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March, 1994 The Use of Polynomial Splines and Their Tensor Products in Multivariate Function Estimation
Charles J. Stone
Ann. Statist. 22(1): 118-171 (March, 1994). DOI: 10.1214/aos/1176325361

Abstract

Let $X_1, \ldots, X_M, Y_1,\ldots, Y_N$ be random variables, and set $\mathbf{X} = (X_1, \ldots, X_M)$ and $\mathbf{Y} = (Y_1, \ldots, Y_N)$. Let $\varphi$ be the regression or logistic or Poisson regression function of $\mathbf{Y}$ on $\mathbf{X}(N = 1)$ or the logarithm of the density function of $\mathbf{Y}$ or the conditional density function of $\mathbf{Y}$ on $\mathbf{X}$. Consider the approximation $\varphi^\ast$ to $\varphi$ having a suitably defined form involving a specified sum of functions of at most $d$ of the variables $x_1, \ldots, x_M, y_1,\ldots, y_N$ 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 $\varphi$. Let $p$ be a suitably defined lower bound to the smoothness of the components of $\varphi^\ast$. Consider a random sample of size $n$ from the joint distribution of $\mathbf{X}$ and $\mathbf{Y}$. 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 $\varphi^\ast$ and its components having the $L_2$ rate of convergence $n^{-p/(2p + d)}$.

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Charles J. Stone. "The Use of Polynomial Splines and Their Tensor Products in Multivariate Function Estimation." Ann. Statist. 22 (1) 118 - 171, March, 1994. https://doi.org/10.1214/aos/1176325361

Information

Published: March, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0827.62038
MathSciNet: MR1272079
Digital Object Identifier: 10.1214/aos/1176325361

Subjects:
Primary: 62G07
Secondary: 62G20

Rights: Copyright © 1994 Institute of Mathematical Statistics

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Vol.22 • No. 1 • March, 1994
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