Beta negative binomial distribution

Using the properties of the Beta function, the PMF with integer ${\displaystyle r}$ can be rewritten as:

${\displaystyle f(k|\alpha ,\beta ,r)={\binom {r+k-1}{k}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}}$.

More generally, the PMF can be written as

${\displaystyle f(k|\alpha ,\beta ,r)={\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\Gamma (\alpha +r)\Gamma (\beta +k)\Gamma (\alpha +\beta )}{\Gamma (\alpha +r+\beta +k)\Gamma (\alpha )\Gamma (\beta )}}}$.

The PMF is often also presented in terms of the Pochammer symbol for integer ${\displaystyle r}$

${\displaystyle f(k|\alpha ,\beta ,r)={\frac {r^{(k)}\alpha ^{(r)}\beta ^{(k)}}{k!(\alpha +\beta )^{(r)}(r+\alpha +\beta )^{(k)}}}}$

The beta negative binomial is non-identifiable which can be seen easily by simply swapping ${\displaystyle r}$ and ${\displaystyle \beta }$ in the above density or characteristic function and noting that it is unchanged.

The beta negative binomial distribution contains the beta geometric distribution as a special case when ${\displaystyle r=1}$. It can therefore approximate the geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. It can therefore approximate the Poisson distribution arbitrarily well for large ${\displaystyle \alpha }$, ${\displaystyle \beta }$ and ${\displaystyle r}$.

By Stirling's approximation to the beta function, it can be easily shown that

${\displaystyle f(k|\alpha ,\beta ,r)\sim {\frac {\Gamma (\alpha +r)}{\Gamma (r)\mathrm {B} (\alpha ,\beta )}}{\frac {k^{r-1}}{(\beta +k)^{r+\alpha }}}}$

which implies that the beta negative binomial distribution is heavy tailed and that moments less than or equal to ${\displaystyle \alpha }$ do not exist.