Convolution Theorem and Asymptotic Efficiency

  • Bing LiEmail author
  • G. Jogesh Babu
Part of the Springer Texts in Statistics book series (STS)


The optimality of the maximum likelihood estimate developed in Chapters  8 and  9 is in terms of the asymptotic variance of the solution of estimating equations. However, this optimality goes much deeper. In this chapter the maximum likelihood estimate, as well as estimates that are asymptotically equivalent to it, are shown to have the smallest asymptotic variance among all regular estimates, which is a much wider class of estimates than solutions of estimating equations. The theory underlying this general result – the framework of Local Asymptotic Normality and the Convolution Theorem – is systematically developed. This is an amazingly logical system that leads to far-reaching results with a small set of assumptions. Some techniques introduced in this chapter, such as Le Cam’s third lemma and the convolution theorem, will also be useful for developing local alternative distributions for asymptotic hypothesis tests in Chapter  11.


  1. Bahadur, R. R. (1964). On Fisher’s bound for asymptotic variances. The Annals of Mathematical Statistics. 35, 1545–1552.Google Scholar
  2. Bickel, P. J., Klaassen, C. A. J., Ritov, Y. Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. The Johns Hopkins University Press.Google Scholar
  3. Fisher, R. A. (1922). On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society A, 222, 594–604.Google Scholar
  4. Fisher, R. A. (1925). Theory of statistical estimation. Proc. Cambridge Phil. Soc., 22, 700–725.Google Scholar
  5. Hájek, J. (1970). A characterization of limiting distributions of regular estimates. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete. 14, 323–330.Google Scholar
  6. Hall, W. J. and Mathiason, D. J. (1990). On large-sample estimation and testing in parametric models. Int. Statist. Rev., 58, 77–97.Google Scholar
  7. Le Cam, L. (1953). On some asymptotic Properties of maximum likelihood estimates and related Bayes estimates. Univ. California Publ. Statistic.1, 277–330.Google Scholar
  8. Le Cam, L. (1960). Locally asymptotically normal families of distributions. Univ. California Publ. Statistic. 3, 370–98.Google Scholar
  9. Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics: Some Basic Concepts. Second Edition. Springer.Google Scholar
  10. van der Vaart, A. W. (1997). Superefficiency. Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, 397–410. Springer.Google Scholar
  11. van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of StatisticsPenn State UniversityUniversity ParkUSA

Personalised recommendations