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<p>Kernel density estimators with circular data have been studied extensively for decades, as they allow flexible estimation 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sqrt{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msqrt> <mi>n</mi> </msqrt> </math></EquationSource> </InlineEquation>-consistency under the characteristic function of finite support, such as the circular uniform and cardioid distributions. We confirm these theoretical results in 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.</p>

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Wrapped flat-top kernel density estimation with circular data

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月24日

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  • Yasuhito Tsuruta

摘要

Kernel density estimators with circular data have been studied extensively for decades, as they allow flexible estimation 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 \(\sqrt{n}\) n -consistency under the characteristic function of finite support, such as the circular uniform and cardioid distributions. We confirm these theoretical results in 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.