Nonparametric regression function estimation using interaction least squares splines and comlexity regularization

Abstract

Let (X,Y) be a pair of random variables with supp(X) \subseteq [0,1]?I and EY?2 < \infinity. Let m* be the best approximation of the regression function of (X,Y) by sums of functions of at most d variables (formula). Estimation of m* from i.i.d. data is considered. For the estimation interaction least squares splines, which are defined as sums of polynomial tensor product splines of at most d variables, are used. The knot sequences of the tensor product splines are chosen equidistant. Complexity regularization is used to choose the number of the knots and the degree of the splines automatically using only the given data. Without any additional condition on the distribution of (X,Y) the weak and strong L2-consistency of the estimate is shown. Furthermore, for every distribution of (X,Y) with supp(X) \subseteq [0,1]?I, Y bounded and m* p-smooth, the integrated squared error of the estimate achieves up to a logarithmic factor the (optimal) rate n?{-\frac{2p}{2p+d}}. Let (X,Y) be a pair of random variables with supp(X) (formula) [0,1]1 and EY2,(formula). Let m* be the best approximation of the regression function of (X,Y0 by sums of functions of at most d variables (formula). Estimation of m* from i.i.d. data is considered. For the estimation interaction least squares splines, which are defined as sums of polynomial tensor product splines of at most d variables, are used. The knot sequences of the tensor product splines are chosen equidistant. Complexity regularization is used to choose the number of the knots and the degree of the splines automatically using only the given data. Without any additional condition on the distribution of (X,Y) the weak and strong L2-consistency of the estimate is shown. Furthermore, for very (formula) and every distribution of (X,Y) with supp(X) (formula) , Y bounded and m* p-smooth, the integrated squard error of the estimate achieves up to a logarithmic factor the (optimal) rate in (formula).