Smooth discrimination analysis

TLDR

Discriminant analysis for two distributions in ℝ^d can be based on estimating the set G = { x : f(x) ≥ g(x) }, a strategy that is meaningful when discrimination serves as a data‑analytic tool. The authors assume G has a smooth boundary (or lies in a nonparametric class) and study decision rules obtained by minimizing empirical risk over the full class of sets and over sieves. They derive convergence rates, showing that the empirical‑risk minimization rules attain optimal rates for estimating G and for Bayes risks, with Bayes‑risk rates that can exceed the parametric root‑n rate, whereas plug‑in rules cannot guarantee such fast rates.

Abstract

Discriminant analysis for two data sets in $\mathbb{R}^d$ with probability densities $f$ and $g$ can be based on the estimation of the set $G = \{x: f(x) \geq g(x)\}$. We consider applications where it is appropriate to assume that the region $G$ has a smooth boundary or belongs to another nonparametric class of sets. In particular, this assumption makes sense if discrimination is used as a data analytic tool. Decision rules based on minimization of empirical risk over the whole class of sets and over sieves are considered. Their rates of convergence are obtained. We show that these rules achieve optimal rates for estimation of $G$ and optimal rates of convergence for Bayes risks. An interesting conclusion is that the optimal rates for Bayes risks can be very fast, in particular, faster than the “parametric” root-$n$ rate. These fast rates cannot be guaranteed for plug-in rules.