What is the Darmois-Koopman Distribution?
A Darmois-Koopman distribution (or Koopman-Darmois) is a member of the exponential-type class of probability distributions. Exponential-type density functions have the form 
Where A(·), B(·), C(·), and D(·) are arbitrary functions.
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 iid 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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