Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 40 / 3 / i

Use the binomial distribution for independent outcomes in each nonoverlapping batch, and let the CUSUM accumulate the log-likelihood ratio favoring a specified change. The two numbered calculations below treat the batch score and the initial-response design separately.

Let $p_1$ be the alternative death probability. The odds ratio specifiesAssume $0<p_0<1$ and $k>0$. For observed batch count $y_t$, the common combinatorial factor in the two binomial likelihoods cancels, giving the CUSUM incrementThereforeFor $k>1$ large death counts add positive evidence of deterioration; for $k<1$ unusually small death counts add evidence for improvement. The score has nonpositive expectation under the null and nonnegative expectation under the specified alternative, as follows from the nonnegativity of Kullback-Leibler divergence. Independent, nonoverlapping batches avoid counting the same patient's outcome repeatedly.

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.