Nonparametric testing for no-effect with functional responses and functional covariates
Résumé
This paper studies the problem of nonparametric testing for the no-effect of a random functional covariate on a functional response. That means testing whether the conditional expectation of the response given the covariate is almost surely zero or not without imposing any model relating response and covariate. The response and the covariate take values in possibly different separable Hilbert spaces. Hence the situations with scalar response or covariate will be particular cases. Our test is based on the remark that checking the no-effect of the functional covariate is equivalent to checking the nullity of the conditional expectation of the response given a sufficiently rich set of projections of the covariate. Such projections could be on elements from finite-dimension subspaces of the Hilbert space where the covariate takes values. Then, the idea is to search a finite-dimension element of norm 1 that is, in some sense, the least favorable for the null hypothesis. With at hand such a least favorable direction, it remains to check the nullity of the conditional expectation of the functional response given the scalar product between the covariate and the selected direction. We follow these steps using a nearest neighbors (NN) smoothing approach. As a result, our test statistic is a quadratic form involving univariate NN smoothing and the asymptotic critical values are given by the standard normal law. The test is able to detect nonparametric alternatives, not only linear ones. The responses could be heteroscedastic with conditional variance of unknown form. The law of the covariate does not need to be known. An empirical study with simulated data is reported. The cases of functional response and functional or scalar covariate are considered. Our conclusion is that the test could be easily implemented and performs well in simulations and real data applications.