Nonparametric comparison of regression functions

In this work, we provide a new methodology for comparing regression functions m 1 and m 2 from two samples. Since apart from smoothness no other (parametric) assumptions are required, our approach is based on a comparison of nonparametric estimators m ˆ 1 and m ˆ 2 of m 1 and m 2 , respectively. The test statistics T ˆ incorporate weighted differences of m ˆ 1 and m ˆ 2 computed at selected points. Since the design variables may come from different distributions, a crucial question is where to compare the two estimators. As our main results we obtain the limit distribution of T ˆ (properly standardized) under the null hypothesis H 0 : m 1 = m 2 and under local and global alternatives. We are also able to choose the weight function so as to maximize the power. Furthermore, the tests are asymptotically distribution free under H 0 and both shift and scale invariant. Several such T ˆ ’s may then be combined to get Maximin tests when the dimension of the local alternative is finite. In a simulation study we found out that our tests achieve the nominal level and already have excellent power for small to moderate sample sizes.