Brown, Cohen and Strawderman propose curtailed procedures for the $t$-test and Hotelling's $T^2$. In this paper we present the exact forms of these procedures and examine the expected sample size savings under the null hypothesis. The sample size savings can be bounded by a constant which is independent of the sample size. Tables are given for the expected sample size savings and maximum sample size saving under the null hypothesis for a range of significance levels $(\alpha)$, dimensions $(p)$ and sample sizes $(n)$.
"Expected Sample Size Savings from Curtailed Procedures for the $t$-Test and Hotelling's $T^2$." Ann. Statist. 8 (3) 682 - 686, May, 1980. https://doi.org/10.1214/aos/1176345019