An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds

Details An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds An extended Stein-type covariance identity for the Pearson family with applications to lower variance bounds

2010-07-21

arxiv.org/abs/1007.3662v2

Bernoulli 2011, Vol. 17, No. 2, 507-529

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/10-BEJ282 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ282

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 [Houdr\'{e} and Kagan, J. Theoret. Probab. 8 (1995) 23--30]; see also [Houdr\'{e} and P\'{e}rez-Abreu, Ann. Probab. 23 (1995) 400--419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.

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