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Convergence rates for parametric components in a partly linear model. (English) Zbl 0637.62067
Consider the regression model $Y\sb i=X\sb i'\beta +g(t\sb i)+e\sb i$ 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 $e\sb i$ 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 $g\sb n$ is proposed to approximate g. The least- squares estimator ${\hat \beta}$ is obtained based on the model $Y\sb i=X\sb i'\beta +g\sb n(t\sb i)+e\sb i$. It is shown that ${\hat \beta}$ can achieve the usual parametric rates $n\sp{-1/2}$ with the smallest possible asymptotic variance for the case that X and T are correlated.

##### MSC:
 62J05 Linear regression 62J10 Analysis of variance and covariance 62G99 Nonparametric inference 41A15 Spline approximation
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