Optimal third root asymptotic bounds in the statistical estimation of thresholds

TL;DR

This paper establishes optimal asymptotic bounds for estimating the intersection point of two densities, crucial in classification, and introduces estimators that asymptotically achieve these bounds.

Contribution

It provides the first lower bounds on estimation errors on the inverse cube root scale and introduces optimal estimators matching these bounds.

Findings

Lower bounds for estimation error probabilities on inverse cube root scale

Probabilistic bounds for classification prediction error

Introduction of estimators that asymptotically attain the bounds

Abstract

This paper is concerned with estimating the intersection point of two densities, given a sample of both of the densities. This problem arises in classification theory. The main results provide lower bounds for the probability of the estimation errors to be large on a scale determined by the inverse cube root of the sample size. As corollaries, we obtain probabilistic bounds for the prediction error in a classification problem. The key to the proof is an entropy estimate. The lower bounds are based on bounds for general estimators, which are applicable in other contexts as well. Furthermore, we introduce a class of optimal estimators whose errors asymptotically meet the border permitted by the lower bounds.