Elliptically symmetric distributions for directional data of arbitrary dimension

TL;DR

This paper introduces a flexible class of angular Gaussian distributions for directional data of any dimension, with a new parameterization that simplifies inference and allows for meaningful interpretation of parameters.

Contribution

It proposes a novel reparameterization of angular Gaussian distributions, enabling easier maximum likelihood estimation and robust inference for directional data.

Findings

The new distribution family accommodates different degrees of isotropy.

The parameterization improves stability and interpretability of inference.

Application to hydrogeology data demonstrates practical utility.

Abstract

We formulate a class of angular Gaussian distributions that allows different degrees of isotropy for directional random variables of arbitrary dimension. Through a series of novel reparameterization, this distribution family is indexed by parameters with meaningful statistical interpretations that can range over the entire real space of an adequate dimension. The new parameterization greatly simplifies maximum likelihood estimation of all model parameters, which in turn leads to theoretically sound and numerically stable inference procedures to infer key features of the distribution. Byproducts from the likelihood-based inference are used to develop graphical and numerical diagnostic tools for assessing goodness of fit of this distribution in a data application. Simulation study and application to data from a hydrogeology study are used to demonstrate implementation and performance of the inference procedures and diagnostics methods.