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Question: What was the first recorded use of someone calling exponential families (in probability/statistics) for which the dimension of the natural parameter space is strictly less than the dimension of the ambient space containing it?

(When the inequality isn't strict, i.e. the affine hull of the natural parameter space is all of the parameter space's ambient space, then the family is called a full-rank exponential family.)

See, e.g. here: http://yaroslavvb.com/papers/notes/lauritzen-expfams.pdf , or Keener, Theoretical Statistics: Topics for a Core Course.

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According to this paper on the Penn State site, page 5 (PDF page 7/22):

The family of distributions $$P(Y=y) = exp\{ \eta(\theta)^tZ(y) - \psi[\eta(\theta)]\} , \theta \in R^q $$ is called the curved exponential family in the terminology of Efron (1975).

The Efron (1975) reference is then given as :

Efron, B. (1975), Defining the curvature of a statistical problem (with applications to second order efficiency) (with discussion), Annals of Statistics, 3: 1189–1242.


EDIT I've just noticed that the above link does not seem to load correctly. If you have problems view this PDF, here is an alternative link : Penn State site paper - here see page 7.

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