Detecting transients of unknown length

For detection of a permanent and precisely-modeled change in distribution of iid (independent and identically distributed) observations, Page\´s test is optimal. When employed to detect a transient change in distribution, Page\´s test is easy to implement and, since it is a GLRT, it provides reliable performance. However, its application to unknown transient changes is less clear. The situation of interest in this manuscript is of a transient signal of unknown scale parameter: a fixed Page procedure tuned to a "short-and-loud" signal uses heavy biasing and low threshold, a combination ill-suited to a "long-but-quiet" signal. In this paper we offer an easy adaptive alternative to the standard Page: a Page test can be thought of as a repeated sequential test, and here we propose that each of this use a time-varying threshold - specifically one that increases. The idea is that the above short signals are detected quickly before post-termination data has a chance to refute them; and that evidence for a long signal is allowed to build, rather than being summarily discarded too early. The approach works remarkably well: its performance essentially traces the "envelope" of fixed/tuned Page tests