Open Access
2022 Statistical inference on the Hilbert sphere with application to random densities
Xiongtao Dai
Author Affiliations +
Electron. J. Statist. 16(1): 700-736 (2022). DOI: 10.1214/21-EJS1942


The infinite-dimensional Hilbert sphere S has been widely employed to model density functions and shapes, extending the finite-dimensional counterpart. We consider the Fréchet mean as an intrinsic summary of the central tendency of data lying on S. For sound statistical inference, we derive properties of the Fréchet mean on S by establishing its existence and uniqueness as well as a root-n central limit theorem (CLT) for the sample version, overcoming obstructions from infinite-dimensionality and lack of compactness on S. Intrinsic CLTs for the estimated tangent vectors and covariance operator are also obtained. Asymptotic and bootstrap hypothesis tests for the Fréchet mean based on projection and norm are then proposed and are shown to be consistent. The proposed two-sample tests are applied to make inference for daily taxi demand patterns over Manhattan, modeled as densities, of which the square root densities are analyzed on the Hilbert sphere. Numerical properties of the proposed hypothesis tests which utilize the spherical geometry are studied in the real data application and simulations, where we demonstrate that the tests based on the intrinsic geometry compare favorably to those based on an extrinsic or flat geometry.

Funding Statement

The research is supported in part by NSF grant DMS-2113713.


The author is grateful to the reviewers for their constructive comments. He would also like to thank Hans-Georg Müller for bringing up this topic.


Download Citation

Xiongtao Dai. "Statistical inference on the Hilbert sphere with application to random densities." Electron. J. Statist. 16 (1) 700 - 736, 2022.


Received: 1 October 2020; Published: 2022
First available in Project Euclid: 19 January 2022

arXiv: 2101.00527
Digital Object Identifier: 10.1214/21-EJS1942

Primary: 62G20
Secondary: 62G05 , 62G99

Keywords: Functional data analysis , Hilbert geometry , intrinsic mean , large sample property , Riemannian manifold

Vol.16 • No. 1 • 2022
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