Limiting values of large deviation probabilities of quadratic statistics (Q750050)

If T(P) is a functional and \(T_ n=T(\hat P_ n)\) is used as a test statistic then the application of Bahadur efficiency leads to the expression \[ K(\Omega_{\epsilon},P)=\inf \{K(Q,P):\;T(Q)-T(P)>\epsilon \}, \] where K is the Kullback-Leibler information number. But \(K(\Omega_{\epsilon},P)\) is often complicate. Therefore the authors study the behaviour of \(K(\Omega_{\epsilon},P)/\epsilon\) as \(\epsilon\downarrow 0\) for quadratic functionals \(\iint \psi (s,t)dQ(s)dQ(t).\) One typical result is Theorem 3.3, which asserts that \[ \lim_{\epsilon \downarrow 0}K(\Omega_{\epsilon},P)/\epsilon =(2\lambda_ 1)^{-1}, \] where \(\lambda_ 1\) is the largest eigenvalue of the integral operator defined by the kernel \(\psi\). Applications of the results concern the generalized Cramér-von Mises statistic, Neyman's smooth test and likelihood ratio tests.