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.

A note on a variance bound for the multinomial and the negative multinomial distribution

“…Chernoff proved that , provided that is finite, where the equality holds iff g is a linear function; see also the previous papers by Nash , Brascamp and Lieb . This inequality has been generalized and extended by many authors (see, e.g., ).…”

“…In the discrete case, let X be an integer‐valued random variable with probability mass function (pmf) p and finite mean μ and variance ; consider the function w given by Then for any suitable function g , the following inequality and Stein‐type covariance identity hold (see Cacoullos and Papathanasiou [11, Lemma 2.2] and [12, eq. (3.2)]) where Δ is the forward difference operator (for the cases where w is a quadratic polynomial see also Afendras et al ).…”

Orthogonal polynomials in the cumulative Ord family and its application to variance bounds

“…Let X ∼ CO(µ; q). Afendras et al [4] studied the orthogonal polynomials generated by a Rodrigues-type formula (see Theorem 7.3 below) and based on these polynomials, they prove Stein-type covariance identities [see 4, Eq. (2.7), p. 512].…”

“…Independently of Hildebrandt's results, Afendras et al [4] studied the orthogonality of the Rodrigues polynomials in the CO family: For each k = 0, 1, 2, . .…”

A general parametric Stein characterization

“…for Pearson distributions (Afendras et al, 2011), general densities (Ley and Swan, 2013), invariant measures of diffusions (Kusuoka and Tudor, 2013)). …”