Rates of convergence for extremes of geometric random variables and marked point processes

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• 作者：Alessandra Cipriani ; Anne Feidt
• 关键词：Stein ; Chen method ; Maxima of geometric random variables ; Marshall ; Olkin geometric distribution ; Poisson approximation ; Marked point process of extremes ; 60F99 ; 62E20
• 刊名：Extremes
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：19
• 期：1
• 页码：105-138
• 全文大小：611 KB
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• 作者单位：Alessandra Cipriani (1)
  Anne Feidt (2)
  
  1. Weierstrass Institute, Mohrenstrasse 39, 10117, Berlin, Germany
  2. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zurich, Switzerland
  
• 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

文摘

We use the Stein-Chen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author’s PhD thesis (Feidt 2013) under the supervision of Andrew D. Barbour. Keywords Stein-Chen method Maxima of geometric random variables Marshall-Olkin geometric distribution Poisson approximation Marked point process of extremes