期刊
ANNALS OF STATISTICS
卷 45, 期 3, 页码 1059-1095出版社
INST MATHEMATICAL STATISTICS
DOI: 10.1214/16-AOS1475
关键词
Convergence rate; linear operator; handwriting data; reproducing kernel Hilbert space; speech recognition; Sliced Average Variance Estimator; Sliced Inverse Regression
资金
- NSF [DMS-14-07537]
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1407537] Funding Source: National Science Foundation
We propose a general theory and the estimation procedures for nonlinear sufficient dimension reduction where both the predictor and the response may be random functions. The relation between the response and predictor can be arbitrary and the sets of observed time points can vary from subject to subject. The functional and nonlinear nature of the problem leads to construction of two functional spaces: the first representing the functional data, assumed to be a Hilbert space, and the second characterizing nonlinearity, assumed to be a reproducing kernel Hilbert space. A particularly attractive feature of our construction is that the two spaces are nested, in the sense that the kernel for the second space is determined by the inner product of the first. We propose two estimators for this general dimension reduction problem, and establish the consistency and convergence rate for one of them. These asymptotic results are flexible enough to accommodate both fully and partially observed functional data. We investigate the performances of our estimators by simulations, and applied them to data sets about speech recognition and handwritten symbols.
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