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Exponential Families

Statistical Theory and Methods

  1. Marc Hallin1,2

Published Online: 15 JAN 2013

DOI: 10.1002/9780470057339.vnn097

Encyclopedia of Environmetrics

Encyclopedia of Environmetrics

How to Cite

Hallin, M. 2013. Exponential Families. Encyclopedia of Environmetrics. 2.

Author Information

  1. 1

    ECARES, Université libre de Bruxelles, Bruxelles, Belgium

  2. 2

    ORFE, Princeton University, Princeton, NJ, USA

Publication History

  1. Published Online: 15 JAN 2013


Exponential families of distributions are parametric dominated families in which the logarithm of probability densities take a simple bilinear form (bilinear in the parameter and a statistic). As a consequence of that special form, sampling models in those families admit a finite-dimensional sufficient statistic irrespective of the sample size, and optimal solutions exist for a number of statistical inference problems: uniformly minimum risk unbiased estimation, uniformly most powerful one-parameter one-sided tests, and so on. Most traditional families of distributions–binomial, multinomial, Poisson, negative binomial, normal, gamma, chi-square, beta, Dirichlet, Wishart, and many others–constitute exponential families. Note, however, that the uniform, logistic, Cauchy, or Student (for given degrees of freedom) location-scale families are not exponential; the double-exponential or Laplace family is exponential for scale only, at fixed location.


  • sufficient and complete statistic;
  • Lehmann–Scheffé theorem;
  • normal;
  • binomial;
  • Poisson;
  • multinomial;
  • Darmois–Koopman–Pitman theorem;
  • monotone likelihood ratio;
  • efficient estimation;
  • generalized linear model