Takeuchi , Takeuchi and Akahira  and Pfanzagl  among others proved that any first-order efficient estimators are second-order efficient. Many other authors e. g., Ghosh , have conjectured that any third-order efficient estimators are also fourth-order efficient. Based on the concentration probability of estimators about a true parameter, this paper gives a positive answer to the conjecture in a curved exponential family with multi-structural parameters. It is seen that choice of bias-correction factors is critical.
Japan Statistical Society