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On the generalized circular projected Cauchy distribution

Omar Alzeley, Michail Tsagris

Abstract

\cite{tsagris2025a} proposed the generalized circular projected Cauchy distribution, whose special case is the wrapped Cauchy distribution. In this paper we first derive the relationship with the wrapped Cauchy distribution and we propose a log-likelihood ratio test for the equality of two angular means, without assuming equality of thew concentration parameters. Simulation studies illustrate the performance of the test when one falsely assumes that the true underlying distribution is the wrapped Cauchy distribution.

On the generalized circular projected Cauchy distribution

Abstract

\cite{tsagris2025a} proposed the generalized circular projected Cauchy distribution, whose special case is the wrapped Cauchy distribution. In this paper we first derive the relationship with the wrapped Cauchy distribution and we propose a log-likelihood ratio test for the equality of two angular means, without assuming equality of thew concentration parameters. Simulation studies illustrate the performance of the test when one falsely assumes that the true underlying distribution is the wrapped Cauchy distribution.
Paper Structure (7 sections, 1 theorem, 17 equations, 1 figure, 1 table)

This paper contains 7 sections, 1 theorem, 17 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The GCPC distribution
  3. The relationship between the GCPC and CIPC distributions
  4. The mean resultant length
  5. Equality of two angular means without assuming equal concentration parameters
  6. Simulation studies
  7. Conclusions

Key Result

Theorem 2.1

If $\phi$ follows the GCPC distribution, GCPC$(0,\gamma,\lambda)$, $\psi = \arctan{\left(\frac{\tan\phi}{\sqrt{\lambda}}\right)}$ follows the CIPC distribution, CIPC$(0,\gamma,1) \equiv WC(0,\delta)$, where $\phi$ and $\delta$ are defined above.

Figures (1)

  • Figure 1: Values of $\rho$ as a function of $\gamma$ and $\lambda$.

Theorems & Definitions (2)

  • Theorem 2.1
  • proof