Generalization to $N$ Dimensions of Inequalities of the Tchebycheff Type

Abstract

The Tchebycheff statistical inequality and its generalizations are further generalized so as to apply equally well to $n$-dimensional probability distributions. Comparisons may be made with other generalizations [1], [2] that have been developed recently for the two-dimensional case. The inequalities given in this paper are generally as close as the most favorable corresponding inequalities that exist for the one-dimensional case and in many simple cases they are closer than those that have been given heretofore for two dimensions. In a special case the upper bound of our inequality is actually attained. The theory contains also a less important generalization in one dimension.