Tatsuya Kubokawa

An approach to improving the James-Stein estimator

For the mean vector of a p-variate normal distribution (p [greater, double equals] 3), the generalized Bayes estimators dominating the James-Stein estimator under quadratic loss are given based on the methods of Brown, Brewster and Zidek for estimating a normal variance.

Shrinkage and modification techniques in estimation of variance and the related problems: A review

One of the surprising decision-theoretic results Charles Stein discovered is the inadmissibility of the uniformly minimum variance unbiased estirnator(UMVUE) of the variance of a normal distribution with an unknown mean. Some methods for deriving estimators better than the UMVUE were given by Stein. Brown, Brewster and Zidek. Recently Kubokawa established a novel approach, called the IERD method, by use of which one gets a unified class of improved estimators including their previous procedures. This paper gives a review for a series of these decision-theoretical developments as well as surveys the study of the variance-estimation problem from various aspects. Related to this issue, the paper enumerates several topics with the situations where the usual plain estimators are required to be shrunken or modified, and gives reasonable procedures improving the usual ones through the IERD method.

Estimation of noncentrality parameters

We propose some estimators of noncentrality parameters which improve upon usual unbiased estimators under quadratic loss. The distributions we consider are the noncentral chi-square and the noncentral F. However, we give more general results for the family of elliptically contoured distributions and propose a robust dominating estimator.

Double shrinkage estimation of ratio of scale parameters

The problems of estimating ratio of scale parameters of two distributions with unknown location parameters are treated from a decision-theoretic point of view. The paper provides the procedures improving on the usual ratio estimator under strictly convex loss functions and the general distributions having monotone likelihood ratio properties. In particular,double shrinkage improved estimators which utilize both of estimators of two location parameters are presented. Under order restrictions on the scale parameters, various improvements for estimation of the ratio and the scale parameters are also considered. These results are applied to normal, lognormal, exponential and pareto distributions. Finally, a multivariate extension is given for ratio of covariance matrices.

Tests for covariance matrices in high dimension with less sample size

In this article, we propose tests for covariance matrices of high dimension with fewer observations than the dimension for a general class of distributions with positive definite covariance matrices. In the one-sample case, tests are proposed for sphericity and for testing the hypothesis that the covariance matrix @S is an identity matrix, by providing an unbiased estimator of tr[@S^2] under the general model which requires no more computing time than the one available in the literature for a normal model. In the two-sample case, tests for the equality of two covariance matrices are given. The asymptotic distributions of proposed tests in the one-sample case are derived under the assumption that the sample size N=O(p^@d),1/2<@d<1, where p is the dimension of the random vector, and O(p^@d) means that N/p goes to zero as N and p go to infinity. Similar assumptions are made in the two-sample case.

Tests for multivariate analysis of variance in high dimension under non-normality

In this article, we consider the problem of testing the equality of mean vectors of dimension p of several groups with a common unknown non-singular covariance matrix @S, based on N independent observation vectors where N may be less than the dimension p. This problem, known in the literature as the multivariate analysis of variance (MANOVA) in high-dimension has recently been considered in the statistical literature by Srivastava and Fujikoshi (2006) [8], Srivastava (2007) [5] and Schott (2007) [3]. All these tests are not invariant under the change of units of measurements. On the lines of Srivastava and Du (2008) [7] and Srivastava (2009) [6], we propose a test that has the above invariance property. The null and the non-null distributions are derived under the assumption that (N,p)->~ and N may be less than p and the observation vectors follow a general non-normal model.

Estimating the covariance matrix: a new approach

In this paper, we consider the problem of estimating the covariance matrix and the generalized variance when the observations follow a nonsingular multivariate normal distribution with unknown mean. A new method is presented to obtain a truncated estimator that utilizes the information available in the sample mean matrix and dominates the James-Stein minimax estimator. Several scale equivariant minimax estimators are also given. This method is then applied to obtain new truncated and improved estimators of the generalized variance; it also provides a new proof to the results of Shorrock and Zidek (Ann. Statist. 4 (1976) 629) and Sinha (J. Multivariate Anal. 6 (1976) 617).

Conditional and unconditional methods for selecting variables in linear mixed models

In the problem of selecting the explanatory variables in the linear mixed model, we address the derivation of the (unconditional or marginal) Akaike information criterion (AIC) and the conditional AIC (cAIC). The covariance matrices of the random effects and the error terms include unknown parameters like variance components, and the selection procedures proposed in the literature are limited to the cases where the parameters are known or partly unknown. In this paper, AIC and cAIC are extended to the situation where the parameters are completely unknown and they are estimated by the general consistent estimators including the maximum likelihood (ML), the restricted maximum likelihood (REML) and other unbiased estimators. We derive, related to AIC and cAIC, the marginal and the conditional prediction error criteria which select superior models in light of minimizing the prediction errors relative to quadratic loss functions. Finally, numerical performances of the proposed selection procedures are investigated through simulation studies.

Estimating the covariance matrix and the generalized variance under a symmetric loss

For estimating the power of a generalized variance under a multivariate normal distribution with unknown means, the inadmissibility of the best affine equivariant estimator relative to the symmetric loss is shown, and a class of improved estimators is given. The problem of estimating the covariance matrix is also discussed.

Conditional information criteria for selecting variables in linear mixed models

In this paper, we consider the problem of selecting the variables of the fixed effects in the linear mixed models where the random effects are present and the observation vectors have been obtained from many clusters. As the variable selection procedure, we here use the Akaike Information Criterion, AIC. In the context of the mixed linear models, two kinds of AIC have been proposed: marginal AIC and conditional AIC. In this paper, we derive three versions of conditional AIC depending upon different estimators of the regression coefficients and the random effects. Through the simulation studies, it is shown that the proposed conditional AIC fs are superior to the marginal and conditional AIC fs proposed in the literature in the sense of selecting the true model. Finally, the results are extended to the case when the random effects in all the clusters are of the same dimension but have a common unknown covariance matrix.