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.     

Stein's phenomenon in estimation of means restricted to a polyhedral convex cone

This paper treats the problem of estimating the restricted means of normal distributions with a known variance, where the means are restricted to a polyhedral convex cone which includes various restrictions such as positive orthant, simple order, tree order and umbrella order restrictions. In the context of the simultaneous estimation of the restricted means, it is of great interest to investigate decision-theoretic properties of the generalized Bayes estimator against the uniform prior distribution over the polyhedral convex cone. In this paper, the generalized Bayes estimator is shown to be minimax. It is also proved that it is admissible in the one- or two-dimensional case, but is improved on by a shrinkage estimator in the three- or more-dimensional case. This means that the so-called Stein phenomenon on the minimax generalized Bayes estimator can be extended to the case where the means are restricted to the polyhedral convex cone. The risk behaviors of the estimators are investigated through Monte Carlo simulation, and it is revealed that the shrinkage estimator has a substantial risk reduction.     

Estimation, prediction and the Stein phenomenon under divergence loss

We consider two problems: (1) estimate a normal mean under a general divergence loss introduced in [S. Amari, Differential geometry of curved exponential families — curvatures and information loss, Ann. Statist. 10 (1982) 357-387] and [N. Cressie, T.R.C. Read, Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. Ser. B. 46 (1984) 440-464] and (2) find a predictive density of a new observation drawn independently of observations sampled from a normal distribution with the same mean but possibly with a different variance under the same loss. The general divergence loss includes as special cases both the Kullback-Leibler and Bhattacharyya-Hellinger losses. The sample mean, which is a Bayes estimator of the population mean under this loss and the improper uniform prior, is shown to be minimax in any arbitrary dimension. A counterpart of this result for predictive density is also proved in any arbitrary dimension. The admissibility of these rules holds in one dimension, and we conjecture that the result is true in two dimensions as well. However, the general Baranchick [A.J. Baranchick, a family of minimax estimators of the mean of a multivariate normal distribution, Ann. Math. Statist. 41 (1970) 642-645] class of estimators, which includes the James-Stein estimator and the Strawderman [W.E. Strawderman, Proper Bayes minimax estimators of the multivariate normal mean, Ann. Math. Statist. 42 (1971) 385-388] class of estimators, dominates the sample mean in three or higher dimensions for the estimation problem. An analogous class of predictive densities is defined and any member of this class is shown to dominate the predictive density corresponding to a uniform prior in three or higher dimensions. For the prediction problem, in the special case of Kullback-Leibler loss, our results complement to a certain extent some of the recent important work of Komaki [F. Komaki, A shrinkage predictive distribution for multivariate normal observations, Biometrika 88 (2001) 859-864] and George, Liang and Xu [E.I. George, F. Liang, X. Xu, Improved minimax predictive densities under Kullbak-Leibler loss, Ann. Statist. 34 (2006) 78-92]. While our proposed approach produces a general class of predictive densities (not necessarily Bayes, but not excluding Bayes predictors) dominating the predictive density under a uniform prior. We show also that various modifications of the James-Stein estimator continue to dominate the sample mean, and by the duality of estimation and predictive density results which we will show, similar results continue to hold for the prediction problem as well.     

Bayes minimax estimators of the mean of a scale mixture of multivariate normal distributions

Bayes estimation of the mean of a variance mixture of multivariate normal distributions is considered under sum of squared errors loss. We find broad class of priors (also in the variance mixture of normal class) which result in proper and generalized Bayes minimax estimators. This paper extends the results of Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255-264] in a manner similar to that of Maruyama [Admissible minimax estimators of a mean vector of scale mixtures of multivariate normal distribution, J. Multivariate Anal. 21 (2003) 69-78] but somewhat more in the spirit of Fourdrinier et al. [On the construction of bayes minimax estimators, Ann. Statist. 26 (1998) 660-671] for the normal case, in the sense that we construct classes of priors giving rise to minimaxity. A feature of this paper is that in certain cases we are able to construct proper Bayes minimax estimators satisfying the properties and bounds in Strawderman [Minimax estimation of location parameters for certain spherically symmetric distribution, J. Multivariate Anal. 4 (1974) 255-264]. We also give some insight into why Strawderman's results do or do not seem to apply in certain cases. In cases where it does not apply, we give minimax estimators based on Berger's [Minimax estimation of location vectors for a wide class of densities, Ann. Statist. 3 (1975) 1318-1328] results. A main condition for minimaxity is that the mixing distributions of the sampling distribution and the prior distribution satisfy a monotone likelihood ratio property with respect to a scale parameter.     

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.     

Modifying estimators of ordered positive parameters under the Stein loss

This paper treats the problem of estimating positive parameters restricted to a polyhedral convex cone which includes typical order restrictions, such as simple order, tree order and umbrella order restrictions. In this paper, two methods are used to show the improvement of order-preserving estimators over crude non-order-preserving estimators without any assumption on underlying distributions. One is to use Fenchel’s duality theorem, and then the superiority of the isotonic regression estimator is established under the general restriction to polyhedral convex cones. The use of the Abel identity is the other method, and we can derive a class of improved estimators which includes order-statistics-based estimators in the typical order restrictions. When the underlying distributions are scale families, the unbiased estimators and their order-restricted estimators are shown to be minimax. The minimaxity of the restrictedly generalized Bayes estimator against the prior over the restricted space is also demonstrated in the two dimensional case. Finally, some examples and multivariate extensions are given.     

Shrinkage priors for Bayesian estimation of the mean matrix in an elliptically contoured distribution

This paper deals with the problem of estimating the mean matrix in an elliptically contoured distribution with unknown scale matrix. The Laplace and inverse Laplace transforms of the density allow us not only to evaluate the risk function with respect to a quadratic loss but also to simplify expressions of Bayes estimators. Consequently, it is shown that generalized Bayes estimators against shrinkage priors dominate the unbiased estimator.     

Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data

In this article, the Stein-Haff identity is established for a singular Wishart distribution with a positive definite mean matrix but with the dimension larger than the degrees of freedom. This identity is then used to obtain estimators of the precision matrix improving on the estimator based on the Moore-Penrose inverse of the Wishart matrix under the Efron-Morris loss function and its variants. Ridge-type empirical Bayes estimators of the precision matrix are also given and their dominance properties over the usual one are shown using this identity. Finally, these precision estimators are used in a quadratic discriminant rule, and it is shown through simulation that discriminant methods based on the ridge-type empirical Bayes estimators provide higher correct classification rates.     

Variance function estimation in multivariate nonparametric regression with fixed design

Variance function estimation in multivariate nonparametric regression is considered and the minimax rate of convergence is established in the iid Gaussian case. Our work uses the approach that generalizes the one used in [A. Munk, Bissantz, T. Wagner, G. Freitag, On difference based variance estimation in nonparametric regression when the covariate is high dimensional, J. R. Stat. Soc. B 67 (Part 1) (2005) 19-41] for the constant variance case. As is the case when the number of dimensions d=1, and very much contrary to standard thinking, it is often not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean. Instead it is desirable to use estimators of the mean with minimal bias. Another important conclusion is that the first order difference based estimator that achieves minimax rate of convergence in the one-dimensional case does not do the same in the high dimensional case. Instead, the optimal order of differences depends on the number of dimensions.     

On improved estimation of normal precision matrix and discriminant coefficients

The problem of estimating the precision matrix of a multivariate normal distribution model is considered with respect to a quadratic loss function. A number of covariance estimators originally intended for a variety of loss functions are adapted so as to obtain alternative estimators of the precision matrix. It is shown that the alternative estimators have analytically smaller risks than the unbiased estimator of the precision matrix. Through numerical studies of risk values, it is shown that the new estimators have substantial reduction in risk. In addition, we consider the problem of the estimation of discriminant coefficients, which arises in linear discriminant analysis when Fisher's linear discriminant function is viewed as the posterior log-odds under the assumption that two classes differ in mean but have a common covariance matrix. The above method is also adapted for this problem in order to obtain improved estimators of the discriminant coefficients under the quadratic loss function. Furthermore, a numerical study is undertaken to compare the properties of a collection of alternatives to the “unbiased” estimator of the discriminant coefficients.     

