PCA-Kernel Estimation

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 Π̂D. 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 (Π̂DX1, . . . , Π̂DXn) 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.