Distribution optimality and second-order efficiency of test procedures

It has been shown, under certain conditions, by Bahadur, Chandra, and Lambert (1982) that in the non-null case the best possible asymptotic distribution for the level attained by a test statistic is a certain lognormal distribution, and that the level of the likelihood ratio statistic has this optimal asymptotic distribution. We describe a technical generalization of this theory; in the present generalization the best possible asymptotic distribution of the standardized log-level is that of the maximum of a family of normally distributed variables. It is pointed out that these considerations yield a corresponding generalization concerning the asymptotic expansion of the log-size of the best critical region when the power against a given alternative is a specified constant