On the optimal choice of the number of empirical Fourier coefficients for comparison of regression curves

The paper is devoted to the elaboration of an efficient approach for comparison of two regression curves based on the empirical Fourier coefficients of regression functions. For the problem of testing for the equality of the two unknown functions in the case of homoscedastic error structure and observation at equidistant points, we derive a new procedure with adaptive choice of the number of the coefficients used in the hypotheses testing. Our approach is based on approximation of the most powerful test using the full knowledge of the regression functions. The results are justified by theoretical arguments and the superiority of the new procedure is also confirmed by a simulation study.