The chain rule for functionals with applications to functions of moments

TL;DR

This paper extends the chain rule to statistical functionals, enabling advanced approximations of moments, cumulants, and distributional properties of functions of sample moments, leading to improved confidence intervals and bias estimates.

Contribution

It introduces a chain rule for derivatives of functionals, facilitating higher-order approximations for functions of moments in statistical analysis.

Findings

Derived distribution of standardized skewness with $O(n^{-2})$ accuracy

Provided methods for third order confidence intervals

Enhanced bias reduction techniques for moment-based estimators

Abstract

The chain rule for derivatives of a function of a function is extended to a function of a statistical functional, and applied to obtain approximations to the cumulants, distribution and quantiles of functions of sample moments, and so to obtain third order confidence intervals and estimates of reduced bias for functions of moments. As an example we give the distribution of the standardized skewness for a normal sample to magnitude $O(n^{-2})$, where $n$ is the sample size.