FUNCTIONAL SUFFICIENT DIMENSION REDUCTION THROUGH AVERAGE FRECHET DERIVATIVES

• 成果类型: Article
• 署名作者: Lee, Kuang-Yao; Li, Lexin
• 署名单位: Pennsylvania Commonwealth System of Higher Education (PCSHE); Temple University; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/21-AOS2131
• 发表日期: 2022
• 页码: 904-929
• 关键词: sliced inverse regression

摘要

Sufficient dimension reduction (SDR) embodies a family of methods that aim for reduction of dimensionality without loss of information in a regression setting. In this article, we propose a new method for nonparametric function-on-function 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 Frechet derivative estimator, which extends the gradient of the regression function to the operator level and enables us to develop estimators for our functional dimension reduction spaces. We show 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. We establish the uniform convergence of the estimators for the functional dimension reduction spaces, while allowing both the number of Karhunen-Loeve expansions and the intrinsic dimension to diverge with the sample size. We demonstrate the efficacy of the proposed methods through both simulations and two real data examples.