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2017 | OriginalPaper | Chapter

2. Sufficiency

Author : Johann Pfanzagl

Published in: Mathematical Statistics

Publisher: Springer Berlin Heidelberg

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Abstract

This essay describes the historical development of the concepts of exhaustive statistics and of sufficiency of statistics and sub-\(\sigma \)-fields. We discuss the relations between these concepts, Neyman’s factorization theorem, completeness and bounded completeness of families of distributions, completeness of exponential families, minimal sufficiency, trivial sufficiency, and different characterizations of sufficiency: decision theoretic, by power of tests, by concentration of unbiased estimators, and by measures of information.

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Footnotes
1
It is interesting to observe how the authorship of a nontrivial example was lost as time went on. Dieudonné’s example appears in various textbooks as an exercise with reference to the author (e.g. Doob 1953, p. 624; Halmos 1950, p. 210, Exercise 4 and p. 292, Reference to Sect. 48; Dudley 1989, p. 275, Problem 6 and Note Sect. 10.2, p. 298). It also appears in Ash 1972, p. 267, Problem 4, without reference to Dieudonné, and in Romano and Siegel (1986), pp. 138/9, Example 6.13, as an “Example given in Ash”. In Lehmann and Casella (1998, p. 35) it was presented as an “Example due to Ash, presented by Romano and Siegel (1986)”.
 
