Smooth Kernel Estimation of a Circular Density Function: A Connection to Orthogonal Polynomials on the Unit Circle

TL;DR

This paper explores the use of the wrapped Cauchy kernel in circular density estimation, connecting it to orthogonal polynomials on the unit circle, and highlights its significance in circular statistics.

Contribution

It introduces a theoretical motivation for circular kernel density estimation using the wrapped Cauchy kernel and links it to orthogonal series methods on the unit circle.

Findings

Wrapped Cauchy kernel is a natural choice for circular density estimation.

The connection to orthogonal polynomials enhances understanding of circular kernels.

The approach emphasizes the importance of the wrapped Cauchy in circular statistics.

Abstract

In this note we provide a simple 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. This adds further weight to the considerable role of the wrapped Cauchy in circular statistics.