Beta negative binomial distribution

Beta Negative Binomial

Parameters	$\alpha > 0$ shape (real) $\beta > 0$ shape (real) $r > 0$ — number of failures until the experiment is stopped (integer but can be extended to real)
Support	k ∈ { 0, 1, 2, 3, ... }
pmf	$\frac{\Gamma(r+k)}{k!\;\Gamma(r)} \frac{\mathrm{B}(\alpha+r,\beta+k)} {\mathrm{B}(\alpha,\beta)}$
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	undefined
CF	$\frac{\mathrm{B}(\alpha,\beta+r)} {\mathrm{B}(\alpha,\beta)} {}_{2}F_{1}(r,\alpha;\alpha+\beta+r;e^{it})\!$ where B is the beta function and 2F1 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 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.

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

Recurrence relation

$\left\{(k+1) p(k+1) (\alpha +\beta +k+r)+(\beta +k) (-k-r) p(k)=0,p(0)=\frac{(\alpha )_r}{(\alpha +\beta )_r}\right\}$

Definition

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

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

More generally the PMF can be written

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

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 $r$

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

Properties

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 $r$.

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

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

which implies that the beta negative binomial distribution is heavy tailed.

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