Abstract
The goal of this paper is to develop best possible estimates for the higher moments of a distribution of a positive bounded random variable, such as claim amounts, where these estimates are given in terms of the mean. First, upper and lower sharp estimates are developed for the second moment and variance in the case of a discrete random variable. A variety of applications are considered in detail with particular emphasis on claims analysis. Then these upper and lower estimates are generalized to higher-order moments and to continuous random variables, as well as to the associated moment generating functions.