Graphical models are multivariate statistical models in which the joint distribution of a family of random variables is restricted by a list of conditional independence assumptions. This list is conveniently summarised in a graph whose vertices represent the variables and whose edges represent interrelations of these variables. Statistical inference for these graphical models has mainly been limited to the assumption of joint normality as far as continuous variables are concerned. We propose a new type of statistical model that permits non-Gaussian distributions as well as the inclusion of conditional independence assumptions induced by a directed acyclic graph (DAG). This combination of features is achieved by using pair-copula constructions in which a multivariate likelihood is decomposed into (potentially conditional) bivariate likelihoods by iterated application of Sklar's theorem on copulas, see [1] and [2]. This pair-copula approach allows us to specifically construct non-Gaussian distributions in order to capture features such as tail behaviour and non-linear, asymmetric dependence. We demonstrate maximum-likelihood estimation of the parameters of such models and compare them to various competing models from the literature.

References:

[1] K. Aas, C. Czado, A. Frigessi, and H. Bakken (2009). Pair-copula constructions of multiple dependence. Insurance: Mathematics and Economics, 44:182-198.

[2] A. Bauer, C. Czado, and T. Klein (2011). Pair-copula constructions for non-Gaussian DAG models. Submitted for publication.

Keywords: Graphical models; Multivariate dependence; Conditional independence; Copulas

Biography: For the past two years Alexander Bauer has been a PhD student at the Department of Mathematics at Technische Universität München, Germany. He holds a Diplom degree in Mathematics from Universität Augsburg, Germany. His research interests are multivariate statistical models, copula methods and financial mathematics.