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.

本文研究基于空间深度和分位数的非参数回归方法，应用于响应变量和协变量都是函数的场景，以分析功能响应的不同部分受函数协变量影响的程度，探测功能回归中的异方差性，并通过两组数据集分析了人均 GDP 对 125 个国家的储蓄率的影响以及人均可支配收入对美国部分州香烟销售的影响。