On the generalized circular projected Cauchy distribution

TL;DR

This paper thoroughly analyzes the generalized circular projected Cauchy distribution, deriving its properties, establishing testing procedures, and demonstrating its application in regression and data analysis.

Contribution

It characterizes the GCPC distribution, derives key formulas, and develops new statistical tests and estimation methods for this distribution.

Findings

The distribution exhibits unimodality under certain conditions.

Log-likelihood ratio tests perform well even when assuming the wrapped Cauchy distribution.

Regression coefficients converge empirically in simulation studies.

Abstract

\cite{tsagris2025a} proposed the generalized circular projected Cauchy (GCPC) distribution, whose special case is the wrapped Cauchy distribution. In this paper we first derive the relationship with the wrapped Cauchy distribution, and then we attempt to characterize the distribution. We establish the conditions under which the distribution exhibits unimodality. We provide non-analytical formulas for the mean resultant length and the Kullback-Leibler divergence, and analytical form for the cumulative probability function and the entropy of the GCPC distribution. We propose log-likelihood ratio tests for one, or two location parameters without assuming equality of the concentration parameters. We revisit maximum likelihood estimation with and without predictors. In the regression setting we briefly mention the addition of circular and simplicial predictors. Simulation studies illustrate a) the performance of the log-likelihood ratio test when one falsely assumes that the true distribution is the wrapped Cauchy distribution, and b) the empirical rate of convergence of the regression coefficients. Using a real data analysis example we show how to avoid the log-likelihood being trapped in a local maximum and we correct a mistake in the regression setting.