2011/10/28 by Jean-Baptiste Monnier, Monnier, Jean-Baptiste
Mathematics · #FOS: Mathematics #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1110.6427
38 pages, 6 figures
arxiv created 2011/12/13 · arxiv updated 2011/12/14
We focus on the supervised binary classification problem, which consists in guessing the label Y associated to a co-variate X ∈ \Rd, given a set of n independent and identically distributed co-variates and associated labels (Xi,Yi). We assume that the law of the random vector (X,Y) is unknown and the marginal law of X admits a density supported on a set \A. In the particular case of plug-in classifiers, solving the classification problem boils down to the estimation of the regression function η(X) = \Exp[Y|X]. Assuming first \A to be known, we show how it is possible to construct an estimator of η by localized projections onto a multi-resolution analysis (MRA). In a second step, we show how this estimation procedure generalizes to the case where \A is unknown. Interestingly, this novel estimation procedure presents similar theoretical performances as the celebrated local-polynomial estimator (LPE). In addition, it benefits from the lattice structure of the underlying MRA and thus outperforms the LPE from a computational standpoint, which turns out to be a crucial feature in many practical applications. Finally, we prove that the associated plug-in classifier can reach super-fast rates under a margin assumption.