Iterated Jackknives and Two-Sided Variance Inequalities

TL;DR

This paper introduces new variance inequalities for functions of independent variables, extending classical bounds like the Efron-Stein inequality through iterated jackknife statistics, applicable to both symmetric and non-symmetric functions.

Contribution

It generalizes variance bounds using iterated jackknife statistics, broadening the scope of existing inequalities for independent random variables.

Findings

Derived new upper and lower variance bounds.

Extended inequalities to non-symmetric functions.

Generalized Efron-Stein inequality using iterated jackknives.

Abstract

We consider the variance of a function of $n$ independent random variables and provide new inequalities which, in particular, extend previous results obtained for symmetric functions in the i.i.d.~setting. For instance, we obtain various upper and lower variance bounds based on iterated jackknives statistics that can be considered as generalizations of the Efron-Stein inequality.