  We consider the problem of sequentially testing a simple null hypothesis
versus a composite alternative hypothesis that consists of a finite set of
densities. We study sequential tests that are based on thresholding of
mixture-based likelihood ratio statistics and weighted generalized likelihood
ratio statistics. It is shown that both sequential tests have several
asymptotic optimality properties as error probabilities go to zero. First, for
any weights, they minimize the expected sample size within a constant term
under every scenario in the alternative hypothesis and at least to first order
under the null hypothesis. Second, for appropriate weights that are specified
up to a prior distribution, they minimize within an asymptotically negligible
term a weighted expected sample size in the alternative hypothesis. Third, for
a particular prior distribution, they are almost minimax with respect to the
expected Kullback-Leibler divergence until stopping. Furthermore, based on
high-order asymptotic expansions for the operating characteristics, we propose
prior distributions that lead to a robust behavior. Finally, based on
asymptotic analysis as well as on simulation experiments, we argue that both
tests have the same performance when they are designed with the same weights.
