Let $X_1, X_2, \cdots$ be a sequence of random variables, $(X_1, \cdots, X_n) \sim F^n_\theta, \theta \in \Theta$. In a work by Bahadur it was shown that, for some sequential problems, an inference may be based on a sequence of sufficient and transitive statistics $S_n = S_n(X_1, \cdots, X_n)$ without any loss in statistical performance. A simple criterion for transitivity is given in Theorem 1.
"A Note on Bahadur's Transitivity." Ann. Statist. 21 (4) 2163 - 2167, December, 1993. https://doi.org/10.1214/aos/1176349417