# 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

, Volume 40 (3) – Oct 27, 2004
14 pages

/lp/springer_journal/point-asymptotics-for-probabilities-of-large-deviations-of-the-2-B3TkWli6wT
Publisher
Kluwer Academic Publishers-Plenum Publishers
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations