Dependence Structures beyond copulas: A new model of a multivariate regular varying distribution based on a finite von Mises-Fisher mixture model
A multivariate regular varying distribution can be characterized by its marginals and a finite measure on the unit sphere. That measure is referred to as the spectral measure of the distribution. The spectral measure describes the structure of the dependence between the marginal distributions. An important class of multivariate regular varying distributions are multivariate extreme value distributions. Existing models for multivariate regular varying distributions in general and multivariate extreme value distributions in particular do not utilize the spectral measure. They focus on closed form equations of the cumulative distribution function. The resulting models are not flexible enough to give a realistic and adequate description of the dependence structure of real life data. We propose a new model for multivariate regular varying distributions, based on a very flexible parametric model of the spectral measure. We use a finite mixture model to obtain a model with as much flexibility as needed to accurately describe the spectral measure of real life data. Since the spectral measure is a measure on the unit sphere, we chose directional distributions as the distributions of the components of the mixture model. Directional distributions provide models for the distribution of random variables on unit spheres. In particular, we use the von Mises-Fisher distribution. Its properties allow it to be interpreted as an directional analogue of the well known normal distribution on a Euclidian space. We describe how to estimate the parameters of this new model from datasets. We introduce a modified version of the likelihood ratio test to decide on how many components are needed for an accurate model of the spectral measure. We show how our model explains the structure of the spectral measure of several financial time series. We develop a comprehensive model for a multivariate regular varying distribution that is based on our model of the spectral measure. As one particular application of this new model we describe how it can be used for portfolio optimization. We found that our model gives much more accurate results than two other well established models. It significantly improves on the deficiencies of the two existing models.
Multivariate Extreme Value Theory; Multivariate regular variation; Spectral Measure; Finite Mixture Model
dissertation or thesis