In this paper, we propose a modification to the density approach to Stein's method for intervals for the unit circle $\mathbb{S}^1$ which is motivated by the differing geometry of $\mathbb{S}^1$ to Euclidean space. We provide an upper bound to the Wasserstein metric for circular distributions and exhibit a variety of different bounds between distributions; particularly, between the von-Mises and wrapped normal distributions, and the wrapped normal and wrapped Cauchy distributions.

翻译：在本文中,我们建议修改Stein对单位圆的间隔方法的密度方法,$\mathbb{S ⁇ 1$,其动机是给欧几何空间的差别几何制为$\mathbb{S ⁇ 1$至欧几里德。我们为循环分布提供了瓦西斯坦标准上限,并展示了分布之间的不同界限,特别是von-Mises和包装正常分布之间的界限,以及包装正常和包装的Cauchy分布之间的界限。