A sequential procedure for simultaneous detection and state estimation of Markov signals

The problem of joint detection and state estimation of a Markov signal when a variable number of noisy measurements can be taken is here considered. In particular, the signal-observation sequence {X_i, Z_i}_iisinN is a hidden Markov process (HMP) while, if the signal is absent, the measurement {Z_i}_i2isinN is an i.i.d. process. In this framework, two coupled detection and estimation procedures are introduced for the cases of discrete and continuous state space. Bounds on the performance of the proposed procedures in terms of the thresholds are derived, similar to the classical bounds for the sequential probability ratio test (SPRT). Moreover, it is shown that, under a set of rather mild conditions, the procedures end with probability one and the stopping time is almost surely minimized in the class of tests with the same or smaller error probabilities.