The discrete Mohr and Noll inequality with applications to variance bounds
Date
2007Author
Afendras, GeorgiosPapadatos, Nickos
Papathanasiou, Vassilis
Source
Sankhya: The Indian Journal of StatisticsVolume
69Issue
2Pages
162-189Google Scholar check
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In this paper, we provide Poincaré-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 io 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 interest in itself. It has been shown by Johnson (1993, Statist. Decisions, vol. 11, pp. 273-278) that for the commonly used absolutely continuous distributions 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. © 2007, Indian Statistical Institute.