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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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