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Efficiency. of infinite dimensional M‐ estimators

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  • A. W. van der Vaart

Abstract

It is well‐known that maximum likelihood estimators are asymptotically normal with covariance equal to the inverse Fisher information in smooth, finite dimensional parametric models. Thus they are asymptotically efficient. A similar phenomenon has been observed for certain infinite dimensional parameter spaces. We give a simple proof of efficiency, starting from a theorem on asymptotic normality of infinite dimensional M‐estimators. The proof avoids the explicit calculation of the Fisher information. We also address Hadamard differentiability of the corresponding M‐functionals.

Suggested Citation

  • A. W. van der Vaart, 1995. "Efficiency. of infinite dimensional M‐ estimators," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 49(1), pages 9-30, March.
    • Handle: RePEc:bla:stanee:v:49:y:1995:i:1:p:9-30
    DOI: 10.1111/j.1467-9574.1995.tb01452.x
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    Cited by:

    1. Murphy, S. A. & van der Vaart, A. W., 2001. "Semiparametric Mixtures in Case-Control Studies," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 1-32, October.
    2. Feike C. Drost & Ramon van den Akker & Bas J. M. Werker, 2009. "Efficient estimation of auto‐regression parameters and innovation distributions for semiparametric integer‐valued AR(p) models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 467-485, April.
    3. Meredith Goldwasser & Lu Tian & L. J. Wei, 2004. "Statistical Inference for Infinite Dimensional Parameters Via Asymptotically Pivotal Estimating Functions," Harvard University Biostatistics Working Paper Series 1007, Berkeley Electronic Press.
    4. Shen, Pao-sheng, 2009. "An inverse-probability-weighted approach to the estimation of distribution function with doubly censored data," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1269-1276, May.
    5. Pao-sheng Shen, 2010. "Nonparametric analysis of doubly truncated data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(5), pages 835-853, October.

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