Moment inequalities via optimal design theory

Abstract

A unified approach is used to rediscover a class of moment inequalities. In particular, complementary inequalities in respect to ratios and differences of means and also inequalities complementary to both Hölder's and Minkowski's inequalities, each of which have been established in the literature by ad hoc methods, are rediscovered. The impetus of the paper is that this unified method is simple and illuminating. A class of optimization problems is considered in which optimization is with respect to a probability distribution. Moment inequalities are implied by the solutions, equality occuring under the optimizing distribution. However, we do not claim to prove any fundamentally new inequalities at this juncture. The method used is illuminating in that it first discovers the support points of the optimizing distribution. It then proves to be simple to discover the optimizing probabilities. The method employs the differential calculus approach developed to establish optimality conditions for optimal linear regression designs. However, the link with optimal design theory is not just this tenous one. Results in respect of optimal design criteria of themselves have implications concerning moments, including the multivariate generalization of the nonnegativity of a variance.