Phantom distribution functions for some stationary sequences

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• 作者：Paul Doukhan ; Adam Jakubowski ; Gabriel Lang
• 关键词：Strictly stationary processes ; Extremes ; Extremal index ; Phantom distribution function ; α ; mixing ; Weak dependence ; Lindley’s process ; Random walk Metropolis algorithm ; Primary-0G70 ; Secondary-0G10 ; 60F99
• 刊名：Extremes
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：18
• 期：4
• 页码：697-725
• 全文大小：433 KB
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• 作者单位：Paul Doukhan (1)
  Adam Jakubowski (2)
  Gabriel Lang (3)
  
  1. Université de Cergy-Pontoise and Institut Universitaire de France, UMR 8088 Analyse, Géométrie et Modélisation, 2, av. Adolphe Chauvin, 95302, Cergy-Pontoise CEDEX, France
  2. Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100, Toruń, Poland
  3. AgroParisTech, UMR 518 Mathématique et Informatique appliquées, 19 avenue du Maine, 75732, Paris CEDEX 15, France
  
• 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

The notion of a phantom distribution function (phdf) was introduced by O’Brien (Ann. Probab. 15, 281-92 (1987)). We show that the existence of a phdf is a quite common phenomenon for stationary weakly dependent sequences. It is proved that any α-mixing stationary sequence with continuous marginals admits a continuous phdf. Sufficient conditions are given for stationary sequences exhibiting weak dependence, what allows the use of attractive models beyond mixing. The case of discontinuous marginals is also discussed for α-mixing. Special attention is paid to examples of processes which admit a continuous phantom distribution function while their extremal index is zero. We show that Asmussen (Ann. Appl. Probab. 8, 354-74 1998) and Roberts et al. (Extremes. 9, 213-29 2006) provide natural examples of such processes. We also construct a non-ergodic stationary process of this type. Keywords Strictly stationary processes Extremes Extremal index Phantom distribution function α-mixing Weak dependence Lindley’s process Random walk Metropolis algorithm