[Paper Review] Implicit Extremes and Implicit Max-Stable Laws

Hans‐Peter Scheffler, Stilian Stoev|arXiv (Cornell University)|Nov 17, 2014

Financial Risk and Volatility Modeling9 references4 citations

TL;DR

This paper introduces the theory of implicit extremes, where the extremal behavior of a random vector is studied not through its marginal values but through a non-negative loss function f. It establishes that the limiting distribution of the vector achieving the maximum f-value (the implicit maximum) is implicitly max-stable, and characterizes these laws via regular variation on cones, extending classical extreme value theory to complex, coordinated extreme events in multivariate settings.

ABSTRACT

Let $X_1,...,X_n$ be iid random vectors and $f\ge 0$ be a non-negative function. Let also $k(n) = { m Argmax}_{i=1,...,n} f(X_i)$. We are interested in the distribution of $X_{k(n)}$ 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.

Motivation & Objective

• To develop a theoretical framework for understanding the joint extremal behavior of random vectors when only a loss function f(X) is observed, rather than the vector X itself.
• To characterize the limiting distribution of the random vector X_k(n) that achieves the maximum value of a non-negative loss function f(X_i) among i.i.d. copies.
• To establish that all such limit laws are implicitly max-stable, generalizing classical extreme value theory to implicit extremes.
• To connect the theory to regular variation on general cones, providing a natural mathematical foundation for implicit extremes.
• To unify and extend prior work on hidden regular variation, hidden dependence, and peaks-over-threshold models through a new perspective on implicit maxima.

Proposed method

• Uses regular variation on cones in R^d \ {f=0} as the key technical tool, with generalized polar coordinates to decompose the measure of regular variation.
• Applies a disintegration formula to represent the measure of regular variation in terms of its spectral measure on a generalized unit sphere.
• Defines implicit extremes as the random vector X_k(n) = argmax_{i=1,...,n} f(X_i), where f is a non-negative, homogeneous loss function.
• Establishes weak convergence of normalized X_k(n) to a limit law, showing that the limit is f-implicit max-stable under regular variation.
• Derives stochastic representations of the limit laws using spectral measures and the disintegration formula.
• Characterizes the implicit max-domain of attraction via regular variation on cones, proving that these are precisely the laws that are regularly varying on R^d \ {f=0}.

Experimental results

Research questions

• RQ1 What is the limiting distribution of the random vector X_k(n) that maximizes a non-negative loss function f(X_i) among i.i.d. copies?
• RQ2 How can the concept of max-stability be generalized to the case where extremal behavior is defined implicitly through a loss function f?
• RQ3 What is the connection between implicit extremes and regular variation on general cones, and how does this extend classical extreme value theory?
• RQ4 How do implicit extremes relate to existing frameworks such as hidden regular variation and peaks-over-threshold models?
• RQ5 What conditions ensure that a distribution lies in the implicit max-domain of attraction of an f-implicit max-stable law?

Key findings

• All limit laws of implicit extremes are f-implicit max-stable, meaning that the normalized implicit maximum of i.i.d. copies converges to a stable law under the same normalization.
• The implicit max-domain of attraction of f-implicit max-stable laws coincides exactly with the class of regularly varying distributions on the cone R^d \ {f=0}.
• The limit laws admit stochastic representations based on spectral measures via the disintegration formula, linking the geometry of the cone to extremal dependence.
• The convergence in distribution of the normalized implicit maximum holds uniformly on compact sets of the positive real line, under the regular variation assumption.
• The boundary of the level sets {f > u} has vanishing ν-measure for all but countably many u, ensuring the validity of the regular variation condition.
• The tail probability P(f(X) > u) is regularly varying with index -α, where α is the index of regular variation of the measure, confirming the heavy-tailed nature of the loss functional.