Mathematics – Statistics Theory
Scientific paper
2006-05-12
Annals of Statistics 2006, Vol. 34, No. 1, 92-122
Mathematics
Statistics Theory
Published at http://dx.doi.org/10.1214/009053605000000859 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst
Scientific paper
10.1214/009053605000000859
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 $g_{\lambda}$. 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 $f_{\theta}$. 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.
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