  We investigate nonparametric regression methods based on spatial depth and
quantiles when the response and the covariate are both functions. As in
classical quantile regression for finite dimensional data, regression
techniques developed here provide insight into the influence of the functional
covariate on different parts, like the center as well as the tails, of the
conditional distribution of the functional response. Depth and quantile based
nonparametric regressions are useful to detect heteroscedasticity in functional
regression. We derive the asymptotic behaviour of nonparametric depth and
quantile regression estimates, which depend on the small ball probabilities in
the covariate space. Our nonparametric regression procedures are used to
analyse a dataset about the influence of per capita GDP on saving rates for 125
countries, and another dataset on the effects of per capita net disposable
income on the sale of cigarettes in some states in the US.
