A Note on the Bivariate Maximum Entropy Modeling

Let X=(X_1,X_2)$ be a continuous random vector. Under the assumption that the marginal distributions of X_1and X_2 are given, we develop models for vector X when there is partial information about the dependence structure between X_1 and X_2. The models which are obtained based on well-known Principle of Maximum Entropy are called the maximum entropy (ME) models. Our results lead to characterization of some well-known bivariate distributions such as Generalized Gumbel, Farlie-Gumbel-Morgenstern and Clayton bivariate distributions. The relationship between ME models and some well known dependence notions are studied. Conditions under which the mixture of bivariate distributions are ME models are also investigated.