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Convergence rates for parametric components in a partly linear model. (English) Zbl 0637.62067
Consider the regression model \(Y_ i=X_ i'\beta +g(t_ i)+e_ i\) for \(i=1,...,n\). Here g is an unknown Hölder continuous function of known order p in R, \(\beta\) is a \(k\times 1\) parameter vector to be estimated and \(e_ i\) is an unobserved disturbance. Such a model is often encountered in situations in which there is little real knowledge about the nature of g.
A piecewise polynomial \(g_ n\) is proposed to approximate g. The least- squares estimator \({\hat \beta}\) is obtained based on the model \(Y_ i=X_ i'\beta +g_ n(t_ i)+e_ i\). It is shown that \({\hat \beta}\) can achieve the usual parametric rates \(n^{-1/2}\) with the smallest possible asymptotic variance for the case that X and T are correlated.

62J05 Linear regression; mixed models
62J10 Analysis of variance and covariance (ANOVA)
62G99 Nonparametric inference
41A15 Spline approximation
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