A note on optimal detection of a change in distribution

Suppose X-1, X-2,..., Xv-1 are lid random variables with distribution F-0, and X-v, Xv+1,... are are lid with distributed F-1. The change point v is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from F-0 to F-1 (detect the change), but to avoid false alarms. Pollak found a version of the Shiryayev-Roberts procedure to be asymptotically optimal for the problem of minimizing the average run length to detection over all stopping times which satisfy a given constraint on the rate of false alarms. Here we find that this procedure is strictly optimal for a slight reformulation of the problem he considered. Explicit formulas are developed for the calculation of the average run length (both before and after the change) for the optimal stopping time.