Efron's monotonicity property for measures on $\mathbb{R}^2$

TL;DR

This paper extends Efron's monotonicity property from independent log-concave measures to general measures on ^2 using kernel representations for covariance and related functionals, providing new formulas and quantitative estimates.

Contribution

It introduces kernel representations for covariance and functionals, enabling the extension of Efron's monotonicity property to broader measures on ^2.

Findings

Extended Efron's monotonicity property to general measures on ^2

Derived new kernel formulas for covariance and functionals

Provided quantitative estimates related to the property

Abstract

First we prove some kernel representations for the covariance of two functions taken on the same random variable and deduce kernel representations for some functionals of a continuous one-dimensional measure. Then we apply these formulas to extend Efron's monotonicity property, given in Efron [1965] and valid for independent log-concave measures, to the case of general measures on $\mathbb{R}^2$. The new formulas are also used to derive some further quantitative estimates in Efron's monotonicity property.