  Assuming that data are collected sequentially from independent streams, we
consider the simultaneous testing of multiple binary hypotheses under two
general setups; when the number of signals (correct alternatives) is known in
advance, and when we only have a lower and an upper bound for it. In each of
these setups, we propose feasible procedures that control, without any
distributional assumptions, the familywise error probabilities of both type I
and type II below given, user-specified levels. Then, in the case of i.i.d.
observations in each stream, we show that the proposed procedures achieve the
optimal expected sample size, under every possible signal configuration,
asymptotically as the two error probabilities vanish at arbitrary rates. A
simulation study is presented in a completely symmetric case and supports
insights obtained from our asymptotic results, such as the fact that knowledge
of the exact number of signals roughly halves the expected number of
observations compared to the case of no prior information.
