Biau Gérard and
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Mas André: Institut de Math´ematiques et de Modelisation de Montpellier, Equipe de Probabilites et Statistique, Montpellier Cedex 5, Frankreich
Statistics & Risk Modeling, 2012, vol. 29, issue 1, 19-46
Many statistical estimation techniques for high-dimensional or functional data are based on a preliminary dimension reduction step, which consists in projecting the sample X1,...,Xn onto the first D eigenvectors of the Principal Component Analysis (PCA) associated with the empirical projector ^ ΠD. 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 (^ΠDX1,...,^ΠDXn) 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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