Asymptotics of a boundary crossing probability of a Brownian bridge with general trend

Details

Serval ID
serval:BIB_DA0CEF069158
Type
Article: article from journal or magazin.
Collection
Publications
Title
Asymptotics of a boundary crossing probability of a Brownian bridge with general trend
Journal
Methodology And Computing In Applied Probability
Author(s)
Bischoff W., Miller F., Hashorva E., Hüsler J.
ISSN
1387-5841
Publication state
Published
Issued date
2003
Peer-reviewed
Oui
Volume
5
Number
3
Pages
271-287
Language
english
Abstract
Let us consider a signal-plus-noise model
yh(z) + B-0(z), z epsilon [0, 1],
where gamma > 0, h : [0, 1] --> R, and B-0 is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for gamma --> infinity, that is
P(sup(zepsilon(0,1)) w(z) (gammah(z) + B-0(z)) > c), for gamma --> infinity, (1)
where w : [0, 1] --> [0, infinity) is a weight function and c > 0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H-o : h equivalent to 0 against the altemative K : h > 0 in the signal-plus-noise model.
Keywords
Brownian bridge with trend, Boundary crossing probability, Asymptotic results, Large deviations, Signal-plus-noise model, Tests of Kolmogorov type
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Create date
03/09/2010 11:15
Last modification date
20/08/2019 15:59
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