Multihypothesis Invariant Sequential Tests with Applications to the Quickest Detection of Signals in Multiple-Resolution-Element Systems

The sequential testing of many hypotheses is important for a variety of applications including multipleresolution-element radar and infrared systems, signal acquisition, clinical trials, etc. In this paper, we consider the behavior of a sequential test for many composite hypotheses assuming that the exact forms of distributions of observations under each hypothesis are not known (parametric or nonparametric uncertainty). The test is a multiple version of the invariant sequential probability ratio test (a specific combination of onesided invariant SPRT’s) and is referred to as MISPRT. First we show that, under general conditions, the proposed MISPRT is asymptotically optimal relative to any positive moment of the stopping time distribution when error probabilities or, more generally, risks associated with wrong decisions are small. Then these results are applied to two statistical models which frequently arise in the signal detection theory and communications. The first model is related to the detection of a signal in a multi-channel system in the presence of Gaussian noise and clutter with unknown mean and variance. The second example is nonparametric detection of a signal in the clutter with unknown continuous distribution.