# Undergraduate-level proofs of the Pitman–Koopman–Darmois theorem

The Pitman–Koopman–Darmois theorem says that if an i.i.d. sample from a parametrized family of probability distributions admits a sufficient statistic whose number of scalar components does not grow with the sample size, then it is an exponential family.

• Do any textbooks or elementary expository papers give proofs?
• Why is it named after those three persons?

The reason the Lemma is called Pitman-Koopman-Darmois is, unsurprisingly, that the three authors established similar versions of the lemma, independently at about the same time:

• Darmois, G. (1935) Sur les lois de probabilité à estimation exhaustive, Comptes Rendus de l'Académie des Sciences, 200, 1265-1266.
• Koopman, B.O. (1936) On Distributions Admitting a Sufﬁcient Statistic, Transactions of the American Mathematical Society, Vol. 39, No. 3. [link]
• Pitman, E.J.G. (1936) Sufficient statistics and intrinsic accuracy, Proceedings of the Cambridge Philosophical Society, 32, 567-579.

following a one-dimensional result in

• Fisher, R.A. (1934) Two new properties of mathematical likelihood, Proceedings of the Royal Society, Series A, 144, 285-307.

I do not know of a non-technical proof of this result. One proof that does not involve complex arguments is Don Fraser's (p.13-16), based on the argument that the likelihood function is a sufficient statistic,with functional value. But I find the argument disputable because statistics are real vectors that are functions of the sample $x$, not functionals (function valued transforms). By changing the nature of the statistic, Don Fraser changes the definition of sufficiency and hence the meaning of the Darmois-Koopman-Pitman lemma.

• +1. Nitpick on the linked Koopman paper in the paragraph following Eq. (6) proving the everywhere vanishing Jacobian: the neighborhood of should not be picked arbitrarily only so that the Jacobian is nonzero. It has to be argued locally for each point $(x_1^0,x_2^0,x_3^0)$ rather than locally. The (defined) existence of the nonzero differential at that point guarantees that there exists a small enough neighborhood of that point such that the left hand side of Eq. (5) in that neighborhood other than that point is always distinct from that at that point. – Hans Feb 26 at 22:51
• It is not true that nonzero Jacobian leads to global unique values in a domain (manifold) as is implied in the paper. It is only true locally. Also the dimensionality is preserved not by homeomorphism as is claimed in the last sentence of that paragraph, but rather by local diffeomorphism, which is the case here. – Hans Feb 26 at 22:54