Bivariate extreme value theory: Models and estimation

Abstract

SUMMARY Bivariate extreme value distributions arise as the limiting distributions of renormalized componentwise maxima. No natural parametric family exists for the dependence between the marginal distributions, but there are considerable restrictions on the dependence structure. We consider modelling the dependence function with parametric models, for which two new models are presented. Tests for independence, and discriminating between models, are also given. The estimation procedure, and the flexibility of the new models, are illustrated with an application to sea level data. Extreme value theory has recently been an area of much theoretical and practical work. Univariate theory is a well documented area, whereas bivariate/multivariate extreme value theory has, until recently, received surprisingly little attention. In the multivariate case, no natural parametric family exists for the dependence structure, so this must be modelled in some way. In the analysis of environmental extreme value data, there is a need for models of dependence between extremes from different sources: for example at various sea ports, or at various -points of a river. In this paper we consider bivariate extreme value distributions. We assume, without loss of generality, that we have exponential marginal distributions with unit means. The class of bivariate exponential distributions in which we are interested, satisfy a strong stability relation. Exponential variables (X, Y) satisfy the stability relation if and only if W = min (aX, bY) is also exponentially distributed for all a, b > 0 (Pickands, 1981). Therefore, the models we will consider have particular application in reliability and survival analysis. One approach to modelling the dependence structure is via parametric models. This requires a flexible family of models which satisfy certain constraints. Models are of two kinds: either differentiable, or nondifferentiable. All nondifferentiable models give distri- butions which are singular, with nonzero probability concentrated on a certain subspace. The differentiable models have densities, but the existing models are symmetric which leads to the variables being exchangeable. Here, we present two new asymmetric differenti- able models, which have increased flexibility. Properties of the differentiable models are examined. Estimation of the parametric models has previously been by ad hoc methods, because there is a nonregular estimation problem when the margins are independent. For the