Kernel density estimation for circular data: a Fourier series-based plug-in approach for bandwidth selection

Abstract

In this paper, we derive asymptotic expressions for the mean integrated squared error of a class of delta sequence density estimators for circular data. This class includes the class of kernel density estimators usually considered in the literature, as well as a new class that is closer in spirit to the class of Parzen–Rosenblatt estimators for linear data. For these two classes of kernel density estimators, a Fourier series-based direct plug-in approach for bandwidth selection is presented. The proposed bandwidth selector has a $n^{-1/2}$ relative convergence rate whenever the underlying density is smooth enough and the simulation results testify that it presents a very good finite sample performance against other bandwidth selectors in the literature.

Keywords:

• Circular data
• kernel density estimation
• bandwidth selection
• plug-in rule
• Fourier series-based estimators

AMS 2020 subject classifications:

• 62G07
• 62G20

Acknowledgments

The author would like to thank the anonymous reviewers and associate editor for their constructive comments and suggestions that greatly helped to improved this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the Centre for Mathematics of the University of Coimbra – UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.