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Approximation of least squares regression on nested subspaces. (English) Zbl 0669.62047
This paper considers the regression model $$y_ i=\theta (x_ i)+\epsilon_ i$$ $$(i=1,...,n)$$ where $$\theta$$ is an unknown function mapping $${\mathbb{R}}^ d\to {\mathbb{R}}^ q$$. Let $$\theta_{nm}$$ be the least squares estimator of $$\theta$$ obtained from the model assuming that $$\theta$$ belongs to a given subspace of functions span $$\{\psi_ 1,...,\psi_ m\}.$$
Theorems are given for approximating the bias and variance of $$\theta_{nm}$$ in a scale of Hilbert norms natural to the problem, when n and m are large and the design determined by the $$x_ i's$$ is suitably approximated by a design measure. Two examples (with $$d=q=1)$$ illustrate the theory: polynomial and Fourier series regression.
Reviewer: H.Caussinus

##### MSC:
 62J05 Linear regression; mixed models 62F12 Asymptotic properties of parametric estimators 41A10 Approximation by polynomials
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