Two statistics for testing that the mean of a normal distribution is zero are compared to show that the measures of asymptotic relative efficiency proposed by Pitman (cf., e.g., ), Hodges and Lehmann , and Bahadur ,  do not always agree, even locally for alternative means near zero.
R. K. Tsutakawa. "An Example of Large Discrepancy Between Measures of Asymptotic Efficiency of Tests." Ann. Math. Statist. 39 (1) 179 - 182, February, 1968. https://doi.org/10.1214/aoms/1177698516