The Discrete Mohr and Noll Inequality with Applications to Variance Bounds

Abstract

In this paper, we provide Poincare-type upper and lower variance bounds for a function g(X) of a discrete integer-valued random variable (r.v.) X, in terms of the (forward) differences of g up to some order. To this end, we investigate a discrete analogue of the Mohr and Noll inequality (1952, Math. Nachr., vol. 7, pp. 55-59), which may be of some independent inter- est in itself. It has been shown by Johnson (1993, Statist. Decisions, vol. 11, pp. 273-278) that for the commonly used absolutely continuous distribu- tions that belong to the Pearson family, the somewhat complicated variance bounds take a very pleasant and simple form. We show here that this is also true for the commonly used discrete distributions. As an application of the proposed inequalities, we study the variance behaviour of the UMVU estimator of log p in Geometric distributions.