Rates of convergence for spline estimates of additive principal components

Additive principal components (APCs) generalize classical principal component analysis to additive nonlinear transformations. Smallest APCs are additive functions of the vector X = (X-1, ...., X-p) minimizing the variance under orthogonality constraints and are characterized as eigenfunctions of an operator which is compact under a standard condition on the joint distribution of (X-1, ..., X-p). As a by-product, smallest APC nearly satisfies the equation Sigma(j)phi(j)(X-j) = 0 and then provides powerful tools for regression and data analysis diagnostics. The principal aim of this paper is the estimation of smallest APCs based on a sample from the distribution of X. This is achieved using additive splines, which have been recently investigated in several functional estimation problems. The rates of convergence are then derived under mild conditions on the component functions. These rates are the same as the optimal rates for a nonparametric estimate of a univariate regression function. (C) 1999 Academic Press.