Testing for the equality of k regression curves

Assume that (X-j, Y-j) are independent random vectors satisfying the nonparametric regression models Y-j = m(j) (X-j) + sigma(j) (X-j)epsilon(j), for j = 1, ... , k, where mj (X-j) = E(Y-j | X-j) and sigma(2)(j)(X-j) = Var (Y-j | X-j) are smooth but unknown regression and variance functions respectively, and the error variable epsilon(j) is independent of X-j. 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 Cramer-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.