In this paper we give necessary conditions and sufficient conditions for a test to be locally unbiased, we define local admissibility and we characterize local admissibility in hypothesis testing problems with simple null hypotheses. Applications are presented involving same-sign alternatives, ordered alternatives and independence testing of several variables.
"Local Admissibility and Local Unbiasedness in Hypothesis Testing Problems." Ann. Statist. 20 (2) 832 - 852, June, 1992. https://doi.org/10.1214/aos/1176348659