Characterizations of the Existence of UMP Tests for Hypotheses Induced by Groups

Abstract

Let (Ω, A, P _j ), j = 1,2, denote probability spaces, G a group of (A, A)-measurable mappings g:Ω→Ω, and P _j the sets of probability distributions P ^g _j , g∈G, j=1,2. It is shown that the existence of an UMP level a-test for testing P ₁ against P ₂ for any α∈[0,1) is equivalent to one of the following conditions:

• (i) The envelope power function β_α on the convex hull of P ₂ is equal to the constant value of β_α on P ₂ for any α∈[0,1), if P ₁ is dominated by some σ-finite measure.

• (ii) Any optimal level α-test for testing {P ₁|J} against {P ₂|J}, where J denotes the sub-σ-algebra of A consisting of all G-invariant sets, is an UMP level α test for testing P ₁ against P ₂ for any α∈[0,1), if G is a locally compact, σ-compact, (topologically) amenable group acting measurably on Ω and P ₁ is dominated by some σ-finite measure.

• (iii) The P ₁ continuous part P _2P1, of P ₂ is G-invariant, if G is a locally compact, σ-compact, (topologically) amenable group acting measurably on Ω and P ₁ is G-invariant.