Minimax optimal procedures for testing the structure of multidimensional functions

John Aston,Gerda Claeskens,Jean-Marc Freyermuth,Florent Autin,Christophe Pouet

doi:10.1016/j.acha.2017.05.003

Abstract

We present a novel method for detecting some structural characteristics of multidimen-sional functions. We consider the multidimensional Gaussian white noise model with an anisotropic estimand. Using the relation between the Sobol decomposition and the geometry of multidimensional wavelet basis we can build test statistics for any of the Sobol functional components. We assess the asymptotical minimax optimality of these test statistics and show that they are optimal in presence of anisotropy with respect to the newly determined minimax rates of separation. An appropriate combination of these test statistics allows to test some general structural characteristics such as the atomic dimension or the presence of some variables. Numerical experiments show the potential of our method for studying spatio-temporal processes.

Highlights

Multidimensional data often exhibit a simpler underlying structure, meaning that their effective dimension is smaller than the observed dimension
Results given for adaptation in Theorem 4.2 can be compared to usual results obtained for adaptation in hypothesis testing problems
In this paper we describe how the Sobol decomposition in relation to the geometry of the multidimensional hyperbolic wavelet basis allows to build simple test statistics with impressive theoretical performance in the presence of anisotropic estimand

Summary

Introduction

Multidimensional data often exhibit a simpler underlying structure, meaning that their effective dimension is smaller than the observed dimension. Detecting the presence of such structure can enhance the understanding of the data and allows for more effective modeling and inferential strategies. There is a flourishing literature dealing with nonparametric methods for structure detection. These contributions are concerned with different types of structures (such as additivity, small atomic dimension and variable selection) and with different modeling approaches according to the nature of the noise, the smoothness assumptions, etc. Following the Sieve estimation of Birgeand Massart (1998), we build a data-driven procedure. It is first tested in ’idealistic’ numerical experiments before being applied to a more sophisticated context of time series data analysis

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