The Use of Polynomial Splines and Their Tensor Products in Multivariate Function Estimation

Let X₁, , XM, Y₁, , YN be random variables, and set X = (X₁, , XM) and Y = (Y₁, , YN). Let be the regression or logistic or Poisson regression function of Y on X (N = 1) or the logarithm of the density function of Y or the conditional density function of Y on X. Consider the approximation ^ to having a suitably defined form involving a specified sum of functions of at most d of the variables x₁, , xM, y₁, , yN and, subject to this form, selected to minimize the mean squared error of approximation or to maximize the expected log-likelihood or conditional log-likelihood, as appropriate, given the choice of. Let p be a suitably defined lower bound to the smoothness of the components of ^. Consider a random sample of size n from the 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 to construct estimates of ^ and its components having the L₂ rate of convergence n^-p/ (2p + d).