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Some asymptotic theory for functional regression with stationary regressor. (English) Zbl 1347.60016

Banerjee, M. (ed.) et al., From probability to statistics and back: high-dimensional models and processes. A Festschrift in honor of Jon A. Wellner. Including papers from the conference, Seattle, WA, USA, July 28–31, 2010. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-83-6). Institute of Mathematical Statistics Collections 9, 291-302 (2013).
Summary: The general asymptotic distribution theory for the functional regression model in [the first author et al., Some asymptotic theory for functional regression and classification. Tech. Rep., Texas Tech University (2009)] simplifies considerably if an extra assumption on the random regressor is made. In the special case where the regressor is a stochastic process on the unit interval, Johannes (privileged communication in 2008) assumes the regressor to be stationary, in which case the eigenfunctions of their covariance operator turn out to be known, so that only the eigenvalues are to be estimated. In the present paper, we will also assume the eigenvectors to be known, but within an abstract setting. The simplification mentioned above is due to the circumstance that the covariance operator of the regressor commutes with its estimator as it can be constructed under the current conditions. Moreover, it is now possible to test linear hypotheses for the regression parameter that correspond to linear subspaces spanned by a finite number of the known eigenvectors.
For the entire collection see [Zbl 1319.62002].

MSC:

60F05 Central limit and other weak theorems
62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
60G10 Stationary stochastic processes
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