Generalizations of Tchebycheff’s Inequalities

Abstract

Summary We may often obtain the lower moments of a sampling distribution when the distribution itself is intractable. Tchebycheff’s celebrated inequalities for the cumulative distribution function are of very little use in practice, being “much too wide”. The present work obtains generalizations of these inequalities by restricting the distribution function to be “smooth” in two senses: firstly, by restricting the number of changes of sign of some derivative, and secondly by also restricting the size of this derivative. These conditions lead to certain “extremal” distributions, and hence to the required inequalities. These have been proved correct only in the simpler cases, but are conjectured to hold in general. In Part I some of the simpler results are given explicitly.