Testing for the equality of k regression curves

Abstract

Assume that (Xj , Yj ) are independent random vectors satisfying the nonparametric regression models Yj = mj (Xj ) + σj (Xj )εj , for j = 1, . . . , k, where mj (Xj ) = E(Yj |Xj ) and σ 2 j (Xj ) = Var (Yj |Xj ) are smooth but unknown regression and variance functions respectively, and the error variable εj is independent of Xj . In this article we introduce a procedure to test the hypothesis of equality of the k regression functions. The test is based on the comparison of two estimators of the distribution of the errors in each population. Kolmogorov-Smirnov and Cram´er-von Mises type statistics are considered, and their asymptotic distributions are obtained. The proposed tests can detect local alternatives converging to the null hypothesis at the rate n −1/2 . We describe a bootstrap procedure that approximates the critical values, and present the results of a simulation study in which the behavior of the tests for small and moderate sample sizes is studied. Finally, we include an application to a data set