2014/11/17 by Hans‐Peter Scheffler, Hans-Peter Scheffler, Scheffler, Hans-Peter +2
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probabilistic and Robust Engineering Design #Statistics Theory (math.ST) #Stochastic processes and financial applications #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1411.4688
arxiv created 2014/11/17 · openalex publication_date 2014/11/17 · arxiv updated 2014/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let X1,...,Xn be iid random vectors and f≥ 0 be a non-negative function. Let also k(n) = \rm Argmaxi=1,...,n f(Xi). We are interested in the distribution of Xk(n) and their limit theorems. In other words, what is the distribution the random vector where a function of its components is extreme? 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 to 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.