Wrapped flat-top kernel density estimation with circular data

TL;DR

This paper introduces wrapped flat-top kernel density estimators for circular data, achieving faster convergence rates and $\

Contribution

It proposes a novel bias correction method using characteristic functions and wrapped flat-top kernels, improving estimation accuracy for circular data.

Findings

Faster convergence rates than previous estimators.

Achieves $\

Effectively captures data shape in empirical analyses.

Abstract

Kernel density estimators with circular data have been studied extensively for decades, as they allow flexible estimations even when the shape of the underlying density is complex. Many recent studies have examined bias correction methods; however, these methods are limited by the order when trying to improve the convergence rate of the bias, even if the true density is sufficiently smooth. To overcome this limitation, the present study considers a new bias correction approach based on the characteristic functions of the underlying circular density. We introduce wrapped flat-top kernels, which are generated by wrapping the standard flat-top kernels defined on the real line onto the circumference of a unit circle. The asymptotic mean squared errors of the wrapped flat-top kernel density estimators are then derived. The results show that the convergence rate of these estimators is faster than that of previously introduced estimators. Furthermore, wrapped flat-top kernel density estimators achieve $\sqrt{n}$-consistency under the characteristic function of finite support, such as the circular uniform and cardioid distributions. We confirm these theoretical results in the numerical experiments. In empirical analyses, we also show that wrapped flat-top kernel density estimators effectively capture the shape of data. Therefore, such estimators are expected to allow flexible and accurate estimation in circular data analysis.