Asymptotic minimax robust and misspecified Lorden quickest change detection for dependent stochastic processes

The quick detection of an abrupt unknown change in the conditional distribution of a dependent stochastic process has numerous applications. In this paper, we pose a minimax robust quickest change detection problem for cases where there is uncertainty about the post-change conditional distribution. Our minimax robust formulation is based on the popular Lorden criteria of optimal quickest change detection. Under a condition on the set of possible post-change distributions, we show that the widely known cumulative sum (CUSUM) rule is asymptotically minimax robust under our Lorden minimax robust formulation as a false alarm constraint becomes more strict. We also establish general asymptotic bounds on the detection delay of misspecified CUSUM rules (i.e. CUSUM rules that are designed with post-change distributions that differ from those of the observed sequence). We exploit these bounds to compare the delay performance of asymptotically minimax robust, asymptotically optimal, and other misspecified CUSUM rules. In simulation examples, we illustrate that asymptotically minimax robust CUSUM rules can provide better detection delay performance at greatly reduced computation effort compared to competing generalised likelihood ratio procedures.