Living on the Multidimensional Edge: Seeking Hidden Risks Using Regular Variation

Abstract

Multivariate regular variation plays a role in assessing tail risk in diverse applications such as finance, telecommunications, insurance, and environmental science. The classical theory, being based on an asymptotic model, sometimes leads to inaccurate and useless estimates of probabilities of joint tail regions. This problem can be partly ameliorated by using hidden regular variation (see Resnick (2002) and Mitra and Resnick (2011)). We offer a more flexible definition of hidden regular variation that provides improved risk estimates for a larger class of tail risk regions.

Conditional extreme value models: fallacies and pitfalls

“…This may be due to the fact that in the standard case where (X, Y ) attains values in [0, ∞) 2 and both components are divided by the same factor, CEVM, classical extreme value models and the additional convergence (4.9) considered in models with hidden regular variation can all be considered special cases of a general concept of regular variation on cones; see e.g. Das et al (2013).…”

“…Moreover, even if there is only one point of accumulation, different multiplicative normalizing functions α may lead to different limits if mass on {±∞} ×Ē (γY ) is allowed; see e.g. Example 5.4 of Das et al (2013). So while in some cases omitting the condition (ii) in Definition 2.1 allows for a more detailed analysis of X for large values of Y , one pays the price of an enormously increased complexity.…”

Are extreme value estimation methods useful for network data?

“…where ν(·) ∈ M(R 2 + \ {0}) is called the limit or tail measure [7,14,19]. Using the power transformation I → I a with a = ι in /ι out , the vector (I a , O) becomes standard regularly varying, i.e., y).…”

Heavy-tailed random walks, buffered queues and hidden large deviations

“…Furthermore our goal is to use such results in the context of queueing to understand the behavior of what we call long intense periods in a large-deviation-type event. Analysis of hidden behavior of regularly varying sequences on R d (Resnick, 2002;Das, Mitra and Resnick, 2013) and more recently on R ∞ and Lévy processes on D[0, 1] (Lindskog, Resnick and Roy, 2014) has been conducted under the name hidden regular variation. Connections between hidden regular variation and elements of the classical large deviations framework have been established recently in Rhee, Blanchet and Zwart (2016).…”

“…We establish that the most probable way a large deviation event occurs, which is not the result of only one random variable being large, is actually when two random variables are large; resulting in a non-null limit measure as in (2) concentrating on processes having two jump discontinuities. For our analysis we use the framework proposed in Lindskog, Resnick and Roy (2014) and the notion of convergence used here is known as M O -convergence which is closely related to the w #convergence of boundedly finite measures and developed in Hult and Lindskog (2006); Das, Mitra and Resnick (2013); Lindskog, Resnick and Roy (2014). We briefly recall the required background on regular variation and M O -convergence in Section 2.…”