Extreme value theory for stochastic integrals of Legendre polynomials |
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Authors: | Alexander Aue Lajos Horváth |
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Institution: | a Department of Statistics, University of California, One Shields Avenue, Davis, CA 95616, USA b Department of Mathematics, University of Utah, 155 South 1440 East, Salt Lake City, UT 84112, USA c Department of Statistics, Charles University, Sokolovská 83, CZ-18600 Praha, Czech Republic |
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Abstract: | 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→∞. 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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Keywords: | primary 60G70 secondary 62J02 62J12 |
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