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

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.

LPD Communication: A Sequential Change-Point Detection Perspective

“…Regarding the detection threshold η r , for ease of discussion, we only consider the case η r ≥ 1 q . This is because when η r < 1 q , the SR detector always raises a false alarm before the transmission begins provided ν > − ln(q( 1 q −ηr)) ln(1+q) , which has also been pointed out in [30]. Nevertheless, the method used in this part to evaluate Q SR L (q) can also be extended to the case with η r < 1 q .…”

“…, which has also been pointed out in [30]. Nevertheless, the method used in this part to evaluate Q SR L (q) can also be extended to the case with η r < 1 q .…”

“…where P SR 0 (y|x) is defined in Lemma 2. In fact, (30) provides a recursive integral formula for Q SR n (x; q). Unfortunately, we are unable to obtain explicit expressions for Q SR n (x; q) for n ≥ 2 due to the complicated form of (30).…”

“…In fact, (30) provides a recursive integral formula for Q SR n (x; q). Unfortunately, we are unable to obtain explicit expressions for Q SR n (x; q) for n ≥ 2 due to the complicated form of (30). Nevertheless, we can obtain Q SR n (x; q) numerically by using trapezoidal quadrature rule, see e.g.…”

A note on efficient performance evaluation of the Cumulative Sum chart and the Sequential Probability Ratio Test

“…It is worth recalling that the need to evaluate the performance of the CUSUM chart (or that of the SPRT, or any other control chart for that matter) numerically is dictated by the fact that the corresponding characteristics (e.g., the zerostate ARL, the ASN function, or the OC function) are governed by integral (renewal) equations that seldom allow for an analytical solution; cases where an analytic closed-form solution is possible are offered, for example, in [27][28][29][30][31][32][33] for the CUSUM chart, in [34][35][36][37][38] for the SPRT, in [39][40][41] for the Exponentially Weighted Moving Average chart (introduced by Roberts [17]), and in [21,[42][43][44][45][46][47] and [48,Chapter 4] for the Generalized Shiryaev-Roberts procedure. As control charts' performance evaluation is a persistent problem in applied sequential analysis (notably in quality control), numerical treatment of the corresponding integral equations has de facto become a separate research field, and the literature on the subject is vast indeed.…”