monte-Carlo simulation study indicates that the risk improvement of the orthogonal invariant estimators over the James-Stein type estimators, the Haff (1979) estimator, and the “testimator” given by Sinha and Ghosh (1987) is substantial.



Soit Sp × p une matrice aleatoire suivant une loi de Wishart avec n degres de liberte et matrice des parametres Σ. On etudie le probleme de l'estimation de Σ−1 lorsqu'un utilise les fonctions de perte: L1(σ) tr (σ) - log |σ| et L2(σ) = tr (σ). Des estimateurs de type James-Stein sont definis pour n'importe quelle valeur de p. Relativement aux deux fonctions de perte considereaes, on obtient des estimateurs minimax qui sont invariants sous les transformations orthogonales ou celles dont la matrice est triangulaire (infearieure). Une eatude de simulation Monte-Carlo indique qu'on obtient une diminution importante du risque en utilisant les estimateurs invariants sous les transformations orthogonales plutot que les estimateurs de type James-Stein, ou de Haff (1979), ou le “test-estimateur” propose par Sinha et Ghosh (1987).

Improved minimax estimation of a normal precision matrix

The problem of estimating ordered parameters is encountered in biological, agricultural, reliability and various other experiments. Consider two populations with densities f1(x1-ω1) and f2(x2-ω2) where ω1#ω2. The estimation of ω1,ω2) with the loss function, the sum of squared errors, is studied. when fi is the fi(,i,,i 2) density with ,i known, i=1,2; we obtain a class of minimax estimators. When ω1 #ω2 we show some of these estimators are improved by the maximum likelihood estimator. For a general fi we give sufficient conditions for the minimaxity of the analogue of the Pitman estimator.

Simultaneous estimation of ordered parameters

The scale and orthogonal equivariant minimax estimators are obtained for the bivariate normal covariance matrix and precision matrix under Selliah's (1964} and Stein's (1961) loss functions. These new estimators are better than Selliah's and Stein's minimax estimators. An unbiased estimator of the risk of the new estimator is obtained under Selliah's loss function using Haff's (1979) identity for the Wishart distribution. Simulation results seem to indicate that the new estimators dominate the corresponding Hatf's [(1979), (1980)] estimators. We also prove that, for p-2, Haff's estimators are not minimax.

Orthogonal Equivariant Minimax Estimatorsof Bivariate Normal Covariance Matrix and Precision Matrix

Let л1 and л2 denote two independent gamma populations G(α1, p) and G(α2, p) respectively. Assume α(i=1,2)are unknown and the common shape parameter p is a known positive integer. Let Yi denote the sample mean based on a random sample of size n from the i-th population. For selecting the population with the larger mean, we consider, the natural rule according to which the population corresponding to the larger Yi is selected. We consider? in this paper, the estimation of M, the mean of the selected population. It is shown that the natural estimator is positively biased. We obtain the uniformly minimum variance unbiased estimator(UMVE) of M. We also consider certain subclasses of estikmators of the form c1x(1) +c1x(2) and derive admissible estimators in these classes. The minimazity of certain estimators of interest is investigated. Itis shown that p(p+1)-1x(1) is minimax and dominates the UMVUE. Also UMVUE is not minimax.

Estimation of the mean of the selected gamma population

Let be independent normal random variables with means and the common variance unity. It is assumed that We show that the Pitman estimator the generalized Bayes estimator of with respect to the uniform prior on is minimax. When k = 2, the components of for estimating θ1and θ2 are minimax (Cohen and Sackrowitz (1970)). However, for k = 3, we prove that the similar result does not hold for θ1 and θ3 Admississibility of in certain classes of estimators is also discussed.

On the pitman estimator op ordered normal means

Let be k independent gamma populations with unknown scale parameters and a common known shape parameter p > o. For the case k -2 and p a positive integer, Vellaisamy and Sharma (1988) obtained the uniformly minimum variance unbiased estimator (UMVUE) of M, the mean of the selected gamma population, and showed that the estimator dominates the natural estimator T -X(1) for squared error loss. In this note, the above two results are shown to be true for an arbitrary k and p > o, using the (u,V) -method of Bobbins (1988).

Divakar Sharma

Papers

Simultaneous estimation of ordered parameters

Orthogonal Equivariant Minimax Estimatorsof Bivariate Normal Covariance Matrix and Precision Matrix

Estimation of the mean of the selected gamma population

On the pitman estimator op ordered normal means

A note on the estimation op the mean op the selected gamma population