The coefficients a[tau][varrho], sometimes called "generalized binomial coefficients" in the expansion C[varrho]*(V +I) = [Sigma][tau]a[varrho][tau]C[tau]*(V), are computed explicitly when t = r + 1, where [varrho] is a partition of r and [tau] a partition of t. A recursion formula permits the...

Consider a standard row-column-exchangeable array X = (Xij : i,j = 1), i.e., Xij = f(a, [xi]i, [eta]j, [lambda]ij) is a function of i.i.d. random variables. It is shown that there is a canonical version of X, X', such that X', and [alpha]', [xi]'1, [xi]'2,..., [eta]'1, [eta]'2,..., are...

The purpose of this note is to establish the connection between the ultraspherical polynomials and distributions on the m-sphere. Certain functions defined on the m-sphere have an ultraspherical decomposition which can be used to advantage. Examples of their use are given.

Let x be a random sample with a distribution depending on a vector parameter [theta] [set membership, variant] m. The description of distributions and generalized prior densities on m is given, for which the generalized Bayes estimator of [theta], based on x, is the same for all symmetric loss...

L[alpha] (0 [less, double equals] [alpha] [less, double equals] 1) is a class of infinitely divisible distributions defined by restricting the measure in the Lévy-Khinchin formula to a special form. When [alpha] = 1, L[alpha] is just the classical class L. Several properties for L[alpha]...

If E is an ordered set, we study the processes Yt, t [set membership, variant] E, for which the vectorial spaces t generated by all the conditional expectations E(Ys[beta]t) for s = t have finite dimensions d(t) = N. (t is some convenient filtration.) We first develop a geometrical approach in...

Let Fn(x) be the empirical distribution function based on n independent random variables X1,...,Xn from a common distribution function F(x), and let be the sample mean. We derive the rate of convergence of to normality (for the regular as well as nonregular cases), a law of iterated logarithm,...

The density of the Langevin (or Fisher-Von Mises) distribution is proportional to exp ?[mu]'x, where x and the modal vector [mu] are unit vectors in q. ? (=0) is called the concentration parameter. The distribution of statistics for testing hypotheses about the modal vectors of m distributions...

The limiting distribution of the likelihood ratio statistic Wq for testing the hypothesis of equality of q characteristic roots of a covariance matrix is studied in the case of nonnormal populations. It is shown, both theoretically and empirically, that the limiting distribution of Wq is not...

A random evolution process constructed from regular step processes with a common state space and indexed on an evolution rule space is shown to be a regular step process on the product space. Conversely, it is shown that under mild conditions, any regular step process on a product space is...