Nonparametric Estimation, III. Statistically Equivalent Blocks and Multivariate Tolerance Regions--The Discontinuous Case

Abstract

In Paper II of this series [2, 1947] it was shown that if $n$ functions and a sample of $n$ were used to divide the population space into $n + 1$ blocks in a particular way, and if the joint cumulative of the functions were continuous, then the $n + 1$ fractions of the population, corresponding to the $n + 1$ blocks, were distributed symmetrically and simply. In Paper I of this series [1, 1945] it was shown that the one-dimensional theory of tolerance regions could be extended to the discontinuous case, if equalities were replaced by inequalities. In this paper the results of Paper II will be extended to the discontinuous case with the same weakening of the conclusion. The devices involved are more complex, but the nature of the results is the same (See Section 5). As a tool, it is shown that any $n$-variate distribution can be represented in terms of an $n$-variate distribution with a continuous joint cumulative (in fact, with uniform univariate marginals), where each variate of the given distribution is a different monotone function of the corresponding variate from the continuous distribution.