In this short note, we present a refined approximation for the log-ratio of the density of the von Mises$(\mu,\kappa)$ distribution (also called the circular normal distribution) to the standard (linear) normal distribution when the concentration parameter \k{appa} is large. Our work complements the one of Hill (1976), who obtained a very similar approximation along with quantile couplings, using earlier approximations by Hill & Davis (1968) of Cornish-Fisher type. One motivation for this note is to highlight the connection between the circular and linear normal distributions through their circular variance and (linear) variance.

翻译：在本文中，我们针对集中参数$\kappa$较大时，von Mises$(\mu,\kappa)$分布（亦称圆形正态分布）与标准（线性）正态分布密度之对数比，提出一种精细近似。本研究补充了Hill（1976）的工作，他利用Hill & Davis（1968）对Cornish-Fisher型近似的早期成果，获得了极为相似的近似公式及分位数耦合。本文旨在通过圆形方差与（线性）方差的关联，揭示圆形正态分布与线性正态分布之间的内在联系。