The notion of Bahadur efficiency for test statistics is extended to the sequential case and illustrated in the specific context of testing one-sided hypotheses about a normal mean. An analog of Bahadur's theorem on the asymptotic optimality of the likelihood ratio statistic is seen to hold in the normal case. Some possible definitions of attained level for a sequential experiment are considered.
"Sequential Bahadur Efficiency." Ann. Statist. 6 (3) 567 - 581, May, 1978. https://doi.org/10.1214/aos/1176344201