Point Asymptotics for Probabilities of Large Deviations of the ω2 Statistics in Verification of the Symmetry Hypothesis

Point Asymptotics for Probabilities of Large Deviations of the ω2 Statistics in Verification of... We consider the ω2 statistic, destined for testing the symmetry hypothesis, which has the form $$\omega _n^{\text{2}} = n\;\int\limits_{ - \infty }^\infty {[F_n (x)\; + F_n ( - x)\; - 1]^2 dF_n (x),}$$ where F n (x) is the empirical distribution function. Based on the Laplace method for empirical measures, exact asymptotic (as n → ∞) of the probability $$P\{ \omega _n^2 > nv\} $$ for 0 < v < 1/3 is found. Constants entering the formula for the exact asymptotic are computed by solving the extreme value problem for the rate function and analyzing the spectrum of the second-order differential equation of the Sturm–Liouville type. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

Point Asymptotics for Probabilities of Large Deviations of the ω2 Statistics in Verification of the Symmetry Hypothesis

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Publisher
Kluwer Academic Publishers-Plenum Publishers
Copyright
Copyright © 2004 by MAIK “Nauka/Interperiodica”
Subject
Engineering; Communications Engineering, Networks; Electrical Engineering; Information Storage and Retrieval; Systems Theory, Control
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1023/B:PRIT.0000044257.66680.e7
Publisher site
See Article on Publisher Site

Abstract

We consider the ω2 statistic, destined for testing the symmetry hypothesis, which has the form $$\omega _n^{\text{2}} = n\;\int\limits_{ - \infty }^\infty {[F_n (x)\; + F_n ( - x)\; - 1]^2 dF_n (x),}$$ where F n (x) is the empirical distribution function. Based on the Laplace method for empirical measures, exact asymptotic (as n → ∞) of the probability $$P\{ \omega _n^2 > nv\} $$ for 0 < v < 1/3 is found. Constants entering the formula for the exact asymptotic are computed by solving the extreme value problem for the rate function and analyzing the spectrum of the second-order differential equation of the Sturm–Liouville type.

Journal

Problems of Information TransmissionSpringer Journals

Published: Oct 27, 2004

References

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