Testing for additivity and joint effects in multivariate nonparametric regression using Fourier and wavelet methods |
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Authors: | Daniela De Canditiis Theofanis Sapatinas |
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Institution: | (1) Istituto per le Applicazioni del Calcolo Mauro Picone , Sezione di Roma, Consiglio Nazionale delle Ricerche, Italy;(2) Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, CY, 1678 Nicosia, Cyprus |
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Abstract: | We consider the problem of testing for additivity and joint effects in multivariate nonparametric regression when the data are modelled as observations of an unknown response function observed on a d-dimensional (d 2) lattice and contaminated with additive Gaussian noise. We propose tests for additivity and joint effects, appropriate for both homogeneous and inhomogeneous response functions, using the particular structure of the data expanded in tensor product Fourier or wavelet bases studied recently by Amato and Antoniadis (2001) and Amato, Antoniadis and De Feis (2002). The corresponding tests are constructed by applying the adaptive Neyman truncation and wavelet thresholding procedures of Fan (1996), for testing a high-dimensional Gaussian mean, to the resulting empirical Fourier and wavelet coefficients. As a consequence, asymptotic normality of the proposed test statistics under the null hypothesis and lower bounds of the corresponding powers under a specific alternative are derived. We use several simulated examples to illustrate the performance of the proposed tests, and we make comparisons with other tests available in the literature. |
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Keywords: | additive models Fourier transform hypothesis testing joint effects nonparametric regression tensor product Hilbert spaces wavelet transform |
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