Second-order optimal sequential tests

An asymptotic lower bound is derived involving a second additive term of order root\lnalpha\ as alpha --> 0 for the mean length of a controlled sequential strategy 3 for discrimination between two statistical models in a very general nonparametric setting. The parameter a is the maximal error probability of s. A sequential strategy is constructed attaining (or almost attaining) this asymptotic bound uniformly over the distributions of models including those from the indifference zone. These results are extended for a general loss function g(N) with the power growth of the strategy length N. Applications of these results to change-point detection and testing homogeneity are outlined.