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A fast initial response CUSUM starts a CUSUM with a positive head start below its signalling boundary. Before its first reset, it accumulates a fixed-start log-likelihood ratio like a sequential probability ratio test; after resetting to zero it behaves as an ordinary CUSUM. It gives faster initial detection when the process may already be out of control, at the cost of changed initial false-alarm behavior.

A classical sequential probability ratio test addresses a fixed-start choice between two simple statistical hypotheses. It sums log-likelihood ratio increments until an upper boundary accepts the alternative or a lower boundary accepts the null; it then terminates that test. A CUSUM instead monitors indefinitely for a change at an unknown time. Its reset at zero discards sustained evidence from a previously satisfactory period, so a late change need not first undo all earlier negative scores. An upper boundary $h$ signals when the accumulated change evidence is sufficiently large.

A fast initial response CUSUM takes $0<X_0<h$, often $X_0=h/2$, in place of a zero start. Until its first reset or signal, it acts like a fixed-start sequential probability ratio test with a lower boundary at $-X_0$ and an upper boundary at $h-X_0$ for the subsequent cumulative increments. Upon reaching the lower boundary it resets and continues as an ordinary CUSUM. The head start gives faster detection if a change is already present when monitoring begins or restarts. It is useful after an intervention, a shutdown or a previous alarm when immediate renewed surveillance is desired. Its initial false-alarm behavior differs from the zero-start scheme, so the head start and signalling boundary must be calibrated together; the preceding numerical choice is a design convention, not a universal optimum.

A sequential probability ratio test accumulates the log-likelihood ratio for two specified simple statistical hypotheses until it reaches an upper or lower boundary. An upper crossing favors the alternative, a lower crossing favors the null, and observations continue between them. Error-calibrated boundaries are approximately $\log [(1-\beta )/\alpha ]$ and $\log [\beta /(1-\alpha )]$; discrete overshoot means these approximations need calibration.