Problems of Information Transmission, Vol. 40, No. 3, 2004, pp. 212–225. Translated from Problemy Peredachi Informatsii, No. 3, 2004, pp. 33–48.
Original Russian Text Copyright
2004 by Fatalov.
METHODS OF SIGNAL PROCESSING
Point Asymptotics for Probabilities of Large
Deviations of the ω
Statistics in Veriﬁcation of the
V. R. Fatalov
M.V. Lomonosov Moscow State University
Received January 10, 2002; in ﬁnal form, March 4, 2004
Abstract—We consider the ω
statistic, destined for testing the symmetry hypothesis, which
has the form
(−x) − 1]
(x) is the empirical distribution function. Based on the Laplace method for empirical
measures, exact asymptotic (as n →∞) of the probability
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.
1. INTRODUCTION AND FORMULATION OF THE MAIN RESULT
The Cram´er–von Mises–Smirnov ω
statistic is widely used for testing various statistical hy-
potheses [1–7]. This is accounted for both a simple representation of the statistic and a convenient
form of its limiting distributions. These limiting distributions are distributions of integral func-
tionals of Gaussian processes and are rather diﬃcult to compute precisely. Yet more sophisticated
approaches are required in problems where one should take into account the dependence of the
distribution of the ω
statistic on n.
In the present paper, we apply the general method developed in [8–10] for the computation of
the exact asymptotic of large deviation probabilities of the ω
statistic in testing the symmetry
hypothesis. We show that the corresponding functional and the family of empirical distributions
satisfy the conditions of the main general theorem [9, Theorem 1], which involves the solution of
the extreme value problem for the rate function in a Banach space, and this requires individual
analysis in each particular case.
Note also that, to ﬁnd the exact (but not logarithmic) asymptotic, one has to assume that the
original functional possesses a stronger property, namely, is twice Frechet diﬀerentiable.
be a sample obtained from a distribution with an unknown continuous distribution
function F (x). Consider testing the hypothesis of the symmetry of F about zero:
: F (−x)=1− F (x), for all x ∈ R
Supported in part by the Russian Foundation for Basic Research, project no. 01-01-00649.
2004 MAIK “Nauka/Interperiodica”