Literature
Ash, R. B. (1972). Real Analysis and Probability. New York: Academic Press.
Bahadur, R. R. (1955a). A characterization of sufficiency. The Annals of Mathematical Statistics, 26, 289–293.MATHMathSciNet
Bahadur, R. R. (1957). On unbiased estimates of uniformly minimum variance. Sankhyā, 18, 211–224.MATHMathSciNet
Bahadur, R. R., & Lehmann, E. L. (1955). Two comments on "sufficiency and statistical decision functions". The Annals of Mathematical Statistics, 26, 139–142.CrossRefMATHMathSciNet
Barankin, E. W. (1960). Sufficient parameters: solution of the minimal dimensionality problem. Annals of the Institute of Statistical Mathematics, 12, 91–118.
Barankin, E. W. (1961). A note on functional minimality of sufficient statistics. Sankhyā Series A, 23, 401–404.MATHMathSciNet
Barankin, E. W., & Katz, M, Jr. (1959). Sufficient statistics of minimal dimension. Sankhyā, 21, 217–246.
Barankin, E. W., & Maitra, A. P. (1963). Generalization of the Fisher-Darmois-Koopman-Pitman theorem on suficient statistics. Sankhyā Series A, 25, 217–244.MATH
Barndorff-Nielsen, O., & Pedersen, K. (1969). Sufficient data reduction and exponential families. Mathematica Scandinavica, 22, 197–202.CrossRefMATHMathSciNet
Bartlett, M. S. (1937). Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London. Series A, 160, 268–282.
Blackwell, D. (1951). Comparison of experiments. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (pp. 93–102). University of California Press.
Blackwell, D., & Dubins, L. E. (1975). On existence and non-existence of proper, regular, conditional distributions. Annals of Probability, 3, 741–752.CrossRefMATHMathSciNet
Blackwell, D., & Ryll-Nardzewski, C. (1963). Non-existence of everywhere proper conditional distributions. The Annals of Mathematical Statistics, 34, 223–225.CrossRefMATHMathSciNet
Bomze, I. M. (1990). A functional analytic approach to statistical experiments (Vol. 237). Pitman research notes in mathematics series. Harlow: Longman.
Brown, L. D. (1964). Sufficient statistics in the case of independent random variables. The Annals of Mathematical Statistics, 35, 1456–1474.CrossRefMATHMathSciNet
Csiszár, I. (1963). Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoff’schen Ketten. Magyar Tud. Akad. Mat. Kutató Int. Közl., 8, 85–108.MATH
Denny, J. L. (1970). Cauchy’s equation and sufficient statistics on arcwise connected spaces. The Annals of Mathematical Statistics, 41, 401–411.CrossRefMATHMathSciNet
Dieudonné, J. (1948). Sur le théorème de Lebesgue-Nikodym (III). Annales de l’Institut Fourier (Grenoble), 23, 25–53.MATH
Doob, J. L. (1938). Stochastic processes with an integral-valued parameter. Transactions of the American Mathematical Society, 44, 87–150.MATHMathSciNet
Dudley, R. M. (1989). Real analysis and probability. Pacific Grove: Wadsworth & Brooks/Cole.MATH
Dudley, R. M. (1999). Uniform central limit theorems (Vol. 63). Cambridge studies in advanced mathematics. Cambridge: Cambridge University Press.
Dynkin, E. B. (1951). Necessary and sufficient statistics for a family of probability distributions. Uspeh. Matem. Nauk, 6 (Russian); Selected Translations in Mathematical Statistics and Probability, 1, 23–41 (1961).
Fisher, R. A. (1920). A mathematical examination of the methods of determining the accuracy of an observation by the mean error, and by the mean square error. Monthly Notices of the Royal Astronomical Society, 80, 758–770.
Fisher, R. A. (1922). On the mathematical foundation of theoretical statistics. Philosophical Transactions of the Royal Society of London. Series A, 222, 309–368.
Fisher, R. A. (1934). Two new properties of mathematical likelihood. Proceedings of the Royal Society of London. Series A, 144, 285–307.
Hald, A. (1998). A history of mathematical statistics from 1750 to 1930. New York: Wiley.MATH
Halmos, P. R. (1941). The decomposition of measures. Duke Mathematical Journal, 8, 386–392.
Halmos, P. R., & Savage, L. J. (1949). Application of the Radon-Nikodym theorem to the theory of sufficient statistics. The Annals of Mathematical Statistics, 20, 225–241.CrossRefMATHMathSciNet
Hasegawa, M., & Perlman, M. D. (1974). On the existence of a minimal sufficient subfield. Annals of Statistics, 2, 1049–1055; Correction Note, 3, 1371/2 (1975).
Heyer, H. (1969). Erschöpftheit und Invarianz beim Vergleich von Experimenten. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 12, 21–55.CrossRefMATHMathSciNet
Heyer, H. (1972). Zum Erschöpftheitsbegriff von D. Blackwell. Metrika, 19, 54–67.
Heyer, H. (1982). Theory of statistical experiments. Springer series in statistics. Berlin: Springer.CrossRefMATH
Hipp, Ch. (1972). Characterization of exponential families by existence of sufficient statistics. Thesis: University of Cologne.
Isenbeck, M., & Rüschendorf, L. (1992). Completeness in location families. Probability and Mathematical Statistics, 13, 321–343.MATHMathSciNet
Kolmogorov, A. N. (1933). Grundbegriffe der Wahrscheinlichkeitstheorie. Berlin: Springer.CrossRef
Koopman, B. O. (1936). On distributions admitting a sufficient statistic. Transactions of the American Mathematical Society, 39, 399–409.CrossRefMATHMathSciNet
