Seminar at SMU Delhi

In this talk, I will provide a short review for smooth estimation of density and distribution functions for circular data and provide some new insights. It has been shown that the usual kernel density estimator used for linear data may not be appropriate in the context of circular data. Fisher (1989: J. Structural Geology, 11, 775-778) presents an adaptation of the linear kernel estimator, 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. I provide an approximation theory motivation for the circular kernel density estimation and further explore the usefulness of the wrapped Cauchy kernel in this context. It is seen that the wrapped Cauchy kernel appears as a natural candidate in 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.