Multivariate generalized S-estimators

In this paper we introduce generalized S-estimators for the multivariate regression model. This class of estimators combines high robustness and high efficiency. They are defined by minimizing the determinant of a robust estimator of the scatter matrix of differences of residuals. In the special case of a multivariate location model, the generalized S-estimator has the important independence property, and can be used for high breakdown estimation in independent component analysis. Robustness properties of the estimators are investigated by deriving their breakdown point and the influence function. We also study the efficiency of the estimators, both asymptotically and at finite samples. To obtain inference for the regression parameters, we discuss the fast and robust bootstrap for multivariate generalized S-estimators. The method is illustrated on a real data example.     

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.     

Other Classes of Minimax Estimators of Variance Covariance Matrix in Multivariate Normal Distribution

It is well known that the best equivariant estimator of the variance covariance matrix of the multivariate normal distribution with respect to the full affine group of transformation is not even minimax. Some minimax estimators have been proposed. Here we treat this problem in the framework of a multivariate analysis of variance (MANOVA) model and give other classes of minimax estimators.     

Improved estimation of a covariance matrix in an elliptically contoured matrix distribution

In this paper, the problem of estimating the covariance matrix of the elliptically contoured distribution (ECD) is considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean is proposed. It is shown that this new estimator dominates the unbiased estimator under the squared error loss function. Two special classes of ECD, namely, the multivariate-elliptical t distribution and the ε-contaminated normal distribution are considered. A simulation study is carried out and indicates that this new shrinkage estimator provides a substantial improvement in risk under most situations.     

On Bayes and unbiased estimators of loss

We consider estimation of loss for generalized Bayes or pseudo-Bayes estimators of a multivariate normal mean vector, θ. In 3 and higher dimensions, the MLEX is UMVUE and minimax but is inadmissible. It is dominated by the James-Stein estimator and by many others. Johnstone (1988, On inadmissibility of some unbiased estimates of loss,Statistical Decision Theory and Related Topics, IV (eds. S. S. Gupta and J. O. Berger), Vol. 1, 361–379, Springer, New York) considered the estimation of loss for the usual estimatorX and the James-Stein estimator. He found improvements over the Stein unbiased estimator of risk. In this paper, for a generalized Bayes point estimator of θ, we compare generalized Bayes estimators to unbiased estimators of loss. We find, somewhat surprisingly, that the unbiased estimator often dominates the corresponding generalized Bayes estimator of loss for priors which give minimax estimators in the original point estimation problem. In particular, we give a class of priors for which the generalized Bayes estimator of θ is admissible and minimax but for which the unbiased estimator of loss dominates the generalized Bayes estimator of loss. We also give a general inadmissibility result for a generalized Bayes estimator of loss. Research supported by NSF Grant DMS-97-04524.     

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.     

Linear minimax estimation with ellipsoidal constraints

We consider the simultaneous linear minimax estimation problem in linear models with ellipsoidal constraints imposed on an unknown parameter. Using convex analysis, we derive necessary and sufficient optimality conditions for a matrix to define the linear minimax estimator. For certain regions of the set of characteristics of linear models and constraints, we exploit these optimality conditions and get explicit formulae for linear minimax estimators.     

Statistical inference in a panel data semiparametric regression model with serially correlated errors

We consider a panel data semiparametric partially linear regression model with an unknown vector β of regression coefficients, an unknown nonparametric function g(·) for nonlinear component, and unobservable serially correlated errors. The correlated errors are modeled by a vector autoregressive process which involves a constant intraclass correlation. Applying the pilot estimators of β and g(·), we construct estimators of the autoregressive coefficients, the intraclass correlation and the error variance, and investigate their asymptotic properties. Fitting the error structure results in a new semiparametric two-step estimator of β, which is shown to be asymptotically more efficient than the usual semiparametric least squares estimator in terms of asymptotic covariance matrix. Asymptotic normality of this new estimator is established, and a consistent estimator of its asymptotic covariance matrix is presented. Furthermore, a corresponding estimator of g(·) is also provided. These results can be used to make asymptotically efficient statistical inference. Some simulation studies are conducted to illustrate the finite sample performances of these proposed estimators.     

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.