I am fairly new to this topic but here is my problem:
I have stumbled across a paper (Robinson and Smyth, 2008) stating that the sample sum is a sufficient statistic for NB-distributed random variables.
I have tried to verify this by using the Fisher–Neyman factorization theorem $f_\theta(x)=h(x) \, g_\theta(T(x))$.
This is how far I have come: $\frac{\prod\Gamma(x_i+r)}{\prod x_i! \Gamma(r)^n}(1-p)^{n*r}p^{\sum{x_i}}$
It would be easy if it weren't for the Gamma function in the numerator as then $h(x)=\frac{1}{\prod x_i!}$ and $g_\theta(T(x))=\Gamma(r)^{-n}(1-p)^{n*r}p^{\sum{x_i}}$.
If I am on the wrong path, could somebody please help me solve this?