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

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. © 2011 ISI/BS.