< List of Probability Distributions > Darmois-Koopman Distribution
What is a Darmois-Koopman distribution?
Where A(·), B(·), C(·), and D(·) are arbitrary functions.
As a member of the exponential-type family, the Darmois-Koopman distribution can be defined as a summation :
where k is a set of jointly sufficient statistics for k parameters.
This class of probability distributions was recognized by Darmois  and Koopman  almost simultaneously . Pitman  is often credited with contributions to the introduction of the distribution. In both cases, the form of the PDF described a single sufficient statistic for θ, given n independent and identically distributed (i.i.d.) random variables. A broad subclass of these distributions was presented by Morris . The distribution remains relatively obscure, although it has made relatively recent appearances in estimation theory [7, 8].
The Darmois-Koopman-Pitman Theorem shows that sufficiency sharply restricts the form of the PDF. The theorem is related to the Darmois-Koopman distribution in the following way: given certain regularity conditions on the PDF, a necessary and sufficient condition for the existence of a sufficient statistic (possibly vector-valued) of fixed dimension is that the PDF is a member of the exponential distribution family .
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 Darmois, G. (1935). Sur les lois de probabilitis a estimation exhaustif, Compres Rendus de 1’Acadimie des Sciences, Paris, 200, 1265-1267.
 Koopman, B. 0. (1936). On distributions admitting a sufficient statistic, Transactions of the American Mathematical Society, 39, 339-409.
 Pitman (1936). Sufficient Statistics and Intrinsic Accuracy. Proc. Camb. Phi. Soc. 32: 567-569.
 Morris, C. N. (1983). Natural exponential families with quadratic variance function, Annals of Statistics, 11, 515-529.
 Kubacek, L. (2012). Foundations of Estimation Theory. Elsevier Science.
 American Mathematical Society (2008). Selected Papers on Analysis and Related Topics. American Mathematical Society.