In this paper we present minimal complete class theorems for testing problems with simple null hypotheses and multivariate one-sided alternative hypotheses. The results extend previous ones by not requiring that the tests be based on exponentially distributed random variables or that the null and alternative parameter spaces be topologically separated.
"Minimal Complete Classes of Tests of Hypotheses with Multivariate One-Sided Alternatives." Ann. Statist. 10 (3) 962 - 970, September, 1982. https://doi.org/10.1214/aos/1176345886