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Exponential Family

  1. Donald A. Pierce

Published Online: 15 JUL 2005

DOI: 10.1002/0470011815.b2a15041

Encyclopedia of Biostatistics

Encyclopedia of Biostatistics

How to Cite

Pierce, D. A. 2005. Exponential Family. Encyclopedia of Biostatistics. 3.

Author Information

  1. Radiation Effects Research Foundation, Hiroshima, Japan

Publication History

  1. Published Online: 15 JUL 2005


The overview focuses on aspects of exponential families important in biostatistics, particularly by emphasizing not identically distributed observations involving covariables. This is closely related to generalized linear models. The distinction is made between regular and curved exponential families. In the former, the maximum likelihood estimator is sufficient, the expected and observed information are the same, and there are exact frequency methods for some inferences. In the latter, there is information not contained in the maximum likelihood estimator, much of which is carried by the ratio of observed to expected information, and asymptotic methods must always be used. Of these, likelihood ratio and score methods are preferable to the Wald method. Modern higher-order asymptotic methods, including the modified profile likelihood, are mentioned. The tractability of Bayesian inference in exponential families is indicated.


  • conditional inference;
  • generalized linear models;
  • higher-order asymptotics;
  • modified profile likelihood;
  • observed information;
  • likelihood ratio test;
  • score test;
  • regular and curved exponential families