Likelihood ratio test for the adequacy of a regression model when the noise is given by the Slepian field

Abstract

Abstract In this paper we establish a non parametric likelihood ratio (LR) test procedure for testing the mean of a regression or additive model when the disturbance is represented by the Slepian field. By utilizing the Cameron-Martin-Girsanov formula and the fundamental theorem of projection, we get the likelihood estimator of the mean function under either H 0 or H 1 as the projection of the random observation. By applying the fundamental characteristic of the Slepian field for a typical size of the moving rectangle, it is shown under H 0 that the probability distribution of the test statistic belongs to the family of a central chi-square distributions, whereas under H 1 it is distributed as a noncentral chi-square. Furthermore, by simulating a type of Fourier model, the empirical power function of the test shows that the proposed test is consistent in the sense it maximizes the probability of the rejection when the model is not true.