Limit Laws for Random Vectors with an Extreme Component

By Janet E. Heffernan, Sidney I. Resnick. 2005

174 citations recorded by the source corpus.

Models based on assumptions of multivariate regular variation and hidden regular variation provide ways to describe a broad range of extremal dependence structures when marginal distributions are heavy tailed. Multivariate regular variation provides a rich description of extremal dependence in the case of asymptotic dependence, but fails to distinguish between exact independence and asymptotic independence. Hidden regular variation addresses this problem by requiring components of the random vector to be simultaneously large but on a smaller scale than the scale for the marginal distributions. In doing so, hidden regular variation typically restricts attention to that part of the probability space where all variables are simultaneously large. However, since under asymptotic independence the largest values do not occur in the same observation, the region where variables are simultaneously large may not be of primary interest. A different philosophy was offered in the paper of Heffernan and Tawn [J. R. Stat. Soc. Ser. B Stat. Methodol. 66 (2004) 497–546] which allows examination of distributional tails other than the joint tail. This approach used an asymptotic argument which conditions on one component of the random vector and finds the limiting conditional distribution of the remaining components as the conditioning variable becomes large. In this paper, we provide a thorough mathematical examination of the limiting arguments building on the orientation of Heffernan and Tawn [J. R. Stat. Soc. Ser. B Stat. Methodol. 66 (2004) 497–546]. We examine the conditions required for the assumptions made by the conditioning approach to hold, and highlight simililarities and differences between the new and established methods.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/df27b838c6624ce1cd4f60cc0792a54616142839

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Citations in the corpus

Modelling across extremal dependence classes (2014, background): “Heffernan and Tawn (2004) developed a very general modelling assumption that we present in the adapted form of Heffernan and Resnick (2007).”
Modelling spatial extreme events with environmental applications (2018, methodology): “Under weak assumptions on the joint distribution of X, Heffernan and Resnick (2007) show that, componentwise, a and b must be regularly varying functions satisfying certain constraints, which for Gumbel margins corresponds to each of the components of a (respectively b) being regularly varying funct”
A new representation for multivariate tail probabilities (2013, background): “Summary of important theoretical quantities for representations of Examples 1– 5 under the theory of the current paper, Ledford and Tawn [13] and Heffernan and Tawn [10]/Heffernan and Resnick [9]”
Joint modelling of extreme ocean environments incorporating covariate effects (2013, methodology): “We extend the conditional extremes model of Heffernan and Tawn (2004) to incorporate covariate effects.”
Detecting a conditional extreme value model (2009, methodology): “Refer to Heffernan and Resnick (2007) and Das and Resnick (2008b) for further discussion on conditional extreme value models.”
Geostatistics of extremes (2012, background): “More recent work on these lines is by Heffernan & Resnick (2005, 2007).”
Estimation of Hüsler–Reiss distributions and Brown–Resnick processes (2012, background): “…approach generalizes classical multivariate regular variation and was introduced by Heffernan and Tawn (2004) and further investigated by several researchers (see Hashorva (2006), Balkema and Embrechts (2007), Heffernan and Resnick (2007), Fougères and Soulier (2010) and Das and Resnick (2011)).”
Heavy tailed time series with extremal independence (2013, methodology): “eater than ǫ; a subset Lof (0,∞]×[−∞,∞]h is relatively compact if there exists ǫ>0 such that x ∈ Limplies that the ﬁrst component of x is greater than ǫ. For h= 1, Assumption 1 is Condition (5) in [HR07]. We extend it here to a multidimensional framework and to diﬀerent scaling functions bj, j≥ 1”
Living on the Multidimensional Edge: Seeking Hidden Risks Using Regular Variation (2011, methodology): “This is the restriction to [0,∞)2 of the cone used in the definition of the conditional extreme value model [4], [8], [9].”
A new class of models for bivariate joint tails (2009, background): “Some inroads have been made into this, e.g. Heffernan and Tawn (2004) and Heffernan and Resnick (2007); however, no fully characterized distributional family is available currently.”
Geostatistics of Dependent and Asymptotically Independent Extremes (2013, background): “(1), (2), and (3) have been widely used in applications for decades (Gumbel 1958; Smith 1989; Davison and Smith 1990; Coles 2001; Beirlant et al. 2004), but modeling in the spatial and space-time domains is much more recent.”