 Solution

Iterating the CUSUM recursion shows thatThus it searches all possible starting times for a recent stretch of evidence favoring an increase. If adding an increment makes the current sum negative, that accumulated evidence favors the null; retaining it would force a later real deterioration first to repay a deficit accumulated during healthy years. Starting a new candidate segment gives zero instead. Resetting at zero makes detection responsive to a change with unknown onset. Negative values would be permissible for a fixed-start signed log-likelihood ratio, but that is a different statistic from this one-sided tabular CUSUM.