An introduction on the multivariate normal-ratio distribution

The statistical distribution of the ratio of two normal random variables is characterized by its heavy-tailed nature and absence of finite moments. The shape of its density function is highly variable, capable of exhibiting unimodal or bimodal and symmetrical or asymmetrical structures, and, in certain regions proximal to its mode, it may approximate the characteristics of a normal distribution. Several investigations have proffered diverse representations of the cumulative distribution function utilizing the framework of the bivariate normal distribution and Nicholson's V function. The purpose of this paper is to extend these results to the multivariate case and derive its distribution function. We call this distribution multivariate normal-ratio distribution.