Multivariate smoothing spline functions. (English) Zbl 0581.65012

Cox, Dennis D.

SIAM J. Numer. Anal. 21, 789-813 (1984).

Author’s summary: Given data \(z_ i=g(t_ i)+\epsilon_ i\), \(1\leq i\leq n\), where g is the unknown function, the \(t_ i\) are known d-dimensional variables in a domain \(\Omega\), and the \(\epsilon_ i\) are i.i.d. random errors, the smoothing spline estimate \(g_{n\lambda}\) is defined to be the minimizer over h of \(n^{-1}\sum (z_ i-h(t_ i))^ 2+\lambda J_ m(h)\), where \(\lambda >0\) is a smoothing parameter and \(J_ m(h)\) is the sum of the integrals over \(\Omega\) of the squares of all the m th order derivatives of h. Under the assumptions that \(\Omega\) is bounded and has a smooth boundary, \(\lambda\) \(\to 0\) appropriately, and the \(t_ i\) become dense in \(\Omega\) as \(n\to \infty\), bounds on the rate of convergence of the expected square of p th order Sobolev norm \((L_ 2\) norm of p th derivatives) are obtained. These extend known results in the one dimensional case. The method of proof utilizes an approximation to the smoothing spline based on a Green’s function for a linear elliptic boundary value problem. Using eigenvalue approximation techniques, these rate of convergence results are extended to fairly arbitrary domains including \(\Omega ={\mathbb{R}}^ d\), but only for the case \(p=0\), i.e. \(L_ 2\) norm.

Reviewer: Ch.K.Chui

MSC:

65D10	Numerical smoothing, curve fitting
65C99	Probabilistic methods, stochastic differential equations
41A15	Spline approximation
41A25	Rate of convergence, degree of approximation
62J05	Linear regression; mixed models

Keywords:

nonparametric regression; Sobolev norms; rate of convergence; smoothing spline