Assessing the equivalence of nonparametric regression tests based on spline and local polynomial smoothers

Abstract

It is widely known that, in a certain sense, a smoothing spline estimate of the regression function is asymptotically equivalent to a kernel regression estimate. However, little information has been available about the equivalence between nonparametric regression tests, based on the smoothing spline and local polynomial regression methods. To assess their relative behaviors and to facilitate illustration, we consider in this paper the “generalized likelihood ratio” (GLR) test statistic, constructed from each type of smoother. For the resulting test statistics, we first establish their equivalent forms of the asymptotic distributions, under the null hypothesis. After that, we derive their equivalent optimal rates of smoothing parameters for nonparametric testing. Furthermore, we evaluate their relative asymptotic efficiency and characterize its relation to the magnitude of the smoothing parameters. Finally, we illustrate the large-sample behaviors of the cubic smoothing spline and local linear regression methods, in the GLR tests, with results from small-scale simulations.