Beta negative binomial distribution

In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable X equal to the number of failures needed to get r successes in a sequence of independent Bernoulli trials where the probability p of success on each trial is constant within any given experiment but is itself a random variable following a beta distribution, varying between different experiments. Thus the distribution is a compound probability distribution.

Beta Negative Binomial

Parameters	${\displaystyle \alpha >0}$ shape (real) ${\displaystyle \beta >0}$ shape (real) ${\displaystyle r>0}$ — number of failures until the experiment is stopped (integer but can be extended to real)
Support	k ∈ { 0, 1, 2, 3, ... }
pmf	${\displaystyle {\frac {\Gamma (r+k)}{k!\;\Gamma (r)}}{\frac {\mathrm {B} (\alpha +r,\beta +k)}{\mathrm {B} (\alpha ,\beta )}}}$
Mean	${\displaystyle {\begin{cases}{\frac {r\beta }{\alpha -1}}&{\text{if}}\ \alpha >1\\\infty &{\text{otherwise}}\ \end{cases}}}$
Variance	${\displaystyle {\begin{cases}{\frac {r(\alpha +r-1)\beta (\alpha +\beta -1)}{(\alpha -2){(\alpha -1)}^{2}}}&{\text{if}}\ \alpha >2\\\infty &{\text{otherwise}}\ \end{cases}}}$
Skewness	${\displaystyle {\begin{cases}{\frac {(\alpha +2r-1)(\alpha +2\beta -1)}{(\alpha -3){\sqrt {\frac {r(\alpha +r-1)\beta (\alpha +\beta -1)}{\alpha -2}}}}}&{\text{if}}\ \alpha >3\\\infty &{\text{otherwise}}\ \end{cases}}}$
MGF	undefined
CF	${\displaystyle {\frac {\Gamma (\alpha +r)\Gamma (\alpha +\beta )}{\Gamma (\alpha +\beta +r)\Gamma (\alpha )}}{}_{2}F_{1}(r,\beta$ ;\alpha +\beta +r;e^{it})\!} where ${\displaystyle \Gamma }$ is the gamma function and ${\displaystyle {}_{2}F_{1}}$ is the hypergeometric function.

This distribution has also been called both the inverse Markov-Pólya distribution and the generalized Waring distribution.[1] A shifted form of the distribution has been called the beta-Pascal distribution.[1]

If parameters of the beta distribution are α and β, and if

${\displaystyle X\mid p\sim \mathrm {NB} (r,p),}$

where

${\displaystyle p\sim {\textrm {B}}(\alpha ,\beta ),}$

then the marginal distribution of X is a beta negative binomial distribution:

${\displaystyle X\sim \mathrm {BNB} (r,\alpha ,\beta ).}$

In the above, NB(r, p) is the negative binomial distribution and B(α, β) is the beta distribution.

Definition

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

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

PMF expressed with Gamma

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

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

PMF expressed with the rising Pochammer symbol

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

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

Properties

Non-identifiable

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.

Relation to other distributions

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 ${\displaystyle r}$.

Heavy tailed

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.

See also

• Negative binomial distribution
• Dirichlet negative multinomial distribution

Notes

1. Johnson et al. (1993)

References

• Jonhnson, 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

External links

• Interactive graphic: Univariate Distribution Relationships