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

TL;DR

This paper extends Stein-type covariance identities for Pearson family distributions, providing new lower variance bounds and applications, especially relating to orthogonal polynomials and classical distributions.

Contribution

It introduces a generalized Stein-type covariance identity of order k for Pearson family variables, linking it to orthogonal polynomials and deriving novel variance bounds.

Findings

Derived a generalized covariance identity for Pearson distributions.

Provided new lower variance bounds for functions of Pearson variables.

Applied the identities to classical distributions like Normal, Wiener, and Poisson.

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 [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.