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The prediction problem for a multivariate normal distribution is considered where both mean and variance are unknown. When the Kullback–Leibler loss is used, the Bayesian predictive density based on the right invariant prior, which turns out to be a density of a multivariate t-distribution, is the best invariant and minimax predictive density. In this paper, we introduce an improper shrinkage prior and show that the Bayesian predictive density against the shrinkage prior improves upon the best invariant predictive density when the dimension is greater than or equal to three.

The limiting joint distribution of correlated Hotelling’s T 2 statistics associated with multiple comparisons with a control in multivariate one-way layout model is a multivariate central nonsingular chi-square distribution with one-factorial correlation matrix, which has the distribution function expressed in a closed form as an integral of a product of noncentral chi-square distribution functions with respect to a central chi-square density function. For pairwise comparisons, it is a multivariate central singular chi-square distribution whose distribution function is generally intricate. To overcome the complexity of the (exact or asymptotic) distribution theory of $$T^2_{\rm max}$$ -type statistics appeared in simultaneous confidence intervals of mean vectors, improved Bonferroni-type inequalities are applied to construct asymptotically conservative simultaneous confidence intervals for pairwise comparisons as well as comparisons with a control.

We give a general result to characterize a multivariate distribution from a relationship between the left truncated mean function and the hazard gradient function. This result allows us to obtain new characterizations of multivariate distributions. In particular, we show that, for the multivariate normal distribution, the simple relationship, obtained in standardized form by McGill (1992,Communications in Statistics. Theory Methods,21(11), 3053–3060), actually characterizes the multivariate normal distribution.

The existing model for multivariate skew normal data does not cohere with the joint distribution of a random sample from a univariate skew normal distribution. This incoherence causes awkward interpretation for data analysis in practice, especially in the development of the sampling distribution theory. In this paper, we propose a refined model that is coherent with the joint distribution of the univariate skew normal random sample, for multivariate skew normal data. The proposed model extends and strengthens the multivariate skew model described in Azzalini (1985,Scandinavian Journal of Statistics,12, 171–178). We present a stochastic representation for the newly proposed model, and discuss a bivariate setting, which confirms that the newly proposed model is more plausible than the one given by Azzalini and Dalla Valle (1996,Biometrika,83, 715–726).

Consider the test problem about matrix normal mean M with the null hypothesis M = O against the alternative that M is nonnegative definite. In our previous paper (Kuriki (1993, Ann. Statist., 21, 1379–1384)), the null distribution of the likelihood ratio statistic has been given in the form of a finite mixture of χ2 distributions referred to as X2 distribution. In this paper, we investigate differential-geometric structure such as second fundamental form and volume element of the boundary of the cone formed by real nonnegative definite matrices, and give a geometric derivation of this null distribution by virtue of the general theory on the X2 distribution for piecewise smooth convex cone alternatives developed by Takemura and Kuriki (1997, Ann. Statist., 25, 2368–2387).

In this paper, the problem of estimating the scale matrix and their eigenvalues in a Wishart distribution and in a multivariate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean are proposed. It is shown that the new estimator which dominates the usual unbiased estimator under the squared error loss function. A simulation study was carried out to study the performance of these estimators.

The maximum likelihood estimators are uniquely obtained in a multivariate normal distribution with AR(1) covariance structure for monotone data. The maximum likelihood estimator of mean is unbiased.

Approximations of density functions are considered in the multivariate case. The results are presented with the help of matrix derivatives, powers of Kronecker products and Taylor expansions of functions with matrix argument. In particular, an approximation by the Wishart distribution is discussed. It is shown that in many situations the distributions should be centred. The results are applied to the approximation of the distribution of the sample covariance matrix and to the distribution of the non-central Wishart distribution.

Any correlation matrixR can be mapped to a graph with edges corresponding to the non-vanishing correlations. In particularR is said to be of a “tree type” if the corresponding graph is a spanning tree. The tridiagonal correlation matrices belong to this class. If the accompanying correlation matrixR or its inverse is of a tree type, then some representations of the multivariate gamma distribution are obtained with a much simpler structure than the integral or series representations for the general case.

In the present paper we deal with the characterization of some dependence concepts for the multivariate normal distribution. It turns out that normal distributions have some special properties w.r.t. these dependence concepts and, furthermore, that the characterizations are closely connected to some interesting problems on matrices. Some applications to simultaneous confidence bounds are discussed.

In this paper we present simple approximations for the distributions of the extreme latent roots of three matrices occurring in multivariate analysis. The matrices considered are (i)S1S2−1 whereS1 andS2 are independent Wishart matrices estimating different covarance matrices, (ii)S1S2−1 whereS1 andS2 are independent and estimate the same covariance matrix, withS2 having the Wishart distribution andS1 having the noncentral Wishart distribution, and (iii) the noncentral Wishart matrix. The approximations take the form of upper and lower bounds for the distribution functions of the largest and smallest latent roots respectively. For the three matrices considered above these bounds are expressed very simply in terms of products of (i)F, (ii) noncentralF and (iii) noncentralX2 probabilities.

Expressions for the entries of the information matrix and skewness tensor of a general multivariate elliptic distribution are obtained. From these the coefficients of the a-connections are derived. A general expression for the asymptotic efficiency of the sample mean, when appropriate as an estimator of the location parameter, is obtained. The results are illustrated by examples from the multivariate normal, Cauchy and Student's t-distributions.

Fraser [3], [4] and Fraser and Haq [5] discussed a comprehensive multivariate model: a model with an error variable internal to the system with a known multivariate distribution and a positive affine transformation which generates a response vector from an error vector. Here a multivariate model, with the error variable having a multivariate normal distribution with intra-class covariance structure, has been considered. The analysis of the responses has been carried on in the framework of a transformed structural model and it produces structural distribution for the location parameters and the scale parameter, and the marginal likelihood function for the intra-class correlation coefficient.

The distribution of a quadratic form in a normal sample plays a very important role in multivariate statistical analysis. In many cases, statistics are functions of a quadratic form or special types of it. In the univariate case, the distribution of a quadratic form was treated by many authors and was derived by using the Laguerre polynomials expansion or the Dirichlet series expansion, etc. [6], [7], [8]. In this paper, the distribution of a quadratic form in the multivariate case will be given in terms of zonal polynomials which were developed for multivariate analysis by A. T. James [3], [4] and A. G. Constantine [2]. Recently, the author’s attention was called to C. G. Khatri [9] which deals with the same problem also by using zonal polynomials. However, the present paper treats the problem from another point of view. The distribution of a quadratic form enables us to derive the distribution of a linear combination of several Wishart matrices. The distributions and probability functions of certain statistics of a quadratic form are given.

This paper considers the problem of classifying a multivariate normal population into one of thek multivariate normal populations when the population means are linearly restricted and the common nonsingular covariance matrix is unknown. It is shown that the maximum likelihood rule is an admissible rule. An example is also given to explain the procedure.

New asymptotic expansions of the non-null distributions of the likelihood ratio, Hotelling's and Pillai's statistics for multivariate linear hypothesis are given in terms of normal distribution function and its derivatives, assuming the matrix of noncentrality parameters is of the same order as the sample size.