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!!