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

TL;DR

This paper derives an exact closed-form formula for the average run length to false alarm of the Generalized Shiryaev-Roberts procedure in detecting mean shifts in exponential observations, enhancing theoretical understanding and practical computation.

Contribution

It provides the first exact analytical formula for the GSR procedure's ARL to false alarm without restrictions on headstart or threshold values.

Findings

Exact ARL formula derived for GSR procedure.

Formula is linear in threshold and headstart, simplifying calculations.

The result advances theoretical understanding of change-point detection performance.

Abstract

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 devised 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 range, nor is there a "ceiling" value for the detection threshold. Apart from the theoretical significance (in change-point detection, exact closed-form performance formulae are typically either difficult or impossible to get, especially for the GSR procedure), the obtained formula 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.