Darmois-Koopman Distribution

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What is a 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 [1]:

exponential-type definition

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 [2]:

darmois-koopman distribution definition

where k is a set of jointly sufficient statistics for parameters.

This class of probability distributions was recognized by Darmois [3] and Koopman [4] almost simultaneously [1]. Pitman [5] 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 [6]. The distribution remains relatively obscure, although it has made relatively recent appearances in estimation theory [7, 8].

Darmois-Koopman-Pitman Theorem

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 [2].


[1] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

[2] Commenges , D. et al. (Eds.). (2006). The Oxford Dictionary of Statistical Terms. Oxford University Press.

[3] Darmois, G. (1935). Sur les lois de probabilitis a estimation exhaustif, Compres Rendus de 1’Acadimie des Sciences, Paris, 200, 1265-1267.

[4] Koopman, B. 0. (1936). On distributions admitting a sufficient statistic, Transactions of the American Mathematical Society, 39, 339-409.

[5] Pitman (1936). Sufficient Statistics and Intrinsic Accuracy. Proc. Camb. Phi. Soc. 32: 567-569.

[6] Morris, C. N. (1983). Natural exponential families with quadratic variance function, Annals of Statistics, 11, 515-529.

[7] Kubacek, L. (2012). Foundations of Estimation Theory. Elsevier Science.

[8] American Mathematical Society (2008). Selected Papers on Analysis and Related Topics. American Mathematical Society.

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