Open Access
August 2017 Tests for separability in nonparametric covariance operators of random surfaces
John A. D. Aston, Davide Pigoli, Shahin Tavakoli
Ann. Statist. 45(4): 1431-1461 (August 2017). DOI: 10.1214/16-AOS1495


The assumption of separability of the covariance operator for a random image or hypersurface can be of substantial use in applications, especially in situations where the accurate estimation of the full covariance structure is unfeasible, either for computational reasons, or due to a small sample size. However, inferential tools to verify this assumption are somewhat lacking in high-dimensional or functional data analysis settings, where this assumption is most relevant. We propose here to test separability by focusing on $K$-dimensional projections of the difference between the covariance operator and a nonparametric separable approximation. The subspace we project onto is one generated by the eigenfunctions of the covariance operator estimated under the separability hypothesis, negating the need to ever estimate the full nonseparable covariance. We show that the rescaled difference of the sample covariance operator with its separable approximation is asymptotically Gaussian. As a by-product of this result, we derive asymptotically pivotal tests under Gaussian assumptions, and propose bootstrap methods for approximating the distribution of the test statistics. We probe the finite sample performance through simulations studies, and present an application to log-spectrogram images from a phonetic linguistics dataset.


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John A. D. Aston. Davide Pigoli. Shahin Tavakoli. "Tests for separability in nonparametric covariance operators of random surfaces." Ann. Statist. 45 (4) 1431 - 1461, August 2017.


Received: 1 September 2015; Revised: 1 June 2016; Published: August 2017
First available in Project Euclid: 28 June 2017

zbMATH: 06773279
MathSciNet: MR3670184
Digital Object Identifier: 10.1214/16-AOS1495

Primary: 62G10 , 62G20
Secondary: 62F40

Keywords: Acoustic phonetic data , bootstrap , dimensional reduction , functional data , partial trace , Sparsity

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 4 • August 2017
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