Operations Research ’92 pp 379-383 | Cite as

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

## Abstract

*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 },

*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.

## Preview

Unable to display preview. Download preview PDF.

## References

- Ajne, B. (1968): A simple test for uniformity of a circular distribution. Biometrika. 55: 343–354CrossRefGoogle Scholar
- Beran, R. J. W. (1968): Testing for uniformity on a compact homogeneous space. J. Appl. Prob. 5: 177–195CrossRefGoogle Scholar
- Dunford, N. and T. Schwartz (1964): Linear Operators, Part I. Interscience, New YorkGoogle Scholar
- Ng, V. M. (1976): A note on the best test for discrimination between exponentiality and uniformity. Technometrics 18: 237–238CrossRefGoogle Scholar
- Paterson, A. L. (1988): Amenability. Mathematical Surveys and Monographs 29. Providence, Rhode IslandGoogle Scholar
- Uthoff, V. A. (1970): An optimum test property of two well-known statistics. J. Amer. Statist. Ass. 68: 1597–1600Google Scholar
- Watson, G. S. (1976): Optimal invariant tests for uniformity. In: Studies in Probability and Statistics, ed. by E. J. Williams, New YorkGoogle Scholar