CUSUM

Remarks

Equivalently, $W_n = \max_{1 \le k \le n + 1} \sum_{i=k}^{n} \ell(X_i)$, with the empty sum for $k = n + 1$ equal to $0$. So $W_n$ is the largest log-likelihood ratio of "a change happened at time $k$" against "no change yet", over all candidate times $k$.

Example: for a shift of a normal mean from $\mu_0$ to $\mu_1 > \mu_0$ with variance $\sigma^2$, one finds $\ell(x) = \frac{\mu_1 - \mu_0}{\sigma^2} \left( x - \frac{\mu_0 + \mu_1}{2} \right)$. After rescaling, the statistic becomes $S_n = \max(0, S_{n-1} + X_n - \mu_0 - k)$ with the reference value $k = (\mu_1 - \mu_0)/2$, the form used in quality control.

Non-example: the plain cumulative sum $\sum_{i=1}^n \ell(X_i)$ without the reset at $0$ is not the CUSUM statistic. Before the change it drifts down to $-\infty$, so after a late change it needs a long time to climb back to the threshold.

The mean time to a false alarm satisfies $E_\infty[T_h] \ge e^h$, which guides the choice of $h$.