Multichannel Sequential Detection—Part I: Non-i.i.d. Data

We consider the problem of sequential signal detection in a multichannel system assuming that the number and location of signals are either unknown or only partially known a priori. We focus on the design and analysis of two sequential hypothesis tests: the generalized sequential likelihood ratio test and the mixture sequential likelihood ratio test. We develop an asymptotic theory for a general stochastic model, where the various data streams can be coupled and correlated, and the data in each stream can be dependent and non-identically distributed. Specifically, we show that the two proposed sequential detection procedures asymptotically minimize the expected sample size and even higher moments of the sample size in the class of hypothesis tests with given probabilities of errors under weak distributional assumptions. We also propose efficient importance sampling algorithms for estimating error probabilities of the sequential tests by Monte Carlo simulation. The general theory is illustrated with several practical examples, such as the detection of signals, in Gaussian hidden Markov models, white Gaussian noises with unknown intensity, and testing of the first-order autoregression's correlation coefficient. Finally, we illustrate our asymptotic results and compare the two proposed procedures with a simulation study.