An Exact Formula for the Average Run Length to False Alarm of the Generalized Shiryaev--Roberts Procedure under Exponential Observations

We derive analytically an exact closed-form formula for the standard minimax Average Run Length (ARL) to false alarm delivered by the Generalized Shiryaev--Roberts (GSR) change-point detection procedure when set up to detect a shift in the baseline mean of a sequence of independent exponentially distributed observations. Specifically, the formula is found through direct solution of the respective integral (renewal) equation, and is a general result in that the GSR procedure's headstart is not restricted to a bounded interval, nor is there a "ceiling" for the detection threshold. Apart from the theoretical significance (exact closed-form performance formulae are difficult to get in change-point detection, especially for the GSR procedure), the obtained expression is also useful to a practitioner: in cases of practical interest, the formula is a function linear in both the detection threshold and the headstart, and, therefore, the ARL to false alarm of the GSR procedure can be easily computed by hand.