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

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.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ282 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Strengthened Chernoff-type variance bounds

Let $X$ be an absolutely continuous random variable from the integrated Pearson family and assume that $X$ has finite moments of any order. Using some properties of the associated orthonormal polynomial system, we provide a class of strengthened Chernoff-type variance bounds.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ484 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Unified extension of variance bounds for integrated Pearson family

We use some properties of orthogonal polynomials to provide a class of upper/lower variance bounds for a function g(X) of an absolutely continuous random variable X, in terms of the derivatives of g up to some order. The new bounds are better than the existing ones. © 2012 The Institute of Statistical Mathematics, Tokyo

Moment-based inference for pearson's quadratic q subfamily of distributions

The author uses a Stein-type covariance identity to obtain moment estimators for the parameters of the quadratic polynomial subfamily of Pearson distributions. The asymptotic distribution of the estimators is obtained, and normality and symmetry tests based on it are provided. © 2013 Taylor & Francis Group, LLC

A factorial moment distance and an application to the matching problem

In this note we introduce the notion of factorial moment distance for nonnegative integer-valued random variables, and we compare it with the total variation distance. Furthermore, we study the rate of convergence in the classical matching problem and in a generalized matching distribution. © 2018, Society for Industrial and Applied Mathematics Publications. All rights reserved

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

We prove a Chernoff-type upper variance bound for the multinomial and the negative multinomial distribution. An application is also given. Copyright © 2014 Wiley Periodicals, Inc

Strengthened Chernoff-type variance bounds

Let X be an absolutely continuous random variable from the integrated Pearson family and assume that X has finite moments of any order. Using some properties of the associated orthonormal polynomial system, we provide a class of strengthened Chernoff-type variance bounds. © 2014 ISI/BS

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

This article presents and reviews several basic properties of the Cumulative Ord family of distributions; this family contains all the commonly used discrete distributions. A complete classification of the Ord family of probability mass functions is related to the orthogonality of the corresponding Rodrigues polynomials. Also, for any random variable X of this family and for any suitable function g in L2(R, X),, the article provides useful relationships between the Fourier coefficients of g (with respect to the orthonormal polynomial system associated to X) and the Fourier coefficients of the forward difference of g (with respect to another system of polynomials, orthonormal with respect to another distribution of the system). Finally, using these properties, a class of bounds for the variance of g(X) is obtained, in terms of the forward differences of g. These bounds unify and improve several existing results. © 2017 Informa UK Limited, trading as Taylor & Francis Group