Asymptotic moments of normed sample size savings are derived for one-sided curtailed nonparametric tests for symmetry for testing location shift hypotheses under the null hypothesis and contiguous alternatives. The contiguous alternative results are derived using Feller's (1943) central limit theorem result for large deviations and techniques used in Albers, Bickel and van Zwet (1976).
"Sample Size Savings for Curtailed One-Sample Nonparametric Tests for Location Shift." Ann. Statist. 15 (1) 296 - 313, March, 1987. https://doi.org/10.1214/aos/1176350267