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Kernel density estimation on the torus

Di Marzio, M, Panzera, A and Taylor, CC (2011) Kernel density estimation on the torus. Journal of Statistical Planning & Inference, 141 (6). 2156 - 2173 . ISSN 0378-3758


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Kernel density estimation for multivariate, circular data has been formulated only when the sample space is the sphere, but theory for the torus would also be useful. For data lying on a d-dimensional torus (d >= 1), we discuss kernel estimation of a density, its mixed partial derivatives, and their squared functionals. We introduce a specific class of product kernels whose order is suitably defined in such a way to obtain L-2-risk formulas whose structure can be compared to their Euclidean counterparts. Our kernels are based on circular densities; however, we also discuss smaller bias estimation involving negative kernels which are functions of circular densities. Practical rules for selecting the smoothing degree, based on cross-validation, bootstrap and plug-in ideas are derived. Moreover, we provide specific results on the use of kernels based on the von Mises density. Finally, real-data examples and simulation studies illustrate the findings.

Item Type: Article
Copyright, Publisher and Additional Information: © 2011 Elsevier B.V. This is an author produced version of a paper subsequently published in Journal of Statistical Planning & Inference. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: von Mises density, circular distributions, bandwidth selection, cross-validation, directional-data, choice, family, derivatives, circle, circular symmetric unimodal families, conformation angles, density functionals, efficiency, minimax bounds, mixed derivatives, sin-order, toroidal kernels, twicing
Institution: The University of Leeds
Academic Units: The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Statistics (Leeds)
Depositing User: Symplectic Publications
Date Deposited: 04 May 2011 12:29
Last Modified: 15 Sep 2014 04:00
Published Version: http://dx.doi.org/10.1016/j.jspi.2011.01.002
Status: Published
Publisher: Elsevier Science BV
Identification Number: 10.1016/j.jspi.2011.01.002
URI: http://eprints.whiterose.ac.uk/id/eprint/42947

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