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Validation of linear regression models. (English) Zbl 0930.62041
Summary: A new test is proposed in order to verify that a regression function, say $$g$$ has a prescribed (linear) parametric form. This procedure is based on the large sample behavior of an empirical $$L^2$$-distance between $$g$$ and the subspace $$U$$ spanned by the regression functions to be verified. The asymptotic distribution of the test statistic is shown to be normal with parameters depending only on the variance of the observations and the $$L^2$$-distance between the regression function $$g$$ and the model space $$U$$.
Based on this result, a test is proposed for the hypothesis that “$$g$$ is not in a preassigned $$L^2$$-neighborhood of $$U$$,” which allows the “verification” of the model $$U$$ at a controlled type I error rate. The suggested procedure is very easy to apply because of its asymptotic normal law and the simple form of the test statistic. In particular, it does not require nonparametric estimators of the regression function and hence, the test does not depend on the subjective choice of smoothing parameters.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference 62E20 Asymptotic distribution theory in statistics
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