Beta negative binomial distribution

Beta Negative Binomial

Parameters	$\alpha >0$ shape (real) $\beta >0$ shape (real) $r>0$ — number of successes until the experiment is stopped (integer but can be extended to real)
Support	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle k \in { 0, 1, 2, 3, \dots } }
PMF	${\frac {\mathrm {B} (r+k,\alpha +\beta )}{\mathrm {B} (r,\alpha )}}{\frac {\Gamma (k+\beta )}{k!\;\Gamma (\beta )}}$
Mean	${\begin{cases}{\frac {r\beta }{\alpha -1}}&{\text{if}}\ \alpha >1\\\infty &{\text{otherwise}}\ \end{cases}}$
Variance	${\begin{cases}{\frac {r(\alpha +r-1)\beta (\alpha +\beta -1)}{(\alpha -2){(\alpha -1)}^{2}}}&{\text{if}}\ \alpha >2\\\infty &{\text{otherwise}}\ \end{cases}}$
Skewness	${\begin{cases}{\frac {(\alpha +2r-1)(\alpha +2\beta -1)}{(\alpha -3){\sqrt {\frac {r(\alpha +r-1)\beta (\alpha +\beta -1)}{\alpha -2}}}}}&{\text{if}}\ \alpha >3\\\infty &{\text{otherwise}}\ \end{cases}}$
MGF	does not exist
CF	${\frac {\Gamma (\alpha +r)\Gamma (\alpha +\beta )}{\Gamma (\alpha +\beta +r)\Gamma (\alpha )}}{}_{2}F_{1}(r,\beta ;\alpha +\beta +r;e^{it})\!$ where $\Gamma$ is the gamma function and ${}_{2}F_{1}$ is the hypergeometric function.

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. The probability $p$ of success on each trial stays constant within any given experiment but varies across different experiments following a beta distribution. Thus the distribution is a compound probability distribution.

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

If parameters of the beta distribution are $\alpha$ and $\beta$, and if

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

where

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

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

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

In the above, $\mathrm {NB} (r,p)$ is the negative binomial distribution and ${\textrm {B}}(\alpha ,\beta )$ is the beta distribution.

Definition

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

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

More generally, the PMF can be written

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

or

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

PMF expressed with Gamma

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

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

More generally, the PMF can be written as

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

PMF expressed with the rising Pochammer symbol

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

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

Properties

Non-identifiable

The beta negative binomial is non-identifiable which can be seen easily by simply swapping $r$ and $\beta$ in the above density or characteristic function and noting that it is unchanged. Thus estimation demands that a constraint be placed on $r$, $\beta$ or both.

Relation to other distributions

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

Heavy tailed

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

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

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

Beta geometric distribution

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

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

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

Further, when $\beta =1$ 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 $X\sim BG(\alpha ,1)$ then $X+1\sim YS(\alpha )$.

See also

• Negative binomial distribution
• Dirichlet negative multinomial distribution

References

• 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

External links

• Interactive graphic: Univariate Distribution Relationships