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I am investigating a link between a random walk with drift (call it Brownian process or difusion with drift) and the CUSUM statistic.
The CUSUM procedure accumulates deviations from the process mean over time, thus, if a change in the mean occurs for some reason, then the CUSUM will steadely increase over time, eventually crossing some pre-determined control limit when an alarm is raised.

Can anybody enlighten me as to whether this is in any way similar to calculating the first passage time for the CUSUM, that is the time it takes to cross a given barrier. Is the CUSUM ARL the inverse of the probability of crossing this barrier? How do I go about calculating this probability?

...so many questions!! Any thoughts are appreciated!!

mili
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  • Could you elaborate on the threshold you are using, and the change in drift you expect? – Douglas Zare Jul 23 '12 at 04:53
  • @MichaelChernick to make thgings simple I fix the threshold as a constant positive value, above the initial CUSUM sequence (so the CUSUM will increase in response to a small change in the mean of a process, and eventually cross the threshold). The drif is also fixed and in this case it has a negative value to be able to model the increasing CUSUM. – mili Jul 23 '12 at 13:26
  • Many of these questions have subsequently been answered in the duplicate thread. I am closing this one, despite its historical precedence, because it is broad and (perhaps therefore) it hasn't attracted useful answers. – whuber Jun 10 '15 at 15:47

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Brownian motion with drift is a linear function with Gaussian noise added to it. Generally speaking CUSUM charts are used to detect a process going out of control. So what is assumed is that the process starts out under control and something could go wrong. This would show up in a change in the mean of some measure of the process (e.g. rate of production of defective parts). Such changes are often started at some point in time with alevel shift or a shift occuring over a short period of time. The CUSUM chart crossing a control boundary is a sequential hypothesis testing problem where crossing rejects the hypothesis that the mean has not changed. There are many conditions that could indicate a change in mean. Browian motion with drift is one of several possible ways that a process could go out of control. So the CUSUM deals more generally with a process going out of control and not just Brownian motion with drift.

Michael R. Chernick
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  • Thank you. I agree with all your statements. In short, my problem is: Can a CUSUM be modelled as a Brownian with drift, and by calculating the hitting times, we have the equivalent to alarms? – mili Jul 22 '12 at 18:55
  • From my interpretation of the answer: Yes you can, but CUSUM model can model some more general situations as well. – bdeonovic Dec 17 '13 at 17:10