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

• 成果类型: Article
• 署名作者: STONE, CJ; BUJA, A; HASTIE, T
• 署名单位: Telcordia Technologies; AT&T; Nokia Corporation; Nokia Bell Labs
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176325361
• 发表日期: 1994
• 页码: 118-184
• 关键词: generalized additive-models; regression; CONVERGENCE; rates

摘要

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 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 phi* to phi having a suitably defined form involving 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 the mean squared error of approximation or to maximize the expected log-likelihood or conditional log-likelihood, as appropriate, given the choice of phi. Let p be a suitably defined lower bound to the smoothness of the components of phi*. 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 phi* and its components having the L2 rate of convergence n(-p/(2p+d)).