  There is a need for new models for characterizing dependence in multivariate
data. The multivariate Gaussian distribution is routinely used, but cannot
characterize nonlinear relationships in the data. Most non-linear extensions
tend to be highly complex; for example, involving estimation of a non-linear
regression model in latent variables. In this article, we propose a relatively
simple class of Ellipsoid-Gaussian multivariate distributions, which are
derived by using a Gaussian linear factor model involving latent variables
having a von Mises-Fisher distribution on a unit hyper-sphere. We show that the
Ellipsoid-Gaussian distribution can flexibly model curved relationships among
variables with lower-dimensional structures. Taking a Bayesian approach, we
propose a hybrid of gradient-based geodesic Monte Carlo and adaptive Metropolis
for posterior sampling. We derive basic properties and illustrate the utility
of the Ellipsoid-Gaussian distribution on a variety of simulated and real data
applications. An accompanying R package is also available.
