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This paper examines the problem of nonparametric testing for the no-effect ofa random covariate (or predictor) on a functional response. This means testingwhether the conditional expectation of the response given the covariate isalmost surely zero or not, without imposing any model relating response andcovariate. The covariate could be univariate, multivariate or functional. Ourtest statistic is a quadratic form involving univariate nearest neighborsmoothing and the asymptotic critical values are given by the standard normallaw. When the covariate is multidimensional or functional, a preliminarydimension reduction device is used which allows the effect of the covariate tobe summarized into a univariate random quantity. The test is able to detect notonly linear but nonparametric alternatives. The responses could haveconditional variance of unknown form and the law of the covariate does not needto be known. An empirical study with simulated and real data shows that thetest performs well in applications.

Nonparametric testing for no-effect with functional responses and functional covariates