Berry-Esseen bounds for estimators of entropy and diversity indices on countable alphabets

TL;DR

This paper establishes Berry-Esseen bounds for the convergence rates of estimators of diversity indices, including Shannon entropy, on countable alphabets, providing explicit bounds for various estimators.

Contribution

It introduces non-asymptotic Berry-Esseen bounds for a broad class of diversity index estimators, including bias-corrected variants, on countable alphabets.

Findings

Non-asymptotic convergence rates for plug-in estimators.

Explicit Berry-Esseen bounds for Shannon entropy estimators.

Bounds applicable to bias-corrected estimators like Miller-Madow and jackknife.

Abstract

In the present paper, we derive Berry-Esseen bounds for the estimation of diversity indices on countable alphabets. A general non-asymptotic convergence rate is established for the plug-in estimator of a wide class of indices, including Simpson's index and Re\'{n}yi's entropy. For the practically crucial case of Shannon entropy, we provide explicit Berry-Esseen bounds for the standard plug-in estimator, as well as for two widely used bias-corrected variants, the Miller-Madow and the jackknife estimators.