Discriminant analysis with errors in variables

Details Discriminant analysis with errors in variables Discriminant analysis with errors in variables

2012-01-16

arxiv.org/abs/1201.3283v1

Mathematics

Statistics Theory

Scientific paper

The effect of measurement error in discriminant analysis is investigated. Given observations $Z=X+\epsilon$, where $\epsilon$ denotes a random noise, the goal 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 is to approach the best possible decision rule $G^*$ defined as a minimizer of the Bayes risk. In the free-noise case $(\epsilon=0)$, minimax fast rates of convergence 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 results in the noisy case, i.e. when dealing with errors in variables. In particular, we discuss two possible complexity assumptions that can be set on the problem, which may alternatively concern the regularity of $f-g$ or the boundary of $G^*$. We prove minimax lower bounds for these both problems and explain how can these rates be attained, using in particular Empirical Risk Minimizer (ERM) methods based on deconvolution kernel estimators.

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