Chen and Loh showed that the Box-Cox transformed two-sample $t$-test is more powerful than the ordinary $t$-test under Pitman alternatives where the location shifts appear in the untransformed scale. In this article, we prove that Chen and Loh's result also holds for a general family of transformations. An upper bound on the asymptotic relative efficiency (ARE) is obtained. In addition, we investigate bounds on the ARE under Pitman location shift alternatives in the transformed scale. We find that when the estimate for $\lambda$ is consistent, a lower bound on the ARE is the reciprocal of Fisher information of the standard transformed distribution. This lower bound is close to 1 for commonly used symmetric distributions.
"Tests Following Transformations." Ann. Statist. 23 (5) 1587 - 1593, October, 1995. https://doi.org/10.1214/aos/1176324314