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.

翻译：圆形数据的核密度估计方法已被广泛研究数十年，因其即使在底层密度形状复杂时仍能实现灵活的估计。近年来许多研究探讨了偏差校正方法；然而，即使真实密度足够光滑，这些方法在尝试改善偏差收敛速度时仍受阶数限制。为突破此限制，本研究提出一种基于底层圆形密度特征函数的全新偏差校正方法。我们引入了包裹平坦核——通过将定义在实轴上的标准平坦核包裹至单位圆周长而生成。随后推导了包裹平坦核密度估计量的渐近均方误差。结果表明，该估计量的收敛速度优于以往提出的估计量。此外，在有限支撑特征函数条件下（如圆形均匀分布与心形分布），包裹平坦核密度估计量可实现$\sqrt{n}$-相合性。数值实验验证了这些理论结果。在实证分析中，我们还证明包裹平坦核密度估计量能有效捕捉数据形态。因此，此类估计量有望在圆形数据分析中实现灵活且精确的估计。