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The circular kernel density estimator, with the wrapped Cauchy kernel, is derived from the empirical version of Carathéodory function that is used in the literature on orthogonal polynomials on the unit circle. An equivalence between the resulting circular kernel density estimator, to Fourier series density estimator, has also been established. This adds further weight to the considerable role of the wrapped Cauchy distribution in circular statistics.

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