## The Annals of Statistics

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

Charles J. Stone

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

#### Article information

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

Dates
First available: 11 April 2007

http://projecteuclid.org/euclid.aos/1176325361

JSTOR

Digital Object Identifier
doi:10.1214/aos/1176325361

Mathematical Reviews number (MathSciNet)
MR1272079

Zentralblatt MATH identifier
0827.62038

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

#### Citation

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