Spline smoothing in regression models and asymptotic efficiency in \(L_ 2\)
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 \textit{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.
scientific article; zbMATH DE number 3960784