Exponential families

M. Hallin

    Research output: Chapter in Book/Report/Conference proceedingChapterScientificpeer-review

    Abstract

    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.
    Original languageEnglish
    Title of host publicationEncyclopedia of Environmetrics, 2nd Edition
    EditorsW. Piegorsch, A. El Shaarawi
    PublisherWiley
    Pages106-110
    Number of pages3510
    ISBN (Print)9780470973882
    Publication statusPublished - 2012

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  • Cite this

    Hallin, M. (2012). Exponential families. In W. Piegorsch, & A. El Shaarawi (Eds.), Encyclopedia of Environmetrics, 2nd Edition (pp. 106-110). Wiley.