Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

- E. MAMMEN
**A. B. TSYBAKOV**

**Code(s) de Classification MSC:**

- 62G05 Estimation
- 62G20 Asymptotic properties

**Résumé:** Discriminant analysis for two data sets in $\R^d$ 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 minimisation 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.

**Mots Clés:** *discrimination analysis ; optimal rates ; empirical risk ; Bayes risk ; sieves*

**Date:** 1999-06-22

**Prépublication numéro:** *PMA-511*