Implicit extremes and implicit max–stable laws

Abstract

Let X1, ⋯ , Xn be iid random vectors and f≥0 be a homogeneous non–negative function interpreted as a loss function. Let also k(n)=Argmaxi=1c⋯ , nf(Xi). We are interested in the asymptotic behavior of Xk(n) as n→∞. In other words, what is the distribution of the random vector leading to maximal loss. This question is motivated by a kind of inverse problem where one wants to determine the extremal behavior of X when only explicitly observing f(X). We shall refer to such types of results as implicit extremes. It turns out that, as in the usual case of explicit extremes, all limit implicit extreme value laws are implicit max–stable. We characterize the regularly varying implicit max–stable laws in terms of their spectral and stochastic representations. We also establish the asymptotic behavior of implicit order statistics relative to a given homogeneous loss and conclude with several examples drawing connections to prior work involving regular variation on general cones.