EM-based maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions with fixed λ

Generalized Hyperbolic distribution (Barndorff-Nielsen 1977) is a variance-mean mixture of a normal distribution with the Generalized Inverse Gaussian distribution. Recently subclasses of these distributions (e.g., the hyperbolic distribution and the Normal Inverse Gaussian distribution) have been applied to construct stochastic processes in turbulence and particularly in finance, where multidimensional problems are of special interest. Parameter estimation for these distributions based on an i.i.d. sample is a difficult task even for a specified one-dimensional subclass (subclass being uniquely defined by λ) and relies on numerical methods. For the hyperbolic subclass (λ = 1), computer program ‘hyp’ (Blæsild and Sørensen 1992) estimates parameters via ML when the dimensionality is less than or equal to three. To the best of the author's knowledge, no successful attempts have been made to fit any given subclass when the dimensionality is greater than three. This article proposes a simple EM-based (Dempster, Laird and Rubin 1977) ML estimation procedure to estimate parameters of the distribution when the subclass is known regardless of the dimensionality. Our method relies on the ability to numerically evaluate modified Bessel functions of the third kind and their logarithms, which is made possible by currently available software. The method is applied to fit the five dimensional Normal Inverse Gaussian distribution to a series of returns on foreign exchange rates.