An L2 error test with order selection and thresholding

An L2 error between an estimated function and a true function can be a good means of checking lack of fit of a regression model. Kuchibhatla and Hart (1995) propose an L2 error test based on Fourier series whose order is selected by an AIC type criterion. We call this test the KH test. When the sample size is small and the true function has high frequency behavior, the AIC type criterion often fails in choosing the correct order of a truncated Fourier series regression estimator. This failure results in low power for the KH test. Fan (1996) proposes L2 error tests based on a threshold estimator. These tests do not perform well for low frequency alternatives in comparison to the KH test. As a compromise between the two types of tests and to make up for the failure of the AIC type criterion, we propose a new test which consists of a sum of L2 error of a truncated estimator (KH test) and L2 error of a threshold estimator for higher order Fourier coefficients. The proposed test has similar power to the KH test for low frequency alternatives, and better power than the KH test for high frequency alternatives.