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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.

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