Determination of uniformly most powerful tests in discrete sample spaces

Abstract.

Uniformly most powerful (UMP) tests are known to exist in one-parameter exponential families when the hypothesis H ₀ and the alternative hypothesis H ₁ are given by

(i) H ₀ : θ≤θ₀, H ₁ : θ>θ₀, and

(ii) H ₀ : θ≤θ₁ or θ≥θ₂, H ₁ : θ₁<θ<θ₂, where θ₁<θ₂.

 Likewise, uniformly most powerful unbiased (UMPU) tests do exist when the hypotheses H ₀ and H ₁ take the form

(iii) H ₀ : θ₁≤θ≤θ₂, H ₁ : θ<θ₁ or θ>θ₂, where θ₁<θ₂, and

(iv) H ₀ : θ=θ₀, H ₁: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 c ₁, c ₂ and two randomization 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 straightforward 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).