IDEAS home Printed from
   My bibliography  Save this article

A third-order optimum property of the maximum likelihood estimator


  • Pfanzagl, J.
  • Wefelmeyer, W.


Let [theta](n) denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators [theta](n) + n-1 q([theta](n)), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T(n) there exists q such that the risk of [theta](n) + n-1 q([theta](n)) exceeds the risk of T(n) by an amount of order o(n-1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If q* is chosen such that [theta](n) + n-1 q*([theta](n)) is unbiased up to o(n-1/2), then this estimator minimizes the risk up to an amount of order o(n-1) in the class of all estimators which are unbiased up to o(n-1/2). The results are obtained under the assumption that T(n) admits a stochastic expansion, and that either the distributions have--roughly speaking--densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth.

Suggested Citation

  • Pfanzagl, J. & Wefelmeyer, W., 1978. "A third-order optimum property of the maximum likelihood estimator," Journal of Multivariate Analysis, Elsevier, vol. 8(1), pages 1-29, March.
  • Handle: RePEc:eee:jmvana:v:8:y:1978:i:1:p:1-29

    Download full text from publisher

    File URL:
    Download Restriction: Full text for ScienceDirect subscribers only

    As the access to this document is restricted, you may want to search for a different version of it.


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Michael Creel & Dennis Kristensen, "undated". "Indirect Likelihood Inference," Working Papers 558, Barcelona Graduate School of Economics.
    2. Kunitomo, Naoto & Matsushita, Yukitoshi, 2009. "Asymptotic expansions and higher order properties of semi-parametric estimators in a system of simultaneous equations," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1727-1751, September.
    3. Geert Dhaene & Koen Jochmans, 2015. "Split-panel Jackknife Estimation of Fixed-effect Models," Review of Economic Studies, Oxford University Press, vol. 82(3), pages 991-1030.
    4. Hansen, Christian B., 2007. "Generalized least squares inference in panel and multilevel models with serial correlation and fixed effects," Journal of Econometrics, Elsevier, vol. 140(2), pages 670-694, October.
    5. Andrews, Donald W.K., 2002. "EQUIVALENCE OF THE HIGHER ORDER ASYMPTOTIC EFFICIENCY OF k-STEP AND EXTREMUM STATISTICS," Econometric Theory, Cambridge University Press, vol. 18(5), pages 1040-1085, October.
    6. Susanne M. Schennach, 2007. "Point estimation with exponentially tilted empirical likelihood," Papers 0708.1874,
    7. Whitney K. Newey & Richard J. Smith, 2004. "Higher Order Properties of Gmm and Generalized Empirical Likelihood Estimators," Econometrica, Econometric Society, vol. 72(1), pages 219-255, January.
    8. Masafumi Akahira & Kei Takeuchi, 1989. "Higher order asymptotics in estimation for two-sided Weibull type distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 41(4), pages 725-752, December.
    9. Jeffrey M. Wooldridge, 2004. "Estimating average partial effects under conditional moment independence assumptions," CeMMAP working papers CWP03/04, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    10. Caner, Mehmet, 2008. "Nearly-singular design in GMM and generalized empirical likelihood estimators," Journal of Econometrics, Elsevier, vol. 144(2), pages 511-523, June.
    11. Th. Pfaff, 1983. "Third-order optimum properties of estimator-sequences," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 30(1), pages 125-138, December.
    12. Kyoo Il Kim, 2016. "Higher Order Bias Correcting Moment Equation for M-Estimation and Its Higher Order Efficiency," Econometrics, MDPI, vol. 4(4), pages 1-19, December.
    13. Kano, Yutaka, 1998. "More Higher-Order Efficiency: Concentration Probability," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 349-366, November.
    14. Patrik Guggenberger & Jinyong Hahn, 2005. "Finite Sample Properties of the Two-Step Empirical Likelihood Estimator," Econometric Reviews, Taylor & Francis Journals, vol. 24(3), pages 247-263.
    15. Tan, Zhiqiang, 2014. "Second-order asymptotic theory for calibration estimators in sampling and missing-data problems," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 240-253.
    16. Kundhi, Gubhinder & Rilstone, Paul, 2012. "Edgeworth expansions for GEL estimators," Journal of Multivariate Analysis, Elsevier, vol. 106(C), pages 118-146.
    17. Gubhinder Kundhi & Paul Rilstone, 2015. "Saddlepoint expansions for GEL estimators," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 24(1), pages 1-24, March.
    18. repec:pit:wpaper:211 is not listed on IDEAS
    19. Rilstone, Paul & Srivastava, V. K. & Ullah, Aman, 1996. "The second-order bias and mean squared error of nonlinear estimators," Journal of Econometrics, Elsevier, vol. 75(2), pages 369-395, December.
    20. Michael Creel & Dennis Kristensen, 2013. "Indirect Likelihood Inference (revised)," UFAE and IAE Working Papers 931.13, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:8:y:1978:i:1:p:1-29. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.