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December 1999 Smooth discrimination analysis
Enno Mammen, Alexandre B. Tsybakov
Ann. Statist. 27(6): 1808-1829 (December 1999). DOI: 10.1214/aos/1017939240


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.


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Enno Mammen. Alexandre B. Tsybakov. "Smooth discrimination analysis." Ann. Statist. 27 (6) 1808 - 1829, December 1999.


Published: December 1999
First available in Project Euclid: 4 April 2002

zbMATH: 0961.62058
MathSciNet: MR1765618
Digital Object Identifier: 10.1214/aos/1017939240

Primary: 62G05
Secondary: 62G20

Rights: Copyright © 1999 Institute of Mathematical Statistics


Vol.27 • No. 6 • December 1999
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