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.


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*Do any textbooks or elementary expository papers give proofs?

*Why is it named after those three persons?

 A: 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:

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*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

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*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). With all due respect, by changing the nature of the statistic, Don Fraser changes the definition of sufficiency and hence the meaning of the Darmois-Koopman-Pitman lemma.
