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May 2003 Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks
Hock Peng Chan, Tze Leung Lai
Ann. Appl. Probab. 13(2): 395-429 (May 2003). DOI: 10.1214/aoap/1050689586

Abstract

Saddlepoint approximations are developed for Markov random walks $S_n$ and are used to evaluate the probability that $(j-i) g((S_j - S_i)/(j-i))$ exceeds a threshold value for certain sets of $(i,j)$. The special case $g(x) = x$ reduces to the usual scan statistic in change-point detection problems, and many generalized likelihood ratio detection schemes are also of this form with suitably chosen $g$. We make use of this boundary crossing probability to derive both the asymptotic Gumbel-type distribution of scan statistics and the asymptotic exponential distribution of the waiting time to false alarm in sequential change-point detection. Combining these saddlepoint approximations with truncation arguments and geometric integration theory also yields asymptotic formulas for other nonlinear boundary crossing probabilities of Markov random walks satisfying certain minorization conditions.

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Hock Peng Chan. Tze Leung Lai. "Saddlepoint approximations and nonlinear boundary crossing probabilities of Markov random walks." Ann. Appl. Probab. 13 (2) 395 - 429, May 2003. https://doi.org/10.1214/aoap/1050689586

Information

Published: May 2003
First available in Project Euclid: 18 April 2003

zbMATH: 1029.60058
MathSciNet: MR1970269
Digital Object Identifier: 10.1214/aoap/1050689586

Subjects:
Primary: 60F05 , 60F10 , 60G40
Secondary: 60G60 , 60J05

Keywords: change-point detection , integrals over tubes , Laplace's method , large deviation , Markov additive processes , maxima of random fields

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.13 • No. 2 • May 2003
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