A1017
Title: Functional sufficient dimension reduction through average Frechet 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 for the reduction of dimensionality without loss of information in a regression setting. A new method is proposed for nonparametric function-on-function SDR, where both the response and the predictor are a function. The notions of functional central mean subspace and functional central subspace are first developed, forming the population targets of our functional SDR. An average Frechet derivative estimator is then introduced, which extends the gradient of the regression function to the operator level and enables the development of estimators for the functional dimension reduction spaces. 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 that are commonly imposed by all existing functional SDR methods. The uniform convergence of the estimators for the functional dimension reduction spaces is established while allowing the number of Karhunen-Loeve expansions and the intrinsic dimension to diverge with the sample size. The efficacy of the proposed methods is demonstrated through both simulations and two real data examples.
