Annals of the Institute of Statistical Mathematics

, Volume 34, Issue 1, pp 151–160

# Partitions, sufficiency and undominated families of probability measures

• G. Trenkler
Article

## Summary

This article is concerned with a class of statistical structures which has been introduced by Basu and Ghosh and where the underlying family of probability measures is not dominated. Using the concept of partition-inducible subfields it is shown that the intersection of arbitrarily many subfields is sufficient again. This gives rise to the notion of the coarsest sufficient subfield containing a given family of sets. This generated subfield may be calculated as a function of the minimal sufficient subfield which always exists in these structures. Finally some attention is given to invariance and sufficiency.

## Keywords

Probability Measure Statistical Structure Statistical Decision Underlying Structure Invariance Principle

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## References

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Bahadur, R. R. (1954). Sufficiency and statistical decision functions,Ann. Math. Statist.,25, 423–462.
2. 
Basu, D. (1970). On sufficiency and invariance,Essays in Probability and Statistics, Univ. of North Carolina Press, Chapel Hill, N.C.
3. 
Basu, D. and Ghosh, J. K. (1969). Sufficient statistics in sampling from a finite universe,Proc. 36th Session Internat. Statist. Inst., 850–859.Google Scholar
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Blackwell, D. and Girshick, A. A. (1954).Theory of Games and Statistical Decisions, Wiley, New York.
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Burkholder, D. L. (1961). Sufficiency in the undominated case,Ann. Math. Statist.,32, 1191–1200.
6. 
Hasegawa, M. and Perlman, M. D. (1974). On the existence of a minimal sufficient subfield,Ann. Statist.,2, 1049–1055.
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Morimoto, H. (1972). Statistical structure of the problem of sampling from finite populations,Ann. Math. Statist.,43, 490–497.

© The Institute of Statistical Mathematics, Tokyo 1982

## Authors and Affiliations

• G. Trenkler
• 1
1. 1.University of HannoverHannoverGermany

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