non-exponential family probability distributions and their uses I was actually trying to find information on non-exponential family probability distributions. So many of the distributions that we study in statistics are members of an exponential family of distributions. But this leads to the natural question of whether there exist sets of distributions that are not members of an exponential family? I could not find any references to these. 
Does anyone know of distributions that are not members of an exponential family? Also, what are the applications for these types of distributions? Where are they used?
Thanks.
 A: A simple example is the uniform distribution, which is not a member of the exponential family. Neither is Student's $t$. I would also guess that the F distribution is not a member either (though I haven't checked). 
The $t$-test is, as I'm guessing you know, very well used and evidently distributions that do not belong to the exponential family are still widely used. The uniform distribution is common too, perhaps due to its simplicity. While not so often used in different tests, it's of course still used (quite often as a prior in Bayesian applications, for instance).
A: I assume you're referring to distributions with unbounded tails. That rules out uniform, beta and so forth. The exponential distributions are a good watermark in the distributions manifold, splitting it into 3 subspaces: exponential, subexponential and superexponential. Examples of each are:
Exponential: Normal
Superexponential:
Subexponential: Lévy, Cauchy, Student t, Pareto, Generalised Pareto, Weibull, Burr, Lognormal, Log-Cauchy, Log-Gamma
The latter, the fat-tailed distributions, are particularly useful to Extreme Value Theory and when applied to the measurement of low-frequency/high-impact risks, such as in operational risk.
A: There are many non-exponential families like the uniform, Cauchy, t-distributions, generalized Gaussian, etc.
However, you can always approximate closely any smooth distribution by one exponential family. These are dense models. See
http://arxiv.org/abs/0911.4863
and also
k-MLE for mixtures of generalized Gaussians
http://www.lix.polytechnique.fr/~schwander/articles/icpr2012.pdf
In other words, location-scale, elliptical, exponential families and heavy-tailed distributions are important equally.
