Approximation and estimation of very small probabilities of multivariate extreme events

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• 作者：Cees de Valk
• 关键词：Multivariate extremes ; Large deviation principle ; Residual tail dependence ; Hidden regular variation ; Log ; GW tail limit ; Generalised Weibull tail limit
• 刊名：Extremes
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：19
• 期：4
• 页码：687-717
• 全文大小：1,160 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Statistics
  Statistics
  Quality Control, Reliability, Safety and Risk
  Civil Engineering
  Hydrogeology
  Environmental Management
  Statistics for Business, Economics, Mathematical Finance and Insurance
  
• 出版者：Springer U.S.
• ISSN：1572-915X
• 卷排序：19

文摘

This article discusses modelling of the tail of a multivariate distribution function by means of a large deviation principle (LDP), and its application to the estimation of the probability p_n of a multivariate extreme event from a sample of niid random vectors, with \(p_{n}\in [n^{-\tau _{2}},n^{-\tau _{1}}]\) for some t₁>1 and t₂>t₁. One way to view the classical tail limits is as limits of probability ratios. In contrast, the tail LDP provides asymptotic bounds or limits for log-probability ratios. After standardising the marginals to standard exponential, tail dependence is represented by a homogeneous rate function I. Furthermore, the tail LDP can be extended to represent both dependence and marginals, the latter implying marginal log-Generalised Weibull tail limits. A connection is established between the tail LDP and residual tail dependence (or hidden regular variation) and a recent extension of it. Under a smoothness assumption, they are implied by the tail LDP. Based on the tail LDP, a simple estimator for very small probabilities of extreme events is formulated. It avoids estimation of I by making use of its homogeneity. Strong consistency in the sense of convergence of log-probability ratios is proven. Simulations and an application illustrate the difference between the classical approach and the LDP-based approach.