Relationship between exponential families and moment generating functions

Question

I have recently been playing around with some change of measure arguments for shifting the mean of a sub-gaussian distribution. It occurred to me however, that sub-gaussianity might not be the natural property to consider, but rather being a member of an exponential family (as this simplifies the likelihood transform induced by an exponential tilting).

Now, I must admit that I do not know much about these, but I encountered a rather miraculous claim on wikipedia, https://en.wikipedia.org/wiki/Natural_exponential_family:

"Every distribution possessing a moment-generating function is a member of a natural exponential family."

As I figured it fair to try to read up a bit before asking stupid questions, I familiarized myself with a paper which seems to prove this result partially:

SAMPSON, Allan R. Characterizing exponential family distributions by moment generating functions. The Annals of Statistics, 1975, 747-753.

Here, the author shows that if the moment generating function is of a particular form, then, the distribution is a member of an exponential family. There however still seems to be a leap, one which I am not able to make by myself, to the claim I quoted; in particular, the hypotheses of the claim seem to be weaker and the conclusion stronger (as we have narrowed down to a natural exponential family)!

I would very much appreciate a reference on this result, preferably with proof (as it seems to me an amazing property which I cannot entirely wrap my head around...) Of course, if anyone knows the proof, I would be even happier if that person answered. Thank you in advance!

Answer 1

To link the existence of the moment-generating function and the inclusion in an exponential family is absurd in that the former is a property of a given single distribution while the later is a characteristic of a family of distributions parameterised with a finite dimensional parameter. Note that a given arbitrary distribution with density $f_0$ can always induce an exponential family by considering an arbitrary transform $T$ and the family of densities $$f_\theta(x|)=f_0(x)\exp\{T(x)\cdot\theta-\psi(\theta)\}$$ In particular, the exponentially tilded measure is defined by $$f_θ(x) = \dfrac{\exp\{θ\cdot x\}f_0(x)}{M_X(θ)}$$ where $M_X(\theta)$ is the moment-generating function for $X$.

Furthermore, the moment-generating function is associated with a specific representation of a distribution, namely, $\mathbb{E}^X[\exp\{t X\}]$ differs from $\mathbb{E}^X[\exp\{t \xi(X)\}]$, while exponential families are equivariant under one-to-one transforms. Moment-generating functions can be found for all distributions with compacts support (or alternatively for transforms $\xi(\cdot)$ taking values in a compact set). They also exist for some families of distributions that do not constitute an exponential family, like the family of double-exponential or Laplace distributions, and the family of non-central chi-squared distributions.