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Lee, Kuang-Yao; Li, Lexin

Functional sufficient dimension reduction through average Fréchet derivatives. (English) Zbl 1486.62115

Ann. Stat. 50, No. 2, 904-929 (2022).
Summary: 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 Fréchet 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-Loève 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.

Cited in 3 Documents

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62H12 Estimation in multivariate analysis
62R10 Functional data analysis

Keywords:

consistency; exhaustiveness; functional central mean subspace; functional central subspace; function-on-function regression; reproducing kernel Hilbert space; unbiasedness

Software:

fda (R)
PDFBibTeX XMLCite
\textit{K.-Y. Lee} and \textit{L. Li}, Ann. Stat. 50, No. 2, 904--929 (2022; Zbl 1486.62115)
Full Text: DOI

References:

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