Empirical Bayes estimators are given for the mean of a k-dimensional normal distribution, k = 3. We assume that y ~ Nk([theta], V1), V1 = diag(vi), vi known (i = 1, 2,..., k); also, [theta] ~ Nk(0, V2) - V2 defined by one or more unknown parameters. Of particular interest is V2 generated by an...

A simple relationship is given between the exact null distribution gm,n(J) of the J-th largest latent root of an m - m Wishart matrix on n degrees of freedom, and the distribution fm,n(J) of the ratio of this root to the trace of the matrix. Explicit expressions for certain fm,n(J) may thus be...

Reasonably simple expressions are given for some hypergeometric functions when the size of the argument matrix or matrices is two. Applications of these expressions in connection with the distributions of the latent roots of a 2 - 2 Wishart matrix are also given.

When a system is acted upon by exterior disturbances, its time-development can often be described by a system of ordinary differential equations, provided that the disturbances are smooth functions. But for sound reasons physicists and engineers usually want the theory to apply when the noises...

Let X be a p-variate (p = 3) vector normally distributed with mean [mu] and covariance [Sigma], and let A be a p - p random matrix distributed independent of X, according to the Wishart distribution W(n, [Sigma]). For estimating [mu], we consider estimators of the form [delta] = [delta](X, A)....

Asymptotic expansions of the distributions of two test criteria concerning a covariance matrix are derived under local alternatives in terms of noncentral [chi]2 variates, and under the fixed alternative in terms of standard normal distribution function and its derivatives, respectively. Some...

Let {Xn}n=1 be a sequence of independent and identically distributed random variables. For each integer n = 1 and positive constants r, t, and [epsilon], let Sn = [Sigma]j=1n Xj and E{N[infinity](r, t, [epsilon])} = [Sigma]n=1[infinity] nr-2P{Sn [epsilon]nr/t}. In this paper, we prove that (1)...

In this paper we derive asymptotic expansions for the distributions of some functions of the latent roots of the matrices in three situations in multivariate normal theory, i.e., (i) principal component analysis, (ii) MANOVA model and (iii) canonical correlation analysis. These expansions are...

This paper considers the Maximum Likelihood (ML) estimation of the five parameters of a linear structural relationship y = [alpha] + [beta]x when [alpha] is known. The parameters are [beta], the two variances of observation errors on x and y, the mean and variance of x. When the ML estimates of...

The recently developed strong approximation methods are discussed and applied to the problem of testing whether two independent multivariate samples come from the same population and whether the components of the observations are independent. The usual Cramér-von Mises statistic, as well as one...