Exponential families

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.