Sufficient Dimension Reduction for the Conditional Quantiles of Functional Data

Functional data analysis holds transformative potential across fields but often relies on mean regression, with limited focus on quantile regression.Furthermore, the infinite-dimensional nature of the functional predictors necessitates the use of dimension reduction techniques.Therefore, in this work, we address this gap by developing dimension reduction techniques for the conditional quantiles of functional data.The idea is to replace the functional predictors with a few finite predictors without losing important information on the conditional quantile while maintaining a flexible nonparametric model.We derive the convergence rates of the proposed estimators and demonstrate their finite sample performance using simulations and a real dataset from fMRI studies.