  The sequential multiple testing problem is considered under two generalized
error metrics. Under the first one, the probability of at least $k$ mistakes,
of any kind, is controlled. Under the second, the probabilities of at least
$k_1$ false positives and at least $k_2$ false negatives are simultaneously
controlled. For each formulation, the optimal expected sample size is
characterized, to a first-order asymptotic approximation as the error
probabilities go to 0, and a novel multiple testing procedure is proposed and
shown to be asymptotically efficient under every signal configuration. These
results are established when the data streams for the various hypotheses are
independent and each local log-likelihood ratio statistic satisfies a certain
Strong Law of Large Numbers. In the special case of i.i.d. observations in each
stream, the gains of the proposed sequential procedures over fixed-sample size
schemes are quantified.
