The structure of a CUSUM procedure on an ordinary Poisson jump process is analyzed in terms of stopping rules based on linear boundaries. The total run length is composed of a random number of renewal phases followed by a terminal phase. The exact distributions of the length of these phases are derived, as well as the Wald type approximations. The distribution of the total run length and its moments is given in terms of the phase distributions.
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S. Zacks (2004) studied this question.
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