A class of random processes whose likelihood functions are of exponential type is considered. A necessary and sufficient condition for a stopping time to be efficient (in the Cramer-Rao sense) is proved. This result is obtained after proving a characterization theorem, which asserts that after a suitable random-time transformation such processes become processes with stationary independent increments, by applying the solution of the problem of efficient sequential estimation in the case of processes with stationary independent increments.
"Efficient Sequential Estimation in Exponential-Type Processes." Ann. Statist. 14 (4) 1606 - 1611, December, 1986. https://doi.org/10.1214/aos/1176350181