Nonparametric comparison of several regression functions: Exact and asymptotic theory

Abstract

A new test is proposed for the comparison of two regression curves f and g. We prove an asymptotic normal law under fixed alternatives which can be applied for power calculations, for constructing confidence regions and for testing precise hypotheses of a weighted L2 distance between f and g. In particular, the problem of nonequal sample sizes is treated, which is related to a peculiar formula of the area between two step functions. These results are extended in various directions, such as the comparison of k regression functions or the optimal allocation of the sample sizes when the total sample size is fixed. The proposed pivot statistic is not based on a nonparametric estimator of the regression curves and therefore does not require the specification of any smoothing parameter.