Let the distribution of some random variables depend on real parameters θ₁, ⋯, θₛ and consider the hypothesis H: θᵢ θ^ᵢ, i = 1, ⋯, 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.
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E. L. Lehmann (1952) studied this question.