Unified extension of variance bounds for integrated Pearson family

TL;DR

This paper introduces a unified approach to derive improved variance bounds for functions of absolutely continuous random variables using orthogonal polynomial properties.

Contribution

It provides a new class of variance bounds that are tighter than existing bounds by leveraging orthogonal polynomial techniques.

Findings

New variance bounds outperform previous bounds

Bounds applicable to functions with derivatives up to a certain order

Enhanced understanding of variance estimation for continuous variables

Abstract

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.