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Extreme value theory for stochastic integrals of Legendre polynomials

  • Aue, Alexander
  • Horvth, Lajos
  • Huskov, Marie
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    We study in this paper the extremal behavior of stochastic integrals of Legendre polynomial transforms with respect to Brownian motion. As the main results, we obtain the exact tail behavior of the supremum of these integrals taken over intervals [0,h] with h>0 fixed, and the limiting distribution of the supremum on intervals [0,T] as T-->[infinity]. We show further how this limit distribution is connected to the asymptotic of the maximally selected quasi-likelihood procedure that is used to detect changes at an unknown time in polynomial regression models. In an application to global near-surface temperatures, we demonstrate that the limit results presented in this paper perform well for real data sets.

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    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 100 (2009)
    Issue (Month): 5 (May)
    Pages: 1029-1043

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    Handle: RePEc:eee:jmvana:v:100:y:2009:i:5:p:1029-1043
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    1. Donald W.K. Andrews, 1990. "Tests for Parameter Instability and Structural Change with Unknown Change Point," Cowles Foundation Discussion Papers 943, Cowles Foundation for Research in Economics, Yale University.
    2. Shao, Qi-Man, 1993. "Almost sure invariance principles for mixing sequences of random variables," Stochastic Processes and their Applications, Elsevier, vol. 48(2), pages 319-334, November.
    3. Jandhyala, V. K. & MacNeill, I. B., 1989. "Residual partial sum limit process for regression models with applications to detecting parameter changes at unknown times," Stochastic Processes and their Applications, Elsevier, vol. 33(2), pages 309-323, December.
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