Quickest detection with exponential penalty for delay
研究了在随机序列概率分布发生变化时，如何设计停时以在指数延迟惩罚下平衡检测延迟与虚警，发现经典CUSUM检验在指数惩罚下可能远不如最优检验。
The problem of detecting a change in the probability distribution of a random sequence is considered. Stopping times are derived that optimize the tradeoff between detection delay and false alarms within two criteria. In both cases, the detection delay is penalized exponentially rather than linearly, as has been the case in previous formulations of this problem. The first of these two criteria is to minimize a worst-case measure of the exponential detection delay within a lower-bound constraint on the mean time between false alarms. Expressions for the performance of the optimal detection rule are also developed for this case. It is seen, for example, that the classical Page CUSUM test can be arbitrarily unfavorable relative to the optimal test under exponential delay penalty. The second criterion considered is a Bayesian one, in which the unknown change point is assumed to obey a geometric prior distribution. In this case, the optimal stopping time effects an optimal trade-off between the expected exponential detection delay and the probability of false alarm. Finally, generalizations of these results to problems in which the penalties for delay may be path dependent are also considered.