Uniformly most powerful (UMP) tests are known to exist in one-parameter exponential families when the hypothesis H0 and the alternative hypothesis H1 are given by (i) H0 : θ ≤ θ0, H1 : θ > θ0, and (ii) H0 : θ ≤ θ1 or θ ≥ θ2, H1 : θ1 < θ < θ2, where θ1 < θ2. Likewise, uniformly most powerful unbiased (UMPU) tests do exist when the hypotheses H0 and H1 take the form (iii) H0 : θ1 ≤ θ ≤ θ2, H1 : θ < θ1 or θ > θ2, where θ1 < θ2, and (iv) H0 : θ = θ0, H1 : θ ≠ θ0. To determine tests in case (i), only one critical value c and one randomization constant γ have to be computed. In cases (ii) through (iv) tests are determined by two critical values c1, c2 and two randomized constants γ1, γ2. Unlike determination of tests in case (i), computation of critical values and randomization constants in the remaining cases is rather difficult, unless distributions are symmetric. No straigthforward method to determine two-sided UMP tests in discrete sample spaces seems to be known. The purpose of this note is to disclose a distribution independent principle for the determination of UMP tests in cases (ii) through (iv).
