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

  • D. Plachky
Conference paper

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 j g , gG, 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:
  1. (i)

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

     
  2. (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.

     
  3. (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.

     

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References

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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • D. Plachky
    • 1
  1. 1.Institute for Mathematical StatisticsUniversity of MünsterMünsterGermany

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