Testing for Lack of Fit in Functional Regression Models

Abstract

Motivated by numerous applications, the statistical modeling of functional data received a lot of attention during the last years. However, procedures for functional regression model checks are only little developed. Consider a response variable taking values in a Hilbert space that admits a finite or countable orthonormal basis, and hybrid covariates. That means there could be two sets of observed regressors, one of finite dimension and a second one functional with values in an infinite dimension Hilbert space. Formally, the statistical problem we address is the test of the no-effect of a general covariate. This type of problem occurs in many situations: detecting misspecification of functional generalized linear models, checking the effect of the functional covariate in a semi-functional partial linear regression with scalar responses, significance test for functional regressors in nonparametric regression with hybrid covariates and scalar or functional responses, testing the independence between general random variables. We present several kernel smoothing approaches for detecting the effect of a general covariate. The basic idea underlying these approaches is the fact that checking the no-effect of a general covariate is equivalent to checking the nullity of the conditional expectation of the error term given a sufficiently rich set of projections of that covariate. Two ideas for using the projections of the covariate are proposed: the first one is to search the projection that is, in some sense, the least favorable for the null hypothesis, and the second one is to average, in some sense, over the projections. The resulting test statistics are quadratic forms based on univariate, or low dimension, kernel smoothing and the asymptotic critical values are given by the standard normal law. The test is able to detect general departures from the no-effect condition. The conditional variance of the response could be of general and unknown form. The law of the functional covariate need not to be known. Several illustrations will try to convince that the tests perform well in simulations and real data applications. The talk is based on joint work with Cesar Sanchez Sellero, Pascal Lavergne, Samuel Maistre and Matthieu Saumard.