The Annals of Statistics

Asymptotic efficiency of estimates for models with incidental nuisance parameters

Helmut Strasser

Full-text: Open access

Abstract

In this paper we show that the well-known asymptotic efficiency bounds for full mixture models remain valid if individual sequences of nuisance parameters are considered. This is made precise both for some classes of random (i.i.d.) and nonrandom nuisance parameters. For the random case it is shown that superefficiency of the kind given by an example of Pfanzagl can happen only with low probability. The nonrandom case deals with permutation-invariant estimators under one-dimensional nuisance parameters. It is shown that the efficiency bounds remain valid for individual nonrandom arrays of nuisance parameters whose empirical process, if it is centered around its limit and standardized, satisfies a compactness condition. The compactness condition is satisfied in the random case with high probability. The results make use of basic LAN theory. Regularity conditions are stated in terms of $L^2$-differentiability.

Article information

Source
Ann. Statist., Volume 24, Number 2 (1996), 879-901.

Dates
First available in Project Euclid: 24 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1032894471

Digital Object Identifier
doi:10.1214/aos/1032894471

Mathematical Reviews number (MathSciNet)
MR1394994

Zentralblatt MATH identifier
0860.62028

Subjects
Primary: 62A20 62B15: Theory of statistical experiments
Secondary: 62F05: Asymptotic properties of tests 62F12: Asymptotic properties of estimators

Keywords
Regularity differentiability in quadratic mean sufficiency equivalence of experiments

Citation

Strasser, Helmut. Asymptotic efficiency of estimates for models with incidental nuisance parameters. Ann. Statist. 24 (1996), no. 2, 879--901. doi:10.1214/aos/1032894471. https://projecteuclid.org/euclid.aos/1032894471


Export citation

References

  • BICKEL, P. J. and KLAASSEN, C. A. J. 1986. Empirical Bay es estimation in functional and structural models and uniformly adaptive estimation of location. Adv. in Appl. Math. 7 55 69. Z.
  • BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. 1993. Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press. Z.
  • DIEUDONNE, J. 1960. Foundations of Modern Analy sis. Academic Press, New York. ´Z.
  • HAJEK, J. 1972. Local asy mptotic minimax and admissibility in estimation. In Proc. Sixth ´ Berkeley Sy mp. Math. Statist. Probab. 175 194. Univ. California Press, Berkeley. Z.
  • LE CAM, L. 1953. On some asy mptotic properties of maximum likelihood estimates and related Bay es estimates. University of California Publications in Statistics 1 277 330. Z.
  • LE CAM, L. 1986. Asy mptotic Methods in Statistical Decision Theory. Springer, New York. Z.
  • PFANZAGL, J. 1993. Incidental versus random nuisance parameters. Ann. Statist. 21 1663 1691. Z.
  • PFANZAGL, J. and WEFELMEy ER, W. 1982. Contributions to a General Asy mptotic Statistical Theory. Lecture Notes in Statist. 13 Springer, New York. Z.
  • SHORACK, G. R. and WELLNER, J. A. 1986. Empirical Processes with Applications to Statistics. Wiley, New York. Z.
  • STRASSER, H. 1994. Asy mptotic admissibility and uniqueness of efficient estimates in semiparametric models. In Research Papers in Probability and Statistics: A Festschrift for Z. Lucien Le Cam D. Pollard and G. Yang, eds. Chapter 20. Z.
  • STRASSER, H. 1985. Mathematical Theory of Statistics: Statistical Experiments and Asy mptotic Decision Theory. de Gruy ter, Berlin. Z.
  • STRASSER, H. 1995. Differentiability of statistical experiments. Technical report, Dept. Statistics, Vienna Univ. Economics and Business Administration.