Rates of Fisher information convergence in the central limit theorem for nonlinear statistics

TL;DR

This paper introduces a new method for analyzing the convergence of Fisher information in the central limit theorem for nonlinear statistics, providing explicit rates and broad applicability.

Contribution

The authors develop novel representations for the score function and derive quantitative Fisher information bounds without relying on Poincaré constant finiteness.

Findings

Explicit Fisher information convergence rates for quadratic forms.

Fisher information bounds for functions of sample means.

Applicability to non-central limit theorems.

Abstract

We develop a general method to study the Fisher information distance in central limit theorem for nonlinear statistics. We first construct completely new representations for the score function. We then use these representations to derive quantitative estimates for the Fisher information distance. To illustrate the applicability of our approach, explicit rates of Fisher information convergence for quadratic forms and the functions of sample means are provided. For the sums of independent random variables, we obtain the Fisher information bounds without requiring the finiteness of Poincar\'e constant. Our method can also be used to bound the Fisher information distance in non-central limit theorems.