Nonparametric comparison of regression curves: an empirical process approach

Abstract

We propose a new test for the comparison of two regression curves that is based on a difference of two marked empirical processes based on residuals. The large sample behavior of the corresponding statistic is studied to provide a full nonparametric comparison of regression curves. In contrast to most procedures suggested in the literature, the new procedure is applicable in the case of different design points and heteroscedasticity. Moreover, it is demonstrated that the proposed test detects continuous alternatives converging to the null at a rate $N^{-1/2}$ and that, in contrast to all other available procedures based on marked empirical processes, the new test allows the optimal choice of bandwidths for curve estimation (e.g., $N^{-1/5}$ in the case of twice differentiable regression functions). As a by-product we explain the problems of a related test proposed by Kulasekera [J. Amer. Statist. Assoc. 90 (1995) 1085-1093] and Kulasekera and Wang [J. Amer. Statist. Assoc. 92 (1997) 500-511] with respect to accuracy in the approximation of the level. These difficulties mainly originate from the comparison with the quantiles of an inappropriate limit distribution.

A simulation study is conducted to investigate the finite sample properties of a wild bootstrap version of the new test and to compare it with the so far available procedures. Finally, heteroscedastic data is analyzed in order to demonstrate the benefits of the new test compared to the so far available procedures which require homoscedasticity.