Abstract
Copulas are an important tool to study dependencies for data on the real line (or multivariate extensions of this), referred to as linear data. The analogue of copulas for circular data and data on the torus are circulas. This paper studies kernel estimation of circulas, and discusses important issues such as choice of circular kernels and ‘smoothing’ parameters. This leads to some new insights, and some contrasts with results for linear data. Since a circula is a multivariate cumulative distribution with circular uniform marginals, the paper also contributes to kernel estimation of cumulative distributions for toroidal data.
Funding Statement
The first author was supported by Grant PID2020-116587GB-I00 funded by MCIN/AEI/10.13039/ 501100011033 and the Competitive Reference Groups 2021-2024 (ED431C 2021/24) from the Xunta de Galicia. The second author was supported in part by the FWO research project G0D6619N (Flemish Science Foundation), and the C16/20/002 project (Research Fund KU Leuven).
Acknowledgements
The authors are grateful to Professor Arthur Pewsey, University of Extremadura, Spain, for scientific discussions on the research area, during his visit at the KU Leuven in January 2019. This research was carried out when the first author was a postdoctoral researcher at the KU Leuven.
The authors thank the Editor, an Associate Editor and reviewers for their very valuable comments that led to an improved version of the manuscript.
Citation
Jose Ameijeiras-Alonso. Irène Gijbels. "Smoothed circulas: Nonparametric estimation of circular cumulative distribution functions and circulas." Bernoulli 30 (4) 2747 - 2769, November 2024. https://doi.org/10.3150/23-BEJ1693
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