The Annals of Statistics

Complete order statistics in parametric models

L. Mattner

Full-text: Open access


For a given statistical model $\mathsf{P}$ it may happen that the order statistic is complete for each IID model based on $\mathsf{P}$. After reviewing known relevant results for large nonparametric models and pointing out generalizations to small nonparametric models, we essentially prove that this happens generically even in smooth parametric models.

As a consequence it may be argued that any statistic depending symmetrically on the observations can be regarded as an optimal unbiased estimator of its expectation.

In particular, the sample mean $\overline{X}_n$ is generically an optimal unbiased estimator, but, as it turns out, also generically asymptotically inefficient.

Article information

Ann. Statist. Volume 24, Number 3 (1996), 1265-1282.

First available in Project Euclid: 20 September 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62B05: Sufficient statistics and fields 62F10: Point estimation

Asymptotic efficiency contamination model IID model minimal sufficiency nonparametric neighborhoods optimal unbiased estimation symmetrical completeness UMVU unimodality


Mattner, L. Complete order statistics in parametric models. Ann. Statist. 24 (1996), no. 3, 1265--1282. doi:10.1214/aos/1032526968.

Export citation


  • ANDERSON, T. W. 1962. Least squares and best unbiased estimation. Ann. Math. Statist. 33 266 272. Z.
  • BELL, C. B., BLACKWELL, D. and BREIMAN, L. 1960. On the completeness of order statistics. Ann. Math. Statist. 31 794 797. Z.
  • BILLINGSLEY, P. 1986. Probability and Measure, 2nd ed. Wiley, New York. Z.
  • DIEUDONNE, J. 1960. Foundations of Modern Analy sis. Academic Press, New York. ´ Z.
  • FRASER, D. A. S. 1954. Completeness of order statistics. Canad. J. Math. 6 42 45. Z.
  • HALMOS, P. R. 1946. The theory of unbiased estimation. Ann. Math. Statist. 17 34 43. Z.
  • HEy ER, H. 1982. Theory of Statistical Experiments. Springer, New York. Z.
  • ISENBECK, M. and RUSCHENDORF, L. 1992. Completeness in location families. Probab. Math. ¨ Statist. 13 321 343. Z.
  • LANDERS, D. and ROGGE, L. 1976. A note on completeness. Scand. J. Statist. 3 139. Z.
  • LEHMANN, E. L. 1959. Testing Statistical Hy pothesis, 1st ed. Wiley, New York. Z.
  • LEHMANN, E. L. 1983. Theory of Point Estimation. Wiley, New York. Z.
  • LEHMANN, E. L. 1986. Testing Statistical Hy pothesis, 2nd ed. Wiley, New York. Z.
  • LEHMANN, E. L. and SCHOLZ, F. W. 1992. Ancillarity. In Current Issues in Statistical Inference, Essay s in Honor of D. Basu. IMS, Hay ward, CA. Z.
  • MANDELBAUM, A. and RUSCHENDORF, L. 1987. Complete and sy mmetrically complete families of ¨ distributions. Ann. Statist. 15 1229 1244. Z.
  • MATTNER, L. 1992. Completeness of location families, translated moments, and uniqueness of charges. Probab. Theory Related Fields 92 137 149. Z.
  • MATTNER, L. 1993. Some incomplete but boundedly complete location families. Ann. Statist. 21 2158 2162. Z. MULLER-FUNK, U., PUKELSHEIM, F. and WITTING, H. 1989. On the attainment of the Cramer Rao ¨ ´ bound in -differentiable families of distributions. Ann. Statist. 17 1742 1748. r Z.
  • OXTOBY, J. C. 1980. Measure and Category, 2nd ed. Springer, New York. Z.
  • PFANZAGL, J. 1979. On optimal median unbiased estimators in the presence of nuisance parameters. Ann. Statist. 7 187 193. Z.
  • PFANZAGL, J. 1980. Asy mptotic expansions in parametric statistical theory. In Developments in Z. Statistics P. K. Krishnaiah, ed. 3 1 97. Academic Press, New York. Z.
  • PFANZAGL, J. 1993. Sequences of optimal unbiased estimators need not be asy mptotically optimal. Scand. J. Statist. 20 73 76. Z.
  • PFANZAGL, J. 1994. Parametric Statistical Theory. de Gruy ter, Berlin. Z.
  • PLACHKY, D. 1993. An estimation-theoretical characterization of the Poisson distribution. Statist. Decisions 3 175 178. Z.
  • PORTNOY, S. 1977. Asy mptotic efficiency of minimum variance unbiased estimators. Ann. Statist. 5 522 529. Z.
  • RUSCHENDORF, L. 1987. Estimation in the presence of nuisance parameters. In Contributions to ¨ Z. Stochastics W. Sendler, ed. 190 201. physica, Heidelberg. Z.
  • RUSCHENDORF, L. 1988. Unbiased estimation and local structure. In Probability Theory and ¨ Z Mathematical Statistics with Applications W. Grossmann, J. Magy orodi, I. Vicze and. W. Wertz, eds. 295 306. Reidel, Dordrecht. Z.
  • WITTING, H. 1985. Mathematische Statistik I. Teubner, Stuttgart.