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Spline smoothing in regression models and asymptotic efficiency in $$L_ 2$$. (English) Zbl 0596.62052
Consider the regression model $$y_ j=f(x_ j)+\xi_ j$$, $$1\leq j\leq n$$, where the $$x_ j's$$ are equidistant in [0,1], the $$\xi_ j's$$ are independent random variables with distribution N(0,1) and the function f is to be estimated. Let $$\| \cdot \|_ 2$$ denote the usual norm on $$L^ 2([0,1])$$, $$\| \cdot \|$$ the semi-norm defined by $$\| f\|^ 2=n^{-1}\sum^{n}_{j=1}f^ 2(x_ j)$$. Let, for $$m\in {\mathbb{N}}$$, $$W^ m_ 2$$ denote the usual Sobolev space on [0,1] and, for $$P>0$$, $$W^ m_ 2(P)$$ be defined by $$W^ m_ 2(P)=\{f\in W^ m_ 2| \| D^ mf\|^ 2\leq P\}$$. Let $${\mathcal F}_ n$$ be the class of measurable mappings $$\hat f:$$ $${\mathbb{R}}^ n\times [0,1]\to {\mathbb{R}}.$$
Using minimax filtering result in Gaussian white noise obtained by M. S. Pinsker [Probl. Peredachi Inf. 16, No.2, 52-68 (1980; Zbl 0452.94003)] the author proves the following asymptotic minimax theorem: $\lim_{n\to +\infty}\inf_{\hat f\in {\mathcal F}_ n}\sup_{f\in W^ m_ 2(P)}n^{2m/\quad (2m+1)}E_ f(\|| \hat f-f\||^ 2)=$
$[P(2m+1)]^{1/(2m+1)}(mn^{-1}(m+1)^{-1})^{2m/(2m+1)}.$ The notation $$\|| \cdot \||$$ stands for either $$\| \cdot \|_ 2$$ or $$\| \cdot \|$$. As part of the proof some eigenvalue estimates in spline theory are achieved.
Reviewer: A.Berlinet

##### MSC:
 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation 65D10 Numerical smoothing, curve fitting 41A15 Spline approximation
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