## Smoothing splines: Regression, derivatives and deconvolution.(English)Zbl 0535.41019

The paper deals with statistical properties of smoothing splines and their derivatives. Given $$x_ i=(Af)(t_ i)+\epsilon_ i,$$ A a linear operator, a ’regularized’ estimate of f is the function g which minimizes $\frac{1}{n}\sum^{n}_{i=1}\{x_ i-(Ag)(t_ i)\}^ 2+\lambda \int \{g''(t)\}^ 2\quad dt.$ The case of numerical differentiation, $$(Af)(t)=\int^{1}_{0}f(u)du$$, and deconvolution, $$(Af)(t)=\int^{1}_{0}w(t-s)f(s)ds,$$ is examined.
First, for observations $$x_ k=f(k/n)+\epsilon_ k$$, $$k=0,...,n$$, $$E\epsilon_ k\equiv 0$$, $$E\epsilon_ k\epsilon_ j=\delta_{kj}\sigma^ 2$$, $$\sigma^ 2>0$$, a continuously differentiable function g with $$g''\in L^ 2$$ is to minimize $\frac{1}{n}[\frac{1}{4}\{x_ 0+x_ n-g(0)-g(1)\}^ 2\quad +\sum^{n- 1}_{k=1}\{x_ k-g(\frac{k}{n})\}^ 2]+2\int^{1}_{0}\{g''(t)\}^ 2\quad dt.$ Asymptotic properties for $$\int \sigma^ 2(g(t))dt$$, $$\int \{Eg(t)-f(t)\}^ 2$$ dt, $$\int^{1}_{0}\sigma^ 2(g'(t))dt,\int^{1}_{0}\{Eg'(t)-f'(t)\}^ 2$$ dt are given. In the deconvolution problem the regularized approximation to f is the function g that minimizes $\frac{1}{4n}\{x_ 0+x_ n-G(0)-G(1)\}^ 2\quad +\frac{1}{n}\sum^{n-1}_{k=1}\{x_ k-G(\frac{k}{n})\}^ 2+\lambda \int^{1}_{0}\{g''(t)\}^ 2\quad dt.$ In section 3 follows the derivation of theorems 1 to 4 and of some auxiliary results. The results are derived by developing a Fourier representation for a smoothing spline. It is shown that unless unnatural boundary conditions hold, the integrated squared bias is dominated by local effects near the boundary. Finally, in section 4, the corresponding theorems for the deconvolution problem are derived.
Reviewer: R.Fahrion

### MSC:

 41A15 Spline approximation 62G99 Nonparametric inference 62J99 Linear inference, regression
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