Beta negative binomial distribution

Beta Negative Binomial

Parameters	[math]\displaystyle{ \alpha \gt 0 }[/math] shape (real) [math]\displaystyle{ \beta \gt 0 }[/math] shape (real) [math]\displaystyle{ r \gt 0 }[/math] — number of successes until the experiment is stopped (integer but can be extended to real)
Support	[math]\displaystyle{ k \in \{0,1,2,\ldots\} }[/math]
pmf	[math]\displaystyle{ \frac{\Beta(r+k,\alpha+\beta)}{\Beta(r,\alpha)}\frac{\Gamma(k+\beta)}{k!\;\Gamma(\beta)} }[/math]
Mean	[math]\displaystyle{ \begin{cases} \frac{r\beta}{\alpha-1} & \text{if}\ \alpha\gt 1 \\ \infty & \text{otherwise}\ \end{cases} }[/math]
Variance	[math]\displaystyle{ \begin{cases} \frac{r\beta(r+\alpha-1)(\beta+\alpha-1)}{(\alpha-2){(\alpha-1)}^2} & \text{if}\ \alpha\gt 2 \\ \infty & \text{otherwise}\ \end{cases} }[/math]
Skewness	[math]\displaystyle{ \begin{cases} \frac{(2r+\alpha-1)(2\beta+\alpha-1)}{(\alpha-3)\sqrt{\frac{r\beta(r+\alpha-1)(\beta+\alpha-1)}{\alpha-2}}} & \text{if}\ \alpha\gt 3 \\ \infty & \text{otherwise}\ \end{cases} }[/math]
MGF	does not exist
CF	[math]\displaystyle{ \frac{\Gamma(\alpha+r)\Gamma(\alpha+\beta)} {\Gamma(\alpha+\beta+r)\Gamma(\alpha)} {}_{2}F_{1}(r,\beta;\alpha+\beta+r;e^{it})\! }[/math] where [math]\displaystyle{ \Gamma }[/math] is the gamma function and [math]\displaystyle{ {}_{2}F_{1} }[/math] is the hypergeometric function.

In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable [math]\displaystyle{ X }[/math] equal to the number of failures needed to get [math]\displaystyle{ r }[/math] successes in a sequence of independent Bernoulli trials. The probability [math]\displaystyle{ p }[/math] 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 or simply abbreviated as the BNB distribution. A shifted form of the distribution has been called the beta-Pascal distribution.

If parameters of the beta distribution are [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math], and if

[math]\displaystyle{ X \mid p \sim \mathrm{NB}(r,p), }[/math]

where

[math]\displaystyle{ p \sim \textrm{B}(\alpha,\beta), }[/math]

then the marginal distribution of [math]\displaystyle{ X }[/math] is a beta negative binomial distribution:

[math]\displaystyle{ X \sim \mathrm{BNB}(r,\alpha,\beta). }[/math]

In the above, [math]\displaystyle{ \mathrm{NB}(r,p) }[/math] is the negative binomial distribution and [math]\displaystyle{ \textrm{B}(\alpha,\beta) }[/math] is the beta distribution.

Definition and derivation

Denoting [math]\displaystyle{ f_{X|p}(k|q), f_{p}(q|\alpha,\beta) }[/math] the densities of the negative binomial and beta distributions respectively, we obtain the PMF [math]\displaystyle{ f(k|\alpha,\beta,r) }[/math] of the BNB distribution by marginalization:

[math]\displaystyle{ f(k|\alpha,\beta,r) = \int_0^1 f_{X|p}(k|r,q) \cdot f_{p}(q|\alpha,\beta) \mathrm{d} q = \int_0^1 \binom{k+r-1}{k} (1-q)^k q^r \cdot \frac{q^{\alpha-1}(1-q)^{\beta-1}} {\Beta(\alpha,\beta)} \mathrm{d} q = \frac{1}{\Beta(\alpha,\beta)} \binom{k+r-1}{k} \int_0^1 q^{\alpha+r-1}(1-q)^{\beta+k-1} \mathrm{d} q }[/math]

Noting that the integral evaluates to:

[math]\displaystyle{ \int_0^1 q^{\alpha+r-1}(1-q)^{\beta+k-1} \mathrm{d} q = \frac{\Gamma(\alpha+r)\Gamma(\beta+k)}{\Gamma(\alpha+\beta+k+r)} }[/math]

we can arrive at the following formulas by relatively simple manipulations.

If [math]\displaystyle{ r }[/math] is an integer, then the PMF can be written in terms of the beta function,:

[math]\displaystyle{ f(k|\alpha,\beta,r)=\binom{r+k-1}k\frac{\Beta(\alpha+r,\beta+k)}{\Beta(\alpha,\beta)} }[/math].

