Let the distribution of some random variables depend on real parameters $\theta_1, \cdots, \theta_s$ and consider the hypothesis $H: \theta_i \leqq \theta^\ast_i, i = 1, \cdots, s.$ It is shown under certain regularity assumptions that unbiased tests of $H$ do not exist. Tests of minimum bias and other types of minimax tests are derived under suitable monotonicity conditions. Certain related multidecision problems are discussed and two-sided hypotheses are considered very briefly.
"Testing Multiparameter Hypotheses." Ann. Math. Statist. 23 (4) 541 - 552, December, 1952. https://doi.org/10.1214/aoms/1177729333