Partially complete sufficient statistics are jointly complete

A. Kagan   ; Y. Malinovsky   ; L. Mattner

Teoriâ veroâtnostej i ee primeneniâ, Tome 59 (2014) no. 3, pp. 542-561

Bibliographie

[1] Bahadur R. R., “Sufficiency and statistical decision functions”, Ann. Math. Statist., 25 (1954), 432–462 | DOI | MR | Zbl

[2] Bahadur R. R., “On unbiased estimates of uniformly minimum variance”, Sankhyā, 18 (1957), 211–224 | MR | Zbl

[3] Bahadur R. R., “A note on UMV estimates and ancillary statistics”, Contributions to Statistics, Jaroslav Hájek Memorial Volume, Reidel, Dordrecht, 1979, 19–24 | DOI | MR

[4] Bahadur R. R., R. R. Bahadur's Lectures on the Theory of Estimation, after lectures in 1984/85 at the University of Chicago, IMS, Lecture Notes Monogr. Ser., 39, eds. S. M. Stigler, W. H. Wong, D. Xu, IMS, Beachwood, 2002 | MR | Zbl

[5] Barra Zh.-R., Osnovnye ponyatiya matematicheskoi statistiki, Mir, M., 1974, 275 pp.

[6] Basu D., “On statistics independent of a complete sufficient statistic”, Sankhyā, 15 (1955), 377–380 | MR | Zbl

[7] Basu D., “On statistics independent of sufficient statistics”, Sankhyā, 20 (1958), 223–226 | MR | Zbl

[8] Basu D., “Basu theorems”, Encyclopedia of Statistical Sciences, v. 1, eds. S. Kotz, N. L. Johnson, Wiley, 1982, 193–196; J. K. Ghosh (ed.), Statistical Information and Likelihood, A Collection of Critical Essays of Dr. D. Basu, Lecture Notes in Statistics, 45, Springer, Berlin, 1988, 342–349 | DOI | MR

[9] Bondesson L., “On uniformly minimum variance unbiased estimation when no complete sufficient statistics exist”, Metrika, 30 (1983), 49–54 | DOI | MR | Zbl

[10] Cramer E., Kamps U., Schenk N., “On the joint completeness of independent statistics”, Statist. Decisions, 20:3 (2002), 269–277 | MR | Zbl

[11] D'Haultfœuille X., “On the completeness condition in nonparametric instrumental problems”, Econometric Theory, 27:3 (2011), 460–471 | DOI | MR | Zbl

[12] Feller W., An Introduction to Probability Theory and Its Applications, v. 1, Wiley, New York, 1970

[13] Fichtenholz G., “Sur une fonction de deux variables sans intégrale double”, Fund. Math., 6 (1924), 30–36

[14] Fraser D. A. S., Nonparametric Methods in Statistics, Wiley, New York, 1957 | MR | Zbl

[15] Halmos P. R., “The theory of unbiased estimation”, Ann. Math. Statist., 17 (1946), 34–43 | DOI | MR | Zbl

[16] Hoffmann-Jørgensen J., Probability with a View toward Statistics, v. 1, Chapman Hall, London, 1994, 589 pp.; v. 2, 533 pp.

[17] Kagan A. M., “Dva zamechaniya o kharakterizatsii dostatochnosti”, Predelnye teoremy i statisticheskie vyvody, ed. S. Kh. Sirazhdinov, Fan, Tashkent, 1966, 60–66 | MR

[18] Kagan A. M., “Profile sufficiency”, Austrian J. Statist., 35 (2006), 121–130

[19] Kagan A. M., Konikov M., “The structure of UMVUEs from categorical data”, Teoriya veroyatn. i ee primen., 50:3 (2005), 597–604 | DOI | MR

[20] Kagan A. M., Malinovsky Y., “On the Nile problem of Sir Ronald Fisher”, Electron. J. Statist., 7 (2013), 1968–1982 | DOI | MR | Zbl

[21] Koehn U., Thomas D. L., “On statistics independent of a sufficient statistic: Basu's lemma”, Amer. Statist., 29 (1975), 40–42 | MR | Zbl

[22] Kozek A., “On two necessary $\sigma$-fields and on universal loss functions”, Probab. Math. Statist., 1:1 (1980), 29–47 | MR | Zbl

[23] Landers D., Rogge L., “A note on completeness”, Scand. J. Statist., 3:3 (1976), 139 | MR | Zbl

