Point Asymptotics for Probabilities of Large Deviations of the $\omega^2$ Statistics in Verification of the Symmetry Hypothesis

We consider the $\omega^2$ statistic, destined for testing the symmetry hypothesis, which has the form
$$ \omega^2_n=n\int\limits_{-\infty}^\infty[F_n(x)+F_n(-x)-1]^2\,dF_n(x), $$
where $F_n(x)$ is the empirical distribution function. Based on the Laplace method for empirical measures, exact asymptotic (as $n\to\infty$) of the probability
$$ \mathrm{P}\{\omega_n^2>nv\} $$
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.