Kumar, A., & Pathak, P. K. (1977). Two applications of Basu’s lemma. Scandinavian Journal of Statistics, 4, 37–38.MATHMathSciNet
Landers, D., & Rogge, L. (1972). Minimal sufficient \(\sigma \)-fields and minimal sufficient statistics. Two counterexamples. The Annals of Mathematical Statistics, 43, 2045–2049.CrossRefMATHMathSciNet
Landers, D., & Rogge, L. (1976). A note on completeness. Scandinavian Journal of Statistics, 3, 139.MATHMathSciNet
Laplace, P. S. (1818). Deuxiéme supplément à la theórie analytique des probabilités. Paris: Courcier.
Laube, G., & Pfanzagl, J. (1971). A remark on the functional equation \(\sum _{i=1}^n\varphi (x_i)=\psi (T(x_1,\ldots, x_n))\). Aequationes Mathematicae, 6, 241–242.CrossRefMathSciNet
Lehmann, E. L. (1959). Testing statistical hypotheses. New York: Wiley.MATH
Lehmann, E. L., & Casella, G. (1998). Theory of point estimation (2nd ed.). Berlin: Springer.MATH
Lehmann, E. L., & Scheffé, H. (1950, 1955, 1956). Completeness, similar regions and unbiased estimation. Sankhyā, 10, 305–340; 15, 219–236; Correction, 17, 250.
Mandelbaum, A., & Rüschendorf, L. (1987). Complete and symmetrically complete families of distributions. Annals of Statistics, 15, 1229–1244.CrossRefMATHMathSciNet
Mattner, L. (1999). Sufficiency, exponential families, and algebraically independent numbers. Mathematical Methods of Statistics, 8, 397–406.
Mattner, L. (2001). Minimal sufficient order statistics in convex models. Proceedings of the American Mathematical Society, 129, 3401–3411.CrossRefMATHMathSciNet
Messig, M. A., & Strawderman, W. E. (1993). Minimal sufficiency and completeness for dichotomous quantal response models. Annals of Statistics, 21, 2149–2157.CrossRefMATHMathSciNet
Neyman, J. (1935). Sur un teorema concernente le cosiddette statistiche sufficienti. Giorn. Ist. Ital. Attuari, 6, 320–334.MATH
Neyman, J. (1937). Outline of a theory of statistical estimation based on the classical theory of probability. Philosophical Transactions of the Royal Society of London. Series A, 236, 333–380.MATH
Pfanzagl, J. (1964). On the topological structure of some ordered families of distributions. The Annals of Mathematical Statistics, 35, 1216–1228.CrossRefMATHMathSciNet
Pfanzagl, J. (1970). On a functional equation related to families of exponential probability measures. Aequationes Mathematicae, 4, 139–142; Correction Note, 6, 120.
Pfanzagl, J. (1971b). On the functional equation \(\varphi (x)+\varphi (y)=\psi (T(x, y))\). Aequationes Mathematicae, 6, 202–205.CrossRefMATHMathSciNet
Pitcher, T. S. (1957). Sets of measures not admitting necessary and sufficient statistics or subfields. The Annals of Mathematical Statistics, 28, 267–268.CrossRefMATH
Pitcher, T. S. (1965). A more general property than domination for sets of probability measures. Pacific Journal of Mathematics, 15, 597–611.CrossRefMATHMathSciNet
Pitman, E. J. G. (1936). Sufficient statistics and intrinsic accuracy. Proceedings of the Cambridge Philosophical Society, 32, 567–579.CrossRefMATH
Plachky, D. (1977). A characterization of bounded completeness in the undominated case. In Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes A (pp. 477–480). Reidel.
Rogge, L. (1972). The relations between minimal sufficient statistics and minimal sufficient \(\sigma \)-fields. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 23, 208–215.CrossRefMATHMathSciNet
Romano, J. P., & Siegel, A. F. (1986). Counterexamples in probability and statistics. Monterey: Wadsworth and Brooks.MATH
Roy, K. K., & Ramamoorthi, R. V. (1979). Relationship between Bayes, classical and decision theoretic sufficiency. Sankhyā Series A, 41, 48–58.MATHMathSciNet
Sato, M. (1996). A minimal sufficient statistic and representations of the densities. Scandinavian Journal of Statistics, 23, 381–384.MATHMathSciNet
Schmetterer, L. (1966). On the asymptotic efficiency of estimates. In Research Papers in Statistics. Festschrift for J. Neyman & F. N. David (Eds.) (pp. 301–317). New York: Wiley.
Schmetterer, L. (1974). Introduction to mathematical statistics. Translation of the 2nd German edition 1966. Berlin: Springer.
Stigler, S. M. (1973). Laplace, Fisher, and the discovery of the concept of sufficiency. Biometrika, 60, 439–445. Reprinted in Kendall and Plackett (1977).
Stuart, A., & Ord, K. (1991). Kendall’s advanced theory of statistics. Classical inference and relationship (5th ed., Vol. 2). London: Edward Arnold.
Sverdrup, E. (1953). Similarity, unbiasedness, minimaxity and admissibility of statistical test procedures. Skand. Aktuarietidskr., 36, 64–86.MATHMathSciNet
Torgersen, E. N. (1991). Comparison of statistical experiments (Vol. 36). Encyclopedia of mathematics and its applications. Cambridge: Cambridge University Press.
Witting, H. (1966). Mathematische Statistik. Eine Einführung in Theorie und Methoden: Teubner.MATH
Zacks, S. (1971). The theory of statistical inference. New York: Wiley.
Metadata
Title
Sufficiency
Author
Johann Pfanzagl
Copyright Year
2017
Publisher
Springer Berlin Heidelberg
Book
Mathematical Statistics

Print ISBN: 978-3-642-31083-6

Electronic ISBN: 978-3-642-31084-3

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-642-31084-3_2

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