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A Note on a Theorem of Dynkin on Necessary and Sufficient Statistics

Published online by Cambridge University Press:  20 November 2018

Peter Tan
Peter Tan*
Affiliation:
University of Toronto
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In his paper "Necessary and sufficient statistics for a family of probability distributions", Dynkin (1951) establishes the important concept of rank for such a family with this conclusion: "If the rank is infinite, then the family has no non-trivial sufficient statistic in any size of sample." His concept of rank is based on a theorem, Theorem 2 described below, which has been pointed out by Brown (1964) to be invalid under its hypotheses. This note shows that Dynkin's Theorem 2 remains valid under its original hypotheses provided that the set (in Dynkin's notation) Δ - S is countable.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 3 , June 1969 , pp. 347 - 351
Copyright
Copyright © Canadian Mathematical Society 1969

References

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