A convergence theorem of Billingsley for the empirical process of stationary, real valued radom variables under a mixing condition is generalized to the k-dimensional and nonstationary case. Further a more general empirical process is treated, including the upper summation boundary as argument....

This part is concerned with the applications of the general limit theorems with rates of Part I, achieved by specializing the limiting r.v. X. This leads to new convergence theorems with higher order rates in the one- and multi-dimensional case for the stable limit law, for the central limit...

Set-valued measures whose values are subsets of a Banach space are studied. Some basic properties of these set-valued measures are given. Radon-Nikodym theorems for set-valued measures are established, which assert that under suitable assumptions a set-valued measure is equal (in closures) to...

In three or more dimensions it is well known that the usual point estimator for the mean of a multivariate normal distribution is minimax but not admissible with respect to squared Euclidean distance loss. This paper gives sufficient conditions on the prior distribution under which the Bayes...

In this paper we extend the method of Bernstein using some concepts of differential geometry and prove the existence of a partial differential equation which when solved under suitable boundary conditions leads to a density of a bivariate inverse Gaussian process.

The asymptotic power of the Cramér-von Mises test (CvM test) when parameters are estimated from the data is studied under certain local (contiguous) alternatives. The notion of (asymptotic) direction and distance from the null hypothesis of alternatives is introduced, and it is shown that there...

Let p, q be arbitrary parameter sets, and let 9 be a Hilbert space. We say that x = (xi)i[epsilon]q, xi [epsilon] 9, is a bounded operator-forming vector ([epsilon]9Fq) if the Gram matrix <x, x> = [(xi, xj)]i[epsilon]q,j[epsilon]q is the matrix of a bounded (necessarily = 0) operator on 6, the Hilbert...</x,>

Union-intersection is a heuristic method of test construction developed by S. N. Roy. Among the well-known applications of this principle is the test for independence between two sets of variates which leads directly to the concept of canonical correlation. Another multivariate application of...

In the first part of this paper, nonlinear prediction theory of vector valued random variables in Orlicz spaces is presented. The spaces need not be reflexive and the results of this part are essentially best possible for these spaces. The second part considers operator valued martingales in the...

For the several sample problem, a vector of estimable parameters is considered. For a fixed total sample size, a multistage (sequential) procedure based on generalized U-statistics is developed for choosing a partition of this sample size into individual sample size for which the generalized...