Optimal online detection of parameter changes in two linear models

Shiryaev has obtained the optimal sequential rule for detecting the instant of a distributional change in an independent sequence using the theory of optimal stopping of Markov processes. This paper considers the problem of sequential detection of certain parameter changes in two dependent sequences: an autoregressive process, and a regression model with serially correlated error terms. It is shown that the rule that is optimal in the sense of minimizing the expected positive delay is the one which declares a change to have occured as soon as the posterior probability of a change crosses a threshold. This rule also permits control of the probability of a false-declaration of change, just as in the independent sequence case.