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Sequential change-point detection when unknown parameters are present in the pre-change distribution. (English) Zbl 1091.62064

Summary: In the sequential change-point detection literature, most research specifies a required frequency of false alarms at a given pre-change distribution \(f_\theta\) and tries to minimize the detection delay for every possible post-change distribution \(g_\lambda\). 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 \(f_\theta\). 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 \(f_\theta\) and the post-change distribution \(g_\lambda\) involve unknown parameters. We also apply our approach to the special case of detecting shifts in the mean of independent normal observations.

MSC:

62L10 Sequential statistical analysis
62L15 Optimal stopping in statistics
62F05 Asymptotic properties of parametric tests
62P30 Applications of statistics in engineering and industry; control charts
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