Family of probability distributions related to the normal distribution
Not to be confused with the exponential distribution.
"Natural parameter" redirects here. For use of this term in differential geometry, see Natural parametrization.
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate expectations, covariances using differentiation based on some useful algebraic properties, as well as for generality, as exponential families are in a sense very natural sets of distributions to consider. The term exponential class is sometimes used in place of "exponential family",[1] or the older term Koopman–Darmois family.
Sometimes loosely referred to as "the" exponential family, this class of distributions is distinct because they all possess a variety of desirable properties, most importantly the existence of a sufficient statistic.
The concept of exponential families is credited to[2] E. J. G. Pitman,[3] G. Darmois,[4] and B. O. Koopman[5] in 1935–1936. Exponential families of distributions provide a general framework for selecting a possible alternative parameterisation of a parametric family of distributions, in terms of natural parameters, and for defining useful sample statistics, called the natural sufficient statistics of the family.
^Kupperman, M. (1958). "Probabilities of hypotheses and information-statistics in sampling from exponential-class populations". Annals of Mathematical Statistics. 9 (2): 571–575. doi:10.1214/aoms/1177706633. JSTOR 2237349.
^Andersen, Erling (September 1970). "Sufficiency and Exponential Families for Discrete Sample Spaces". Journal of the American Statistical Association. 65 (331). Journal of the American Statistical Association: 1248–1255. doi:10.2307/2284291. JSTOR 2284291. MR 0268992.
^Pitman, E.; Wishart, J. (1936). "Sufficient statistics and intrinsic accuracy". Mathematical Proceedings of the Cambridge Philosophical Society. 32 (4): 567–579. Bibcode:1936PCPS...32..567P. doi:10.1017/S0305004100019307. S2CID 120708376.
^Darmois, G. (1935). "Sur les lois de probabilites a estimation exhaustive". C. R. Acad. Sci. Paris (in French). 200: 1265–1266.
^Koopman, B. (1936). "On distribution admitting a sufficient statistic". Transactions of the American Mathematical Society. 39 (3). American Mathematical Society: 399–409. doi:10.2307/1989758. JSTOR 1989758. MR 1501854.
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