Sharp bounds on the survival function of exchangeable min-stable multivariate exponential sequences

Abstract

We derive lower and upper bounds for the survival function of an exchangeable sequence of random variables, for which the scaled minimum of each finite subgroup has a univariate exponential distribution. These bounds are sharp in the sense that both bounds themselves are attained by exchangeable sequences of the same kind, for which the (non-scaled) minimum of each subgroup has the same univariate exponential distribution as the original sequence. This result is equivalent to inequalities between infinite-dimensional stable tail dependence functions, which leads to inequalities between multivariate extreme-value copulas. In addition, it is explained how an infinite-dimensional symmetric stable tail dependence function can be obtained from its upper bound by censoring certain distributional information. This technique is applied to derive new parametric families.

Keywords

• exchangeable sequence
• min-stable multivariate exponential
• extreme-value copula
• bernstein function
• lévy subordinator
• stable tail dependence function
• supermodular order
• 60g09
• 60g70
• 60e07