Poisson likelihood-ratio CUSUM ($W_t=Y_t\log \theta -(\theta -1){\lambda }_{0t}$)

For independent Poisson observations, comparing mean ${\lambda }_{0t}$ with $\theta {\lambda }_{0t}$ gives the displayed log-likelihood ratio. Updating a CUSUM with these scores targets the specified multiplicative increase. Its expected score is negative under the null and positive under the alternative when $\theta >1$, so resetting at zero discards earlier evidence against an increase.