On Comparing Different Methods of Estimation for the Parameters of a Pathological Distribution with Application to Climate Data

Abstract

In statistical literature, various probability distributions exist with advantageous properties,
while others are considered pathological since their properties are counterintuitive. A well-known pathological
probability distribution is the Cauchy distribution, and it has applications in areas related to environmental
and financial research. Both the log-Cauchy and half-Cauchy distributions, which have close connections
to the Cauchy distribution, are pathological distributions. This paper considers another pathological model
called the Cauchy Birnbaum-Saunders distribution. Some of the statistical properties of this distribution
are discussed briefly, and its parameters are estimated using eight frequentist estimation methods, including
the maximum likelihood, least-squares-based, and minimum distance estimation methods. Monte Carlo simulations
are carried out to compare and examine the performance of each estimator numerically. Furthermore,
a recent climate data set is analyzed to show the practical applicability of this model.