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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?
Michael Hardy
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1 Answers1

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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). 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.

Xi'an
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    +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 '19 at 22:51
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    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 '19 at 22:54
  • @Hans So, strictly speaking, is Koopman's proof wrong? I can't figure this out right away. – paperskilltrees May 23 '21 at 21:18
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    @paperskilltrees: It has been 2.25 years since I wrote my comment. I will review that paper. Maybe you can remind me of your question next week? – Hans May 23 '21 at 21:45
  • @Hans No worries if you don't have time for this. I hoped this might be something you deal with on a daily basis and still have fresh in your head. – paperskilltrees May 29 '21 at 02:44
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    @paperskilltrees: It is not a topic I deal with daily. But I intend to review it. Thank you for the reminder. I will get to it this week or the next. – Hans Jun 03 '21 at 13:49