On the point process of near-record values

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• 作者：Raúl Gouet ; F. Javier López ; Gerardo Sanz
• 关键词：Record ; Near ; record ; Poisson cluster process ; Law of large numbers ; Central limit theorem ; 60G70 ; 60G55 ; 60F05 ; 60F15
• 刊名：TEST
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：24
• 期：2
• 页码：302-321
• 全文大小：526 KB
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• 作者单位：Raúl Gouet (1)
  F. Javier López (2)
  Gerardo Sanz (2)
  
  1. Dpto. Ingeniería Matemática y Centro de Modelamiento Matemático (UMI 2807, CNRS), Universidad de Chile, Casilla 170/3, Santiago, Chile
  2. Dpto. Métodos Estadísticos and BIFI, Facultad de Ciencias, Universidad de Zaragoza, C/ Pedro Cerbuna, 12, 50009, Zaragoza, Spain
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Statistics
  Statistics
  Statistical Theory and Methods
  Statistics for Business, Economics, Mathematical Finance and Insurance
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1863-8260

文摘

Let \((X_n)\) be a sequence of independent and identically distributed random variables, with common absolutely continuous distribution \(F\). An observation \(X_n\) is a near-record if \(X_n\in (M_{n-1}-a,M_{n-1}]\), where \(M_{n}=\max \{X_1,\ldots ,X_{n}\}\) and \(a>0\) is a parameter. We analyze the point process \(\eta \) on \([0,\infty )\) of near-record values from \((X_n)\), showing that it is a Poisson cluster process. We derive the probability generating functional of \(\eta \) and formulas for the expectation, variance and covariance of the counting variables \(\eta (A), A\subset [0,\infty )\). We also obtain strong convergence and asymptotic normality of \(\eta (t):=\eta ([0,t])\), as \(t\rightarrow \infty \), under mild tail-regularity conditions on \(F\). For heavy-tailed distributions, with square-integrable hazard function, we show that \(\eta (t)\) grows to a finite random limit \(\eta (\infty )\) and compute its probability generating function. We apply our results to Pareto and Weibull distributions and include an example of application to real data.