A two-dimensional smoothing spline and a regression problem.(English)Zbl 0584.62104

Limit theorems in probability and statistics, 2nd Colloq., Veszprém/Hung. 1982, Vol. II, Colloq. Math. Soc. János Bolyai 36, 915-931 (1984).
[For the entire collection see Zbl 0561.00017.]
In the last few years a great deal of work has been done studying so- called smoothing splines applied to the following regression model: $$X_ j=f(t_ j)+\epsilon_ j$$, $$1\leq j\leq n$$, $$0\leq t_ 1<...<t_ n\leq 1$$, where $$(\epsilon_ j)^ n_{j=1}$$ are orthogonal or i.i.d. random errors with expectation 0 and f, the unknown regression function, is assumed to be smooth [see e.g. G. Wahba, Numer. Math. 24, 383-393 (1975; Zbl 0299.65008), for the beginning and B. W. Silverman, Some aspects of the spline smoothing approach to non-parametric regression curve fitting. J. R. Stat. Soc., Ser. B 47, 1-21 (1985), for an overview on the developed theory].
In this paper the two dimensional regression model $$X_{i,j}=f(t_ i,t_ j)+\epsilon_{i,j}$$, $$1\leq i,j\leq n$$, $$t_ i=i/n$$, with orthogonal random errors $$(\epsilon_{i,j})^ n_{i,j=1}$$, having expectation zero, is considered. Minimizing the functional $n^{- 2}\sum^{n}_{j,k=1}(g(t_ j,t_ k)-X_{j,k})^ 2+\lambda \cdot (2\pi)^{-2}\int^{1}_{0}\int^{1}_{0}(g^ 2_ s(s,t)+g^ 2_ t(s,t))ds\quad dt,$ where $$\lambda =\lambda (n)$$ is a smoothing parameter, w.r. to smooth and periodic functions g we end with a two- dimensional smoothing spline.
Under smoothness assumptions and periodic boundary conditions on f resp. conditions on the rate of decay of the Fourier-coefficients of f the author develops the asymptotic behaviour of the integrated variance and bias and hence of the integrated mean squared error (IMSE) via Fourier analysis.
For example if $$f\in C^ 3([0,1]^ 2)$$ and f and f’ satisfy circular boundary conditions the optimal smoothing parameter $$\lambda$$ (n) is of order $$n^{-2/3}$$ which gives $$IMSE(g)\cong n^{-4/3}$$. (Observe we have $$n^ 2$$ data points).