Multivariate Weibull Distributions for Asset Returns : I

We present a characterization of the non-Gaussian properties of the distributions of the asset returns and introduce a general parameterization of the multivariate distribution of returns based on two steps: (i) the projection of the empirical marginal distributions onto Gaussian laws via nonlinear mappings; (ii) the use of an entropy maximization to construct the corresponding most parsimonious representation of the multivariate distribution. The entropy maximization principle amounts to choosing a Gaussian copula for the representation of the dependence of the assets. The marginal distributions are parameterized in terms of so-called modified Weibull distributions, which encompass both sub-exponentials and super-exponentials. We present an empirical calibration of the two key parameters (the exponent c and the characteristic scale chi) of the modified Weibull distribution, and discuss statistical tests of this parameterization. This prepares the foundation for higher-moment portfolio theory developed in companion letters (Malevergne and Sornette, 2005a,b)