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) over cap (1) and (m) over cap (2) of m(1) and m(2), respectively. The test statistics (T) over cap incorporate weighted differences of (m) over cap (1) and (m) over cap (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) over cap (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 Ho and both shift and scale invariant. Several such (T) over cap '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. (c) 2010 Elsevier Inc. All rights reserved.