Study, Implementation, and Application of the Bivariate Cosine Gaussian Distribution

Abstract

This article presents a comprehensive study of a newly introduced probability distribution, the bivariate cosine Gaussian distribution, ranging from theory to practical application. This distribution is characterized by a single parameter, which controls a trigonometric component. It allows for the modelling of nonlinear structural dependencies that are inaccessible to classical Gaussian models. In this study, we translate this mathematical framework into a suite of statistical tools implemented in R, developing a data generator that uses accept-reject sampling and a maximum likelihood estimation method secured by particle swarm optimization. We also develop a robust goodness-of-fit test based on the energy distance via bootstrap. We validate the relevance of our approach using a real-world dataset from electric motor engineering. Combining the energy test with model selection via information criteria demonstrates that the oscillatory nature of this distribution is particularly well-suited to capturing threshold effects and discrete variations imposed by motor control algorithms.