Statistica Sinica 17(2007), 713-724

A TEST FOR CONSTANCY OF ISOTONIC REGRESSIONS

USING THE $\mbox{\boldmath $L_2$}$-NORM

Ana Colubi, J. Santos Domínguez-Menchero and Gil González-Rodríguez

Universidad de Oviedo

Abstract: In this paper we present a nonparametric procedure for testing the constancy of an isotonic regression. We introduce a family of statistics based on the $L_2$-norm of the difference of an isotonic estimate of the regression and the estimate under the (null) hypothesis that it is a constant. We propose to choose as isotonic estimate a tail-smoothed version of the usual least squares isotonic regression estimate. We write the selected statistic in terms of a certain functional in order to analyze its asymptotic distribution from the continuity properties of the functional. Finally, we show some simulations to compare the stated procedure with the well-known parametric $\overline{\chi}^2_{01}$.

Key words and phrases: Isotonic regression, L₂-norm, testing constant regression.