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.^[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 $r$ .

By Sterling'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.

Notes

1 2 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

Probability distributions

List of probability distributions

Discrete univariate with finite support	Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory-Schulz Fréchet Gamma Gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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