Motivated by the first-order Pitman closeness of best asymptotically normal estimators and some recent developments on higher-order asymptotic efficiency of estimators, a second-order asymptotic theory is developed for comparison of estimators under the Pitman closeness criterion, covering both the cases without and with nuisance parameters. The notion of second-order Pitman admissibility is also developed.
"Second-Order Pitman Closeness and Pitman Admissibility." Ann. Statist. 22 (3) 1133 - 1141, September, 1994. https://doi.org/10.1214/aos/1176325621