On the point process of near-record values

• Raúl Gouet ^[1] ; F. Javier López ^[2] ; Gerardo Sanz ^[2]
  1. [1] Universidad de Chile

     Universidad de Chile

     Santiago, Chile

     
  2. [2] Universidad de Zaragoza

     Universidad de Zaragoza

     Zaragoza, España

     
• Localización: Test: An Official Journal of the Spanish Society of Statistics and Operations Research, ISSN-e 1863-8260, ISSN 1133-0686, Vol. 24, Nº. 2, 2015, págs. 302-321
• Idioma: inglés
• DOI: 10.1007/s11749-014-0408-0
• Texto completo no disponible (Saber más ...)
• Resumen

  • Let (Xn) be a sequence of independent and identically distributed random variables, with common absolutely continuous distribution F. An observation Xn is a near-record if Xn∈(Mn−1−a,Mn−1], where Mn=max{X1,…,Xn} and a>0 is a parameter. We analyze the point process η on [0,∞) of near-record values from (Xn), showing that it is a Poisson cluster process. We derive the probability generating functional of η and formulas for the expectation, variance and covariance of the counting variables η(A),A⊂[0,∞). We also obtain strong convergence and asymptotic normality of η(t):=η([0,t]), as t→∞, under mild tail-regularity conditions on F. For heavy-tailed distributions, with square-integrable hazard function, we show that η(t) grows to a finite random limit η(∞) and compute its probability generating function. We apply our results to Pareto and Weibull distributions and include an example of application to real data.