Consider the 2-sample problem where the null cdf $F(x)$ satisfies $F(0) = 0$ and the alternative is $F_\theta(x) = F(x/(1 + \theta))$ with $\theta > 0$. An asymptotically optimum statistic $z$ is obtained for a parametric model where $F(x)$ is a gamma distribution. The Mann-Whitney $U$ and Savage $T$ statistics are compared to $z$ for several null densities. It is shown that the Pitman asymptotic relative efficiency, ARE $(U/z)$, can approach zero if $\mu/\sigma\rightarrow 0$, where $\mu$ is the mean and $\sigma^2$ the variance of the null distribution. However, a lower bond on ARE $(U/z)$ is obtained as a function of $\mu/\sigma$ for general $F(x)$. Using the bound a composite test is constructed which has a specified minimum ARE of any desired value between 0 and .864. Densities exist for the composite test which result in arbitrarily large values of efficiency.
Melvin N. Woinsky. "A Composite Nonparametric Test for a Scale Slippage Alternative." Ann. Math. Statist. 43 (1) 65 - 73, February, 1972. https://doi.org/10.1214/aoms/1177692701