A0242
Title: Functional dimension reduction via average Fr'echet derivatives
Authors: Kuang-Yao Lee - Temple University (United States) [presenting]
Lexin Li - University of California Berkeley (United States)
Abstract: Sufficient dimension reduction (SDR) embodies a family of methods that aim to reduce dimensionality without loss of information in a regression setting. We propose a new method for nonparametric SDR, where both the response and the predictor are a function. We first develop the notions of functional central mean subspace and functional central subspace, which form the population targets of our functional SDR. We then introduce an average Fr\'echet derivative estimator, which extends the gradient of the regression function to the operator level, and use it to develop estimators for our functional dimension reduction spaces. We show that the resulting functional SDR estimators are unbiased and exhaustive, and more importantly, without imposing any distributional assumptions such as the linearity or the constant variance conditions commonly imposed by all existing functional SDR methods. We establish the uniform convergence of the estimators for the functional dimension reduction spaces while allowing both the number of Karhunen-Lo\`eve expansions and the intrinsic dimension to diverge with the sample size. We demonstrate the efficacy of the proposed methods through both simulations and a real data example.
