THE USE OF POLYNOMIAL SPLINES AND THEIR TENSOR-PRODUCTS IN MULTIVARIATE FUNCTION ESTIMATION

Let X1,...,X(M),Y1,...,Y(N) be random variables, and set X = (X1,...,X (M)) and Y = (Y1,...,Y(N)). Let phi be the regression or logistic or P oisson regression function of Y on X (N = 1) or the logarithm of the d ensity function of Y or the conditional density function of Y on X Con sider the approximation phi to phi having a suitably defined form inv olving a specified sum of functions of at most d of the variables x1,. ..,X(M), Y1,...,Y(N) and, subject to this form, selected to minimize t he mean squared error of approximation or to maximize the expected log -likelihood or conditional log-likelihood, as appropriate, given the c hoice of phi. Let p be a suitably defined lower bound to the smoothnes s of the components of phi. Consider a random sample of size n from t he joint distribution of X and Y. Under suitable conditions, the least squares or maximum likelihood method is applied to a model involving nonadaptively selected sums of tensor products of polynomial splines t o construct estimates of phi and its components having the L2 rate of convergence n(-p/(2p+d)).