Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T05:48:10.610Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Brown, L., (1964), Sufficient statistics in the case of independent random variables. Annals Math. Statist. 35, 1456–1474. Daniel Dumas De Rauly, (1966), L'Estimation Statistique. (Gauthier-Villars, Paris).Google Scholar
Dynkin, E.B., (1951), Necessary and sufficient statistics for a family of probability distributions. Selected Transi. Math. Statist, and Prob. 1, 17–40.Google Scholar
Fraser, D.A.S., (1963), On sufficiency and exponential family. J. Roy. Stat. Soc. B, 25, 115–123.Google Scholar
Fraser, D.A.S., (1966), Sufficiency for regular models. Sankhya A, 28, 137–144.Google Scholar
Linnik, Ju. V., (1968), Statistical problems with nuisance parameters. (Amer. Math. Soc., Providence, R.I.)Google Scholar
Monroe, M. E., (1953), Introduction to measure and integration. (Addison-Wesley, Reading, Massachusetts).Google Scholar
Tan, P. C., (1968), R-regular statistical models, sufficiency and conditional sufficiency. (Ph.D. thesis, University of Toronto).Google Scholar
Zhuravlev, O.G., (1966), Minimal sufficient statistics for a sequence of independent random variables. Theor. Probability Appl. XI, 282–291.Google Scholar