Abstract

Fisher (1989: J. Structural Geology, 11, 775-778) outlined an adaptation of the linear kernel estimator for density estimation that is commonly used in applications. However, better alternatives are now available based on circular kernels; see e.g. Di Marzio, Panzera, and Taylor, 2009: Statistics & Probability Letters, 79(19), 2066-2075. This paper provides a short review on modern smoothing methods for density and distribution functions dealing with the circular data. We highlight the usefulness of circular kernels for smooth density estimation in this context and contrast it with smooth density estimation based on orthogonal series. It is seen that the wrapped Cauchy kernel as a choice of circular kernel appears as a natural candidate as it has a close connection to orthogonal series density estimation on a unit circle. In the literature, the use of von Mises circular kernel is investigated (see Taylor, 2008: Computational Statistics & Data Analysis, 52(7), 3493-3500), that requires numerical computation of Bessel function. On the other hand, the wrapped Cauchy kernel is much simpler to use. This adds further weight to the considerable role of the wrapped Cauchy distribution in circular statistics.