Família de distribuições log-skew-multivariadas: definição,entropia e outras propriedades

Abstract

The study of asymmetric families is very important, since the normality assumption often considered in describing the statistical behavior of data is, in some cases, inadequate. Moreover, sometimes, we are interested in variables that do not take values in the negative reals and thus the skew-normal distributions may not be appropriate. This work introduces a generalization of the family of multivariate log-skew normal distributions defined by Marchenko and Genton (2010), called canonical fundamental log-skew normal distribution, where the function used to asymmetrise the normal distribution becomes a m-variate cumulative distribution function. It is obtained via a transformation of the family of multivariate canonical fundamental skew normal distribution, defined by Arellano-Valle and Genton (2005). Some of its properties, such as moments, marginal and conditional distributions and stochastic representations, are discussed in this text. Another contribution of this work is the study of entropy in asymmetric families. We calculate the entropy and mutual information for the canonical fundamental log-skew normal distribution, relating these results with existing ones for multivariate normal and skew-normal distributions. Aiming to compare this new family and the one introduced by Marchenko and Genton (2010), we also find an expression for the relative entropy between them. The new distribution is used to model databases that take only positive values, using bayesian inference. Different models are proposed and compared by the goodness of fit when we change the dimension m of the function responsible for asymmetrise normal distribution in the family of canonical fundamental log-skew normal distributions.