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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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    Abstract

    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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    Bibliographic Info

    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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    Related research

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

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    Cited by:
    1. 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.
    2. 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.
    3. Whitney Newey & Richard Smith, 2003. "Higher order properties of GMM and generalised empirical likelihood estimators," CeMMAP working papers CWP04/03, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    4. 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).
    5. Mehmet Caner, 2005. "Nearly Singular design in gmm and generalized empirical likelihood estimators," Working Papers 211, University of Pittsburgh, Department of Economics, revised Jan 2005.
    6. Naoto Kunitomo & Yukitoshi Matsushita, 2009. "Asymptotic Expansions and Higher Order Properties of Semi-Parametric Estimators in a System of Simultaneous Equations," CIRJE F-Series CIRJE-F-611, CIRJE, Faculty of Economics, University of Tokyo.
    7. Th. Pfaff, 1983. "Third-order optimum properties of estimator-sequences," Metrika, Springer, vol. 30(1), pages 125-138, December.
    8. Michael Creel & Dennis Kristensen, 2011. "Indirect likelihood inference," UFAE and IAE Working Papers 874.11, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
    9. Kano, Yutaka, 1998. "More Higher-Order Efficiency: Concentration Probability," Journal of Multivariate Analysis, Elsevier, vol. 67(2), pages 349-366, November.
    10. Geert Dhaene & Koen Jochmans, 2010. "Split-panel jackknife estimation of fixed-effect models," Sciences Po publications info:hdl:2441/eu4vqp9ompq, Sciences Po.
    11. 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.
    12. 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.
    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. Susanne M. Schennach, 2007. "Point estimation with exponentially tilted empirical likelihood," Papers 0708.1874, arXiv.org.

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