A TEST FOR ADDITIVITY IN NONPARAMETRIC REGRESSION

STEPHAN DERBORT, HOLGER DETTE AND AXEL MUNK

Ruhr-Universität Bochum, Fakultät für Mathematik, Universitätsstr. 150, D–44780 Bochum, Germany

(Received June 22, 1999; revised September 25, 2000)

Abstract. A simple consistent test of additivity in a multiple nonparametric regression model is proposed, where data are observed on a lattice. The new test is based on an estimator of the L^{2}-distance between the (unknown) nonparametric regression function and its best approximation by an additive nonparametric regression model. The corresponding test-statistic is the difference of a classical ANOVA style statistic in a two-way layout with one observation per cell and a variance estimator in a homoscedastic nonparametric regression model. Under the null hypothesis of additivity asymptotic normality is established with a limiting variance which involves only the variance of the error of measurements. The results are extended to models with an approximate lattice structure, a heteroscedastic error structure and the finite sample behaviour of the proposed procedure is investigated by means of a simulation study.

Key words and phrases: Additive models, dimension reduction, test of additivity.