Sequential change-point detection when unknown parameters are present in the pre-change distribution

The Annals of Statistics · 2006 · Volume 34 · Issue 1

Abstract

In the sequential change-point detection literature, most research specifies a required frequency of false alarms at a given pre-change distribution fθ and tries to minimize the detection delay for every possible post-change distribution gλ. In this paper, motivated by a number of practical examples, we first consider the reverse question by specifying a required detection delay at a given post-change distribution and trying to minimize the frequency of false alarms for every possible pre-change distribution fθ. We present asymptotically optimal procedures for one-parameter exponential families. Next, we develop a general theory for change-point problems when both the pre-change distribution fθ and the post-change distribution gλ involve unknown parameters. We also apply our approach to the special case of detecting shifts in the mean of independent normal observations.

Keywords

62F05, 62L10, 62L15, asymptotic optimality, Change-point, optimizer, Power one tests, quality control, statistical process control, surveillance

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