Pairwise sufficiency

  • Eberhard Siebert
Article

Summary

For an arbitrary experiment E we investigate the relation between its pairwise sufficient subalgebras and its sufficient sublattices in the M-space of E (in the sense of L. LeCam). By exhibiting an experiment without minimal pairwise sufficient subalgebra it is shown that this correspondence is in general not bijective. In view of this we introduce the rather large class of majorized experiments. They have a minimal pairwise sufficient subalgebra which can be described explicitely.

As a natural subclass of the majorized experiments appear the coherent experiments that are distinguished by the coincidence of sufficiency and pairwise sufficiency. It is shown that the coherent experiments are characterized by the fact that they admit a majorizing measure which is localizable. As a consequence we obtain that the class of coherent experiments coincides with classes of experiments previously introduced by T.S. Pitcher, D. Mußmann, M. Hasegawa and M.D. Perlman.

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References

  1. 1.
    Bahadur, R.R.: Statistics and Subfields. Ann. Math. Statist. 26, 490–497 (1955)Google Scholar
  2. 2.
    Burkholder, D.L.: Sufficiency in the undominated case. Ann. Math. Statist. 32, 1191–1200 (1961)Google Scholar
  3. 3.
    Halmos, P.R., Savage, L.J.: Applications of the Radon-Nikodym theorem to the theory of sufficient statistics. Ann. Math. Statist. 20, 225–241 (1949)Google Scholar
  4. 4.
    Hasegawa, M., Perlman, M.D.: On the existence of a minimal sufficient subfield. Ann. Statist. 2, 1049–1055 (1974). Correction: Ann. Statist. 3, 1371–1372 (1975)Google Scholar
  5. 5.
    Heyer, H.: Mathematische Theorie statistischer Experimente. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  6. 6.
    LeBihan, M.-F., Littaye-Petit, M., Petit, J.-L.: Exhaustivité par paire. C.R. Acad. Sci. Paris Sér. A270, 1753–1756 (1970)Google Scholar
  7. 7.
    LeCam, L.: Sufficiency and approximate Sufficiency. Ann. Math. Statist. 35, 1419–1455 (1964)Google Scholar
  8. 8.
    Littaye-Petit, M., Piednoir, J.-L., van Cutsem, B.: Exhaustivité. Ann. Inst. H. Poincaré V, 289–322 (1969)Google Scholar
  9. 9.
    Luschgy, H.: Sur l'existence d'une plus petite sous-tribu exhaustive par paire. Preprint 1977Google Scholar
  10. 10.
    Mußmann, D.: Vergleich von Experimenten im schwach dominierten Fall. Z. Wahrscheinlichkeitstheorie verw. Gebiete 24, 295–308 (1972)Google Scholar
  11. 11.
    Pitcher, T.S.: Sets of measures not admitting necessary and sufficient statistics or subfields. Ann. Math. Statist. 28, 267–268 (1957)Google Scholar
  12. 12.
    Pitcher, T.S.: A more general property than domination for sets of probability measures. Pacific J. Math. 15, 597–611 (1965)Google Scholar
  13. 13.
    Schaefer, H.H.: Banach Lattices and Positive Operators. Berlin-Heidelberg-New York: Springer 1974Google Scholar
  14. 14.
    Siebert, E.: Klasseneigenschaften statistischer Experimente und ihre Charakterisierung durch Kegelmaße. Habilitationsschrift, Tübingen 1976Google Scholar
  15. 15.
    Torgersen, E.N.: Comparison of translation experiments. Ann. Math. Statist. 43, 1383–1399 (1972)Google Scholar
  16. 16.
    Zaanen, A.C.: Integration. Amsterdam: North Holland 1967Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • Eberhard Siebert
    • 1
  1. 1.Mathematisches Institut der UniversitätTübingenFederal Republic of Germany

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