[24] Lehmann E. L., Casella G., Theory of Point Estimation, Springer-Verlag, New York, 1998, 589 pp. | MR

[25] Lehmann E. L., Romano J. P., Testing Statistical Hypotheses, Springer, New York, 2005, 784 pp. | MR

[26] Lehmann E. L., Scheffé H., “On the problem of similar regions”, Proc. Natl. Acad. Sci. USA, 33 (1947), 382–386 | DOI | MR | Zbl

[27] Lehmann E. L., Scheffé H., “Completeness, similar regions, and unbiased estimation. I; II”, Sankhyā, 10 (1950), 305–340 ; 15 (1955), 219–236 ; Correction, 17 (1956), 250 | MR | Zbl | MR | Zbl

[28] Mandelbaum A., Rüschendorf L., “Complete and symmetrically complete families of distributions”, Ann. Statist., 15:3 (1987), 1229–1244 | DOI | MR | Zbl

[29] Mattner L., “Complete order statistics in parametric models”, Ann. Statist., 24:3 (1996), 1265–1282 | DOI | MR | Zbl

[30] Mattner L., “Sufficiency, exponential families, and algebraically independent numbers”, Math. Methods Statist., 8:3 (1999), 397–406 | MR | Zbl

[31] Mattner L., Mattner F., “Confidence bounds for the sensitivity lack of a less specific diagnostic test, without gold standard”, Metrika, 76:2 (2013), 239–263 | DOI | MR | Zbl

[32] Odén A., Wedel H., “Arguments for Fisher's permutation test”, Ann. Statist., 3 (1975), 518–520 | DOI | MR | Zbl

[33] Oosterhoff J., Schriever B. F., “A note on complete families of distributions”, Statist. Neerl., 41 (1987), 183–189 | DOI | MR | Zbl

[34] Pfanzagl J., Parametric Statistical Theory, de Gruyter, Berlin, 1994, 374 pp. | MR | Zbl

[35] Plachky D., “A characterization of bounded completeness in the undominated case” (Prague, 1974), Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, 1977, 477–480 | MR | Zbl

[36] Rao C. R., “Some theorems on minimum variance estimation”, Sankhyā, 12 (1952), 27–42 | MR | Zbl

[37] Rüschendorf L., “Estimation in the presence of nuisance parameters”, Contributions to Stochastics, ed. W. Sendler, Physica-Verlag, Heidelberg, 1997, 190–201 | MR

[38] Patil G. P., “Weighted distributions”, Encyclopedia of Environmetrics, eds. A. H. El-Shaarawi, W. W. Piegorsch, Wiley, Chichester, 2002, 2369–2377

[39] San Martín E., Mouchart M., “On joint completeness: sampling and Bayesian versions, and their connections”, Sankhyā, 69:4 (2007), 780–807 | MR | Zbl

[40] Scheffé H., “On a measure problem arising in the theory of non-parametric tests”, Ann. Math. Statist., 14 (1943), 227–233 | DOI | MR | Zbl

[41] Schmetterer L., Introduction to Mathematical Statistics, Second Edition, Springer, 1974 | MR | Zbl

[42] Schmetterer L., Strasser H., “Zur Theorie der erwartungstreuen Schätzungen”, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl., 76:6 (1974), 59–66 | MR

[43] Smith W. L., “A note on truncation and sufficient statistics”, Ann. Math. Statist., 28 (1957), 247–252 | DOI | MR | Zbl

[44] Strasser H., Mathematical Theory of Statistics, de Gruyter, Berlin, 1985, 492 pp. | MR | Zbl

[45] Torgersen E., “On Bahadur's converse of the Rao–Blackwell theorem. Extension to majorized experiments”, Scand. J. Statist., 15:4 (1988), 273–280 | MR | Zbl

[46] Torgersen E., Comparison of Statistical Experiments, Cambridge Univ. Press, Cambridge, 1991, 675 pp. | MR | Zbl

[47] Wald A., “On the power function of the analysis of variance test”, Ann. Math. Statist., 13 (1942), 434–439 | DOI | MR | Zbl

[48] Wald A., “Note on a lemma”, Ann. Math. Statist., 15 (1944), 330–333 | DOI | MR | Zbl

[49] Witting H., Mathematische Statistik, v. I, Teubner, Stuttgart, 1985, 538 pp. | MR | Zbl

[50] Witting H., Müller-Funk U., Mathematische Statistik, v. II, Teubner, Stuttgart, 1995, 803 pp. | MR | Zbl