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A third-order optimum property of the maximum likelihood estimator


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  • Pfanzagl, J.
  • Wefelmeyer, W.
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    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.

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    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 8 (1978)
    Issue (Month): 1 (March)
    Pages: 1-29

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    Handle: RePEc:eee:jmvana:v:8:y:1978:i:1:p:1-29

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    Keywords: Asymptotic theory Edgeworth-expansions higher-order efficiency complete classes maximum likelihood estimation unbiasedness;


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    Cited by:
    1. Susanne M. Schennach, 2007. "Point estimation with exponentially tilted empirical likelihood," Papers 0708.1874,
    2. 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, 01.
    3. 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.
    4. Kano, Yutaka, 1998. "More Higher-Order Efficiency: Concentration Probability," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 349-366, November.
    5. 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).
    6. Creel, Michael & Kristensen, Dennis, 2011. "Indirect Likelihood Inference," Dynare Working Papers 8, CEPREMAP.
    7. 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.
    8. Geert Dhaene & Koen Jochmans, 2014. "Split-Panel Jackknife Estimation of Fixed-Effect Models," Sciences Po Economics Discussion Papers 2014-03, Sciences Po Departement of Economics.
    9. 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(05), pages 1040-1085, October.
    10. Th. Pfaff, 1983. "Third-order optimum properties of estimator-sequences," Metrika, Springer, vol. 30(1), pages 125-138, December.
    11. Caner, Mehmet, 2008. "Nearly-singular design in GMM and generalized empirical likelihood estimators," Journal of Econometrics, Elsevier, vol. 144(2), pages 511-523, June.
    12. 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.
    13. Masafumi Akahira & Kei Takeuchi, 1989. "Higher order asymptotics in estimation for two-sided Weibull type distributions," Annals of the Institute of Statistical Mathematics, Springer, vol. 41(4), pages 725-752, December.
    14. 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.


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