Tail Behavior for Bivariate Distributions Based on Pareto Mixtures

Abstract

For conditional tail inferences from multivariate distributions, we desire models with positive dependence for which conditional distributions of one variable given others have extreme value indices that can be functions of the values of conditioning variables. That is, the tails of the conditional distributions behave like Pareto distributions with varying tail parameter. It is shown in Arnold (Stat Probab Lett 5:263–266, 1987) and Arnold et al. (Stat Probab Lett 17:361–368, 1993) in the bivariate case that if all conditional distributions are (generalized) Pareto and the tail parameters are non-constant, then tractable solutions have limited range of dependence. To obtain models with our desired properties, we specify one set of conditional Pareto distributions and one marginal distribution: for example, (a) FY |X(⋅|x) is Pareto with tail parameter decreasing in x and/or scale parameter is increasing in x, (b) no specification is made for FX|Y(⋅|x), and (c) X has a distribution with regularly varying upper tail. Based on this construction, we study the following properties: (1) relation of concordance ordering to the tail parameter function, (2) relations of conditional extreme value indices to marginal extreme value indices, and (3) tail dependence.