More generally, the PMF can be written

[math]\displaystyle{ f(k|\alpha,\beta,r)=\frac{\Gamma(r+k)}{k!\;\Gamma(r)}\frac{\Beta(\alpha+r,\beta+k)}{\Beta(\alpha,\beta)} }[/math]

or

[math]\displaystyle{ f(k|\alpha,\beta,r)=\frac{\Beta(r+k,\alpha+\beta)}{\Beta(r,\alpha)}\frac{\Gamma(k+\beta)}{k!\;\Gamma(\beta)} }[/math].

PMF expressed with Gamma

Using the properties of the Beta function, the PMF with integer [math]\displaystyle{ r }[/math] can be rewritten as:

[math]\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)} }[/math].

More generally, the PMF can be written as

[math]\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)} }[/math].

PMF expressed with the rising Pochammer symbol

The PMF is often also presented in terms of the Pochammer symbol for integer [math]\displaystyle{ r }[/math]

[math]\displaystyle{ f(k|\alpha,\beta,r)=\frac{r^{(k)}\alpha^{(r)}\beta^{(k)}}{k!(\alpha+\beta)^{(r+k)}} }[/math]

Properties

Factorial Moments

The k-th factorial moment of a beta negative binomial random variable X is defined for [math]\displaystyle{ k \lt \alpha }[/math] and in this case is equal to

[math]\displaystyle{ \operatorname{E}\bigl[(X)_k\bigr] = \frac{\Gamma(r+k)}{\Gamma(r)}\frac{\Gamma(\beta+k)}{\Gamma(\beta)}\frac{\Gamma(\alpha-k)}{\Gamma(\alpha)}. }[/math]

Non-identifiable

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

Relation to other distributions

The beta negative binomial distribution contains the beta geometric distribution as a special case when either [math]\displaystyle{ r=1 }[/math] or [math]\displaystyle{ \beta=1 }[/math]. It can therefore approximate the geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large [math]\displaystyle{ \alpha }[/math]. It can therefore approximate the Poisson distribution arbitrarily well for large [math]\displaystyle{ \alpha }[/math], [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ r }[/math].

Heavy tailed

By Stirling's approximation to the beta function, it can be easily shown that for large [math]\displaystyle{ k }[/math]

[math]\displaystyle{ f(k|\alpha,\beta,r) \sim \frac{\Gamma(\alpha+r)}{\Gamma(r)\Beta(\alpha,\beta)}\frac{k^{r-1}}{(\beta+k)^{r+\alpha}} }[/math]

which implies that the beta negative binomial distribution is heavy tailed and that moments less than or equal to [math]\displaystyle{ \alpha }[/math] do not exist.

Beta geometric distribution

The beta geometric distribution is an important special case of the beta negative binomial distribution occurring for [math]\displaystyle{ r=1 }[/math]. In this case the pmf simplifies to

[math]\displaystyle{ f(k|\alpha,\beta)=\frac{\mathrm{B}(\alpha+1,\beta+k)} {\mathrm{B}(\alpha,\beta)} }[/math].

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

Further, when [math]\displaystyle{ \beta=1 }[/math] 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 [math]\displaystyle{ X \sim BG(\alpha,1) }[/math] then [math]\displaystyle{ X+1 \sim YS(\alpha) }[/math].

Beta negative binomial as a Pólya urn model

In the case when the 3 parameters [math]\displaystyle{ r, \alpha }[/math] and [math]\displaystyle{ \beta }[/math] are positive integers, the Beta negative binomial can also be motivated by an urn model - or more specifically a basic Pólya urn model. Consider an urn initially containing [math]\displaystyle{ \alpha }[/math] red balls (the stopping color) and [math]\displaystyle{ \beta }[/math] blue balls. At each step of the model, a ball is drawn at random from the urn and replaced, along with one additional ball of the same color. The process is repeated over and over, until [math]\displaystyle{ r }[/math] red colored balls are drawn. The random variable [math]\displaystyle{ X }[/math] of observed draws of blue balls are distributed according to a [math]\displaystyle{ \mathrm{BNB}(r, \alpha, \beta) }[/math]. Note, at the end of the experiment, the urn always contains the fixed number [math]\displaystyle{ r+\alpha }[/math] of red balls while containing the random number [math]\displaystyle{ X+\beta }[/math] blue balls.

By the non-identifiability property, [math]\displaystyle{ X }[/math] can be equivalently generated with the urn initially containing [math]\displaystyle{ \alpha }[/math] red balls (the stopping color) and [math]\displaystyle{ r }[/math] blue balls and stopping when [math]\displaystyle{ \beta }[/math] red balls are observed.

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