Nonparametric F-tests for nested global and local polynomial models

In this paper, we investigate geometric properties of local polynomial regression and show that the class of global polynomial models is nested within the class of functions generated by fitting local polynomials. The geometric properties are then utilized to construct nonparametric F-tests for testing whether a regression relationship is a polynomial function. The proposed F-tests can be seen as a "calculus" extension of the classical F-tests with analysis of variance interpretations. With the normality assumption, the test statistic is shown to have asymptotic F-distributions under the null hypothesis and fixed alternatives. Simulation results illustrate that the asymptotic null F-distribution approximates well in finite sample cases and the proposed tests enjoy robustness against heteroscedasticity and non-normality as do the classical F-tests.