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})\!

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.[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

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.

Notes

  1. ↑ 1.0 1.1 Johnson et al. (1993)

References

External links