I had a homework assignment to express the negative binomial distribution as an exponential family of distributions given that the dispersion parameter was a known constant. This was fairly easy, but I wondered why they would require we held that parameter fixed. I found that I couldn't come up with a way to put it in the right form with the two parameters being unknown.
Looking online, I found claims that it is not possible. However, I have found no proof that this is true. I can't seem to come up with one myself either. Does anybody have a proof of this?
As requested below, I have attached a couple of the claims:
"The family of negative binomial distributions with fixed number of failures (a.k.a. stopping-time parameter) r is an exponential family. However, when any of the above-mentioned fixed parameters are allowed to vary, the resulting family is not an exponential family." http://en.wikipedia.org/wiki/Exponential_family
"The two-parameter negative binomial distribution is not a member of the exponential family. But if we treat the dispersion parameter as a known, fixed constant, then it is a member." http://www.unc.edu/courses/2006spring/ecol/145/001/docs/lectures/lecture21.htm