An e-detector for a pre-change class P is a nonnegative process M such that EP[Mτ]≤EP[τ]
Nearby in the stack
for all stopping times
τ
and all
P∈P
. Thresholding e-detectors controls the average run length (ARL): declaring a change at the first time
Tb
when
M
crosses
b
ensures that
infP∈PEP[T]≥b
. But e-detectors do substantially more than control the ARL; they also satisfy a optional-horizon inequality:
P(Tb≤σ)≤EP[σ]/b
for every data-dependent stopping time (monitoring horizon)
σ
and
P∈P
. In particular, every e-detector-based procedure obeys
P(T≤t)≤t/b
at each fixed
t
, thus avoiding early false alarms. Remarkably, the converse also holds: every stopping time
T
that satisfies the optional-horizon inequality must in fact arise from thresholding an e-detector. We also derive a universal representation of stopping times that satisfy (only) ARL control. These are represented by weak e-detectors, that only require
EP[Mτ]≤EP[τ]
to hold at all threshold stopping times
Tb
. Appendices present universal representations for other (less common) change detection metrics.