Statistica Sinica 17(2007), 1115-1137

TESTING FOR THE EQUALITY OF $\mbox{\boldmath $k$}$ REGRESSION

CURVES

Juan Carlos Pardo-Fernández, Ingrid Van Keilegom

and Wenceslao González-Manteiga

Universidade de Vigo, Université catholique de Louvain

and Universidade de Santiago de Compostela

Abstract: Assume that $(X_j, Y_j)$ are independent random vectors satisfying the non-parametric regression models $Y_j=m_j(X_j)+\sigma_j(X_j)\varepsilon_j$, for $j=1, \dots, k$, where $m_j(X_j)=E(Y_j\vert X_j)$ and $\sigma_j^2(X_j)=\hbox{Var}(Y_j\vert X_j)$ are smooth but unknown regression and variance functions respectively, and the error variable $\varepsilon_j$ 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 Cramér-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.

Key words and phrases: Bootstrap, Comparison of regression curves, Heteroscedastic regression, nonparametric regression.