Sankhya: The Indian Journal of Statistics

2007, Volume 69, Pt. 2, 190--220

The Discrete Mohr and Noll Inequality with Applications to Variance Bounds

G. Afendras, N. Papadatos and V. Papathanasiou, University of Athens, Greece

SUMMARY. In this paper, we provide Poincar\'{e}-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, {\it Math.\ Nachr.}, vol.\ 7, pp.\ 55--59), which may be of some independent interest in itself. It has been shown by Johnson (1993, {\it 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.

AMS (2000) subject classification. Primary 60E15.

Key words and phrases. Poincar\'{e}-type variance bounds, Mohr and Noll inequality, discrete Pearson family, UMVUE of $\log p$ in geometric distribution.

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