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The effect of measurement errors in discriminant analysis is investigated.Given observations $Z=X+\epsilon$, where $\epsilon$ denotes a random noise, thegoal is to predict the density of $X$ among two possible candidates $f$ and$g$. We suppose that we have at our disposal two learning samples. The aim isto approach the best possible decision rule $G^\star$ defined as a minimizer ofthe Bayes risk. In the free-noise case $(\epsilon=0)$, minimax fast rates ofconvergence are well-known under the margin assumption in discriminant analysis(see \cite{mammen}) or in the more general classification framework (see\cite{tsybakov2004,AT}). In this paper we intend to establish similar resultsin the noisy case, i.e. when dealing with errors in variables. We prove minimaxlower bounds for this problem and explain how can these rates be attained,using in particular an Empirical Risk Minimizer (ERM) method based ondeconvolution kernel estimators.

Minimax fast rates for discriminant analysis with errors in variables