PCA-Kernel Estimation

2010/03/26 by Biau, Gérard, Mas, André
#FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1003.5089

Abstract

Many statistical estimation techniques for high-dimensional or functional data are based on a preliminary dimension reduction step, which consists in projecting the sample \bX₁, \hdots, \bX_n onto the first D eigenvectors of the Principal Component Analysis (PCA) associated with the empirical projector Π_D. Classical nonparametric inference methods such as kernel density estimation or kernel regression analysis are then performed in the (usually small) D-dimensional space. However, the mathematical analysis of this data-driven dimension reduction scheme raises technical problems, due to the fact that the random variables of the projected sample ( Π_D\bX₁,\hdots, Π_D\bX_n) are no more independent. As a reference for further studies, we offer in this paper several results showing the asymptotic equivalencies between important kernel-related quantities based on the empirical projector and its theoretical counterpart. As an illustration, we provide an in-depth analysis of the nonparametric kernel regression case

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