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Regression models with infinitely many parameters: Consistency of bounded linear functionals. (English) Zbl 0544.62062
Let $$y=\sum^{\infty}_{i=1}x_ i\theta_ i+\epsilon =<x,\theta>+\epsilon$$ be a linear model, where $$x=(x_ 1,x_ 2,...)\in \ell^ 2$$, $$\theta =(\theta_ 1,\theta_ 2,...)\in \Theta \subset \ell^ 2$$; $$\epsilon$$ is the random error, $$E\epsilon =0$$, Var $$\epsilon \in(0,\infty)$$. Suppose independent observations $$y_ 1,...,y_ n$$ at levels $$x_ 1,...,x_ n$$ and we are interested in estimating $$T(\theta)=\sum^{\infty}_{i=1}c_ i\theta_ i$$ for some $$c=(c_ 1,c_ 2,...)\in \ell^ 2.$$
Under some conditions there exists a sequence of estimators $$\{$$ $$\hat T_ n\}$$ such that $$E(\hat T_ n-T(\theta))^ 2\to 0$$ as $$n\to \infty$$ for any $$\theta \in \Theta$$. The nonparametric regression problem $$y=f(t)+\epsilon =<x,\theta>+\epsilon$$, where $$f\in W^ m_ 2[0,1]$$, $$\theta_ j=<f,f_ j> (f_ 1,f_ 2,..$$. is a complete orthogonal system of $$W^ m_ 2[0,1])$$, is considered, too.
Reviewer: N.Leonenko

##### MSC:
 62J05 Linear regression; mixed models 62J02 General nonlinear regression 62G05 Nonparametric estimation
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