AISM 54, 60-82
© 2002 ISM

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.

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