Application of structural risk minimization to multivariate smoothing spline regression estimates

Abstract

Estimation of regression functions from bounded, independent and identically distributed data is considered. Motivated by Vapnik's principle of structural risk minimization, a data-dependent choice of the smoothing parameter of multivariate smoothing spline estimates is proposed. The corresponding smoothing spline estimates automatically adapt to the unknown smoothness of the regression function and their $L^2$ errors achieve the optimal rate of convergence up to a logarithmic factor. The result is valid without any regularity conditions on the distribution of the design.