  The asymptotic efficiency of a generalized likelihood ratio test proposed by
Cox is studied under the large deviations framework for error probabilities
developed by Chernoff. In particular, two separate parametric families of
hypotheses are considered [Cox, 1961, 1962]. The significance level is set such
that the maximal type I and type II error probabilities for the generalized
likelihood ratio test decay exponentially fast with the same rate. We derive
the analytic form of such a rate that is also known as the Chernoff index
[Chernoff, 1952], a relative efficiency measure when there is no preference
between the null and the alternative hypotheses. We further extend the analysis
to approximate error probabilities when the two families are not completely
separated. Discussions are provided concerning the implications of the present
result on model selection.
