Nonlinear sufficient dimension reduction for Conditional quantiles in scalar-on-function single-index models

초록

Functional data analysis is crucial in many applications, yet its high-dimensional nature necessitates effective dimension reduction techniques. While existing approaches primarily focus on linear reductions, we introduce a nonlinear sufficient dimension reduction framework for conditional quantiles of single-index models when the predictors are random functions. Our approach constructs two nested functional spaces: a Hilbert space representing the functional data and a reproducing kernel Hilbert space that captures nonlinearity. The kernel in the latter is determined by the inner product of the former, leading to a natural hierarchical structure. We begin by characterizing dimension reduction at the general level of sigma$\sigma$-fields and proceed to that of classes of functions, leading to the notion of the central quantile class. We introduce our proposed estimator, called the tau$\tau$th functional generalized central quantile subspace (tau$\tau$-fGCQS), and establish its convergence rate. Finally, we demonstrate the performance of our estimator through simulations and real-world applications to health studies, examining various health indicators, such as ADHD, Parkinson's disease, and BMI.

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