Using Second-Order Probabilities to Make Maximum Entropy Approach to Copulas More Reasonable

Hung T. Nguyen, Vladik Kreinovich, Berlin Wu

Abstract

Copulas are a general way of describing dependence between twoor more random variables. When we only have partial information about thedependence, i.e., when several dierent copulas are consistent with our knowledge,it is often necessary to select one of these copulas. A frequently used method ofselecting this copula is the maximum entropy approach, when we select a copulawith the largest entropy. However, in some cases, the maximum entropy approachleads to an unreasonable selection { e.g., even if we know that the two randomvariables are positively correlated, the maximum entropy approach completelyignores this information. In this paper, we show how to properly modify themaximum entropy approach so that it will lead to more reasonable results: by applying this approach not to the probabilities themselves, but to \second order"probabilities { i.e., probabilities of dierent probability distributions