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Testing the goodness of fit of a linear model via nonparametric regression techniques. (English) Zbl 0702.62037
Summary: This article investigates the use of nonparametric regression methodology to test the adequacy of a parametric linear model. The large-sample properties of parametric goodness-of-fit tests for linearity are considered. The inadequacies of such tests lead to the proposal of new tests that are constructed from nonparametric regression fits to the residuals from linear regression. Large-sample theory is derived for two variants of this type of statistic. The results demonstrate that such tests are consistent against all fixed smooth alternatives to linearity but are incapable of detecting local alternatives converging to a linear model at the parametric rate \(n^{-1/2}\). Simulation experiments involving a test based on fitting cubic smoothing splines to residuals reveals that this test has good power properties against several reasonable alternatives.

62G10 Nonparametric hypothesis testing
62F05 Asymptotic properties of parametric tests
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