The approximate Bahadur efficiency and the Pitman efficiency for hypothesis testing problems are considered. A theorem is stated and proved which gives a condition under which the existence of the limiting (as the alternative approaches the hypothesis) approximate Bahadur efficiency implies the existence of the limiting (as the significance level approaches 0) Pitman efficiency and the equality of the two limits. Several examples are then given to show how the theorem may be used in computing previously unknown limiting Pitman efficiencies using the Bahadur approach.
"A Condition Under Which the Pitman and Bahadur Approaches to Efficiency Coincide." Ann. Statist. 4 (5) 1003 - 1011, September, 1976. https://doi.org/10.1214/aos/1176343600