Abstract

Recent extensions have explored additional flexibility, though these models still face certain limitations, such as restrictions on parameter ranges and the inability to cover half-circular distributions.
Additionally, the absence of explicit solutions for parameter estimation in skewed circular models complicates their application.

Parameter estimation for circular distributions is often complex, with the solution to the likelihood equation seldom available in a simple form, except for the von Mises distribution.
While the maximum likelihood estimate of the von Mises distribution is straightforward, other circular distributions typically require numerical calculation algorithms.

The Expectation-Maximization (EM) algorithm is a stable and widely used method for parameter estimation in statistical models.
However, no explicit solutions currently exist for skewed distributions on the circle.
To address these challenges, we introduce Cauchy-type distributions on the circle and propose an accelerated algorithm using the vector epsilon algorithm.

In this talk, we first introduce Cauchy-type distributions on the circle, highlighting their properties and advantages over existing models.
We then provide detailed elements of the information matrices for these distributions, which are crucial for understanding their statistical properties.
Following this, we present EM algorithms specifically tailored to these distributions.
Additionally, we propose an acceleration method for the EM algorithm, which significantly reduces computational time while maintaining accuracy.

Finally, we give examples of parameter estimation using circular data sets, demonstrating the practical applicability and efficiency of our proposed methods.
These examples will include a comparison with traditional methods, showcasing the improvements in estimation accuracy and computational efficiency.