Limit laws for random vectors with an extreme component

Sidney I. Resnick

114 citations0/0 files storedThe Annals of Applied Probability

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Probability and Risk Models Financial Risk and Volatility Modeling Statistical Methods and Inference

Abstract

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.

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References

2001An Introduction to Statistical Modeling of Extreme Values 1997Modelling Extremal Events 1987Extreme Values, Regular Variation and Point Processes 1990Models for Exceedances Over High Thresholds 2006Extreme Value Theory : An Introduction 1996Statistics for near independence in multivariate extreme values 1989Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone 2004A Conditional Approach for Multivariate Extreme Values (with Discussion)

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2012Estimation of the conditional distribution of a multivariate variable given that one of its components is large: Additional constraints for the Heffernan and Tawn model 2013Efficient inference for spatial extreme value processes associated to log-Gaussian random functions 2014Estimation of Hüsler–Reiss Distributions and Brown–Resnick Processes 2020Advances in statistical modeling of spatial extremes 2008A New Class of Models for Bivariate Joint Tails

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