title: A note on a nonparametric regression test through penalized splines.
abstract: We examine a test of a nonparametric regression function based on penalized spline smoothing. We show that, similarly to a penalized spline estimator, the asymptotic power of the penalized spline test falls into a small-Kor a large-Kscenarios characterized by the number of knotsKand the smoothing parameter. However, the optimal rate ofKand the smoothing parameter maximizing power for testing is different from the optimal rate minimizing the mean squared error for estimation. Our investigation reveals that compared to estimation, some under-smoothing may be desirable for the testing problems. Furthermore, we compare the proposed test with the likelihood ratio test (LRT). We show that when the true function is more complicated, containing multiple modes, the test proposed here may have greater power than LRT. Finally, we investigate the properties of the test through simulations and apply it to two data examples.
authors: New York University.
authors: Huaihou
authors: H
authors: Chen
authors: Columbia University.
authors: Yuanjia
authors: Y
authors: Wang
authors: Pennsylvania State University.
authors: Runze
authors: R
authors: Li
authors: Columbia University.
authors: Katherine
authors: K
authors: Shear
keywords: Goodness of fit; Likelihood ratio test; Nonparametric regression; Partial linear model; Spectral decomposition