Beta negative binomial distribution

Denoting ${\displaystyle f_{X|p}(k|q),f_{p}(q|\alpha ,\beta )}$  the densities of the negative binomial and beta distributions respectively, we obtain the PMF ${\displaystyle f(k|\alpha ,\beta ,r)}$  of the BNB distribution by marginalization:

${\displaystyle {\begin{aligned}f(k|\alpha ,\beta ,r)\;=&\;\int _{0}^{1}f_{X|p}(k|r,q)\cdot f_{p}(q|\alpha ,\beta )\mathrm {d} q\\=&\;\int _{0}^{1}{\binom {k+r-1}{k}}(1-q)^{k}q^{r}\cdot {\frac {q^{\alpha -1}(1-q)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\mathrm {d} q\\=&\;{\frac {1}{\mathrm {B} (\alpha ,\beta )}}{\binom {k+r-1}{k}}\int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q\end{aligned}}}$ 

Noting that the integral evaluates to:

${\displaystyle \int _{0}^{1}q^{\alpha +r-1}(1-q)^{\beta +k-1}\mathrm {d} q={\frac {\Gamma (\alpha +r)\Gamma (\beta +k)}{\Gamma (\alpha +\beta +k+r)}}}$ 

we can arrive at the following formulas by relatively simple manipulations.

If ${\displaystyle r}$  is an integer, then the PMF can be written in terms of the beta function,:

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

More generally, the PMF can be written

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

or

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

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 ${\displaystyle r}$ 

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

The k-th factorial moment of a beta negative binomial random variable X is defined for ${\displaystyle k<\alpha }$  and in this case is equal to

${\displaystyle \operatorname {E} {\bigl [}(X)_{k}{\bigr ]}={\frac {\Gamma (r+k)}{\Gamma (r)}}{\frac {\Gamma (\beta +k)}{\Gamma (\beta )}}{\frac {\Gamma (\alpha -k)}{\Gamma (\alpha )}}.}$ 

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. Thus estimation demands that a constraint be placed on ${\displaystyle r}$ , ${\displaystyle \beta }$  or both.

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

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

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

The beta geometric distribution is an important special case of the beta negative binomial distribution occurring for ${\displaystyle r=1}$ . In this case the pmf simplifies to

${\displaystyle f(k|\alpha ,\beta )={\frac {\mathrm {B} (\alpha +1,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}$ .

This distribution is used in some Buy Till you Die (BTYD) models.

Further, when ${\displaystyle \beta =1}$  the beta geometric reduces to the Yule–Simon distribution. However, it is more common to define the Yule-Simon distribution in terms of a shifted version of the beta geometric. In particular, if ${\displaystyle X\sim BG(\alpha ,1)}$  then ${\displaystyle X+1\sim YS(\alpha )}$ .

In the case when the 3 parameters ${\displaystyle r,\alpha }$  and ${\displaystyle \beta }$  are positive integers, the Beta negative binomial can also be motivated by an urn model - or more specifically a basic Pólya urn model. Consider an urn initially containing ${\displaystyle \alpha }$  red balls (the stopping color) and ${\displaystyle \beta }$  blue balls. At each step of the model, a ball is drawn at random from the urn and replaced, along with one additional ball of the same color. The process is repeated over and over, until ${\displaystyle r}$  red colored balls are drawn. The random variable ${\displaystyle X}$  of observed draws of blue balls are distributed according to a ${\displaystyle \mathrm {BNB} (r,\alpha ,\beta )}$ . Note, at the end of the experiment, the urn always contains the fixed number ${\displaystyle r+\alpha }$  of red balls while containing the random number ${\displaystyle X+\beta }$  blue balls.

By the non-identifiability property, ${\displaystyle X}$  can be equivalently generated with the urn initially containing ${\displaystyle \alpha }$  red balls (the stopping color) and ${\displaystyle r}$  blue balls and stopping when ${\displaystyle \beta }$  red balls are observed.

• Negative binomial distribution
• Dirichlet negative multinomial distribution

• Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions, 2nd edition, Wiley ISBN 0-471-54897-9 (Section 6.2.3)
• Kemp, C.D.; Kemp, A.W. (1956) "Generalized hypergeometric distributions, Journal of the Royal Statistical Society, Series B, 18, 202–211
• Wang, Zhaoliang (2011) "One mixed negative binomial distribution with application", Journal of Statistical Planning and Inference, 141 (3), 1153-1160 doi:10.1016/j.jspi.2010.09.020

• Interactive graphic: Univariate Distribution Relationships