Conditional distribution regression for functional responses

Modeling conditional distributions for functional data extends the concept of a mean response in functional regression settings, where vector predictors are paired with functional responses. This extension is challenging because of the nonexistence of well‐defined densities, cumulative distributions, or quantile functions in the Hilbert space where the response functions are located. To address this challenge, we simplify the problem by assuming that the response functions are Gaussian processes, which means that the conditional distribution of the responses is determined by conditional mean and conditional covariance. We demonstrate that these quantities can be obtained by applying global and local Fréchet regression, where the local version is more flexible and applicable when the covariate dimension is low and covariates are continuous, while the global version is not subject to these restrictions but is based on the assumption of a more restrictive regression relation. Convergence rates for the proposed estimates are obtained under the framework of M‐estimation. The corresponding estimation of conditional distributions is illustrated with simulations and an application to bike‐sharing data, where predictors include weather characteristics and responses are bike rental profiles. We also show that our methods are applicable to the challenging problem to study functional fragments. Such data are observed in accelerated longitudinal studies and correspond to functional data observed over short domain segments. We demonstrate the utility of conditional distributions in this context by using the time (age) at which a subject enters the domain of a fragment in addition to other covariates as predictor and the function observed over the domain of the fragment as response.