Parameter estimation for the Cauchy distribution

In physics, mathematics, and several related disciplines like spectroscopy, the Cauchy probability distribution plays an important role. Therefore, a parameter estimation methodology for data which is distributed according to a Cauchy distribution is of great practical importance. However, the Cauchy distribution is also well known for causing difficulties with standard approaches. It has undefined moments due its heavy tails, effectively modeling a high probability that large outlying events occur, making parameter estimation an exceptionally difficult task. In this paper, we explore possibilities to accurately estimate unknown parameters from data distributed according to a Cauchy distribution for an arbitrary linear model by showing a computationally simple way to iteratively solve the complicated associated likelihood equations. We furthermore state insightful interpretations of the resulting estimators and parameter estimation bounds, revealing a basic structure for robust parameter estimation procedures derived for completely different probability distributions.