The multivariate normal inverse Gaussian heavy-tailed distribution; simulation and estimation

Abstract

The normal inverse Gaussian (NIG) distribution is a recent variance-mean mixture of a Gaussian with an inverse Gaussian distribution. The NIG can serve as a model for data that are heavy-tailed (leptokurtic), and the model was first introduced in empirical finance by Bamdorrf-Nielsen in 1995. In this paper, we present the important extension to multivariate NIG (MNIG) distributions, and we discuss some of the basic properties of the MNIG. We furthermore discuss several new and important properties of the MNIG. An important part of the paper deals with the derivation of a fast and accurate method for generating i.i.d. MNIG-distributed variates. We also present a multivariate Expectation-Maximization (EM) algorithm for the estimation of the scalar, vector, and matrix parameters of the MNIG. Finally, we present a fit of the bivariate NIG to an actual multichannel radar data set, where we have applied our EM parameter estimation algorithm. From the insight we have gained, we conclude that the MNIG has numerous potential applications in multivariate data analysis and modeling, and that the simulation and estimation methods described in this paper may serve as important and useful tools in that respect.