This work considers the problem of detecting signals from multiple sequentially observed data streams, where only one stream can be observed at every time instant. The goal is to detect signals as quickly as possible while controlling the global probabilities of false alarm and missed detection. In this active sampling setup, it is impossible to minimize the expected detection time simultaneously for every signal, so we formulate a novel set of performance criteria that aim to minimize the expectations of the order statistics of the detection times. A novel procedure is proposed, which incorporates an exploration mechanism to a "follow-the-leader" procedure, and is shown to optimize all the criteria asymptotically as the global error probabilities go to zero. Its finite-sample performance is compared with existing and oracle procedures in simulation studies.

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