## Darmois, Koopman, and Pitman

**W**hen [X’ed] seeking a simple proof of the Pitman-Koopman-Darmois lemma [that exponential families are the only types of distributions with constant support allowing for a fixed dimension sufficient statistic], I came across a 1962 Stanford technical report by Don Fraser containing a short proof of the result. Proof that I do not fully understand as it relies on the notion that the likelihood function itself is a minimal sufficient statistic.

April 11, 2022 at 11:02 am

[…] In general it says the Student’s t distribution is not an exponential family and therefore cannot have a conjugate prior. The fact that the Student’s t distribution cannot enjoy a conjugate family (other than the trivial collection of all probability distributions) over the parameter space is connected with the Darmois-Pitman-Koopman lemma. […]