The Annals of Statistics

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

Charles J. Stone

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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 $n^{-p/(2p + d)}$.

Article information

Ann. Statist., Volume 22, Number 1 (1994), 118-171.

First available in Project Euclid: 11 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

Polynomial splines tensor products interactions ANOVA decomposition exponential family generalized linear model log-linear model least squares maximum likelihood rate of convergence AID CART MARS


Stone, Charles J. The Use of Polynomial Splines and Their Tensor Products in Multivariate Function Estimation. Ann. Statist. 22 (1994), no. 1, 118--171. doi:10.1214/aos/1176325361.

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