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May 2011 An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds
G. Afendras, N. Papadatos, V. Papathanasiou
Bernoulli 17(2): 507-529 (May 2011). DOI: 10.3150/10-BEJ282

Abstract

For an absolutely continuous (integer-valued) r.v. $X$ of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order $k$ holds (cf. [Goldstein and Reinert, J. Theoret. Probab. 18 (2005) 237–260]). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. $X$, expressions that seem to be known only in particular cases (for the Normal, see [Houdré and Kagan, J. Theoret. Probab. 8 (1995) 23–30]; see also [Houdré and Pérez-Abreu, Ann. Probab. 23 (1995) 400–419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.

Citation

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G. Afendras. N. Papadatos. V. Papathanasiou. "An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds." Bernoulli 17 (2) 507 - 529, May 2011. https://doi.org/10.3150/10-BEJ282

Information

Published: May 2011
First available in Project Euclid: 5 April 2011

zbMATH: 1345.60009
MathSciNet: MR2787602
Digital Object Identifier: 10.3150/10-BEJ282

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability

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Vol.17 • No. 2 • May 2011
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