Beta Negative Binomial Distribution

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 distributionJohnson et al. (1993) 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 $\alpha$ and $\beta$, and if
:$X \mid p \sim \mathrm(r,p),$
where
:$p \sim \textrm(\alpha,\beta),$
then the marginal distribution of $X$ is a beta negative binomial distribution:
:$X \sim \mathrm(r,\alpha,\beta).$

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

Definition and derivation

Denoting $f_(k, q), f_(q, \alpha,\beta)$ the densities of the negative binomial and beta distributions respectively, we obtain the PMF $f(k, \alpha,\beta,r)$ of the BNB distribution by marginalization:
:$f(k, \alpha,\beta,r) = \int_0^1 f_(k, r,q) \cdot f_(q, \alpha,\beta) \mathrm q = \int_0^1 \binom (1-q)^k q^r \cdot \frac \mathrm q = \frac \binom \int_0^1 q^(1-q)^ \mathrm q$

Noting that the integral evaluates to:
:$\int_0^1 q^(1-q)^ \mathrm q = \frac$
we can arrive at the following formulas by relatively simple manipulations.

If $r$ is an integer, then the PMF can be written in terms of the beta function,:
:$f(k, \alpha,\beta,r)=\binomk\frac$.
More generally, the PMF can be written
:$f(k, \alpha,\beta,r)=\frac\frac$
or
:$f(k, \alpha,\beta,r)=\frac\frac$.

PMF expressed with Gamma

Using the properties of the Beta function, the PMF with integer $r$ can be rewritten as:
:$f(k, \alpha,\beta,r)=\binomk\frac$.

More generally, the PMF can be written as
:$f(k, \alpha,\beta,r)=\frac\frac$.

PMF expressed with the rising Pochammer symbol

The PMF is often also presented in terms of the Pochammer symbol for integer $r$
:$f(k, \alpha,\beta,r)=\frac$

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 either $r=1$ or $\beta=1$. It can therefore approximate the geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large $\alpha$. 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 $k$
:$f(k, \alpha,\beta,r) \sim \frac\frac$
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

:$f(k, \alpha,\beta)=\frac$.

This distribution is used in some 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 $X+1 \sim YS(\alpha)$.

Beta negative binomial as a Pólya urn model

In the case when the 3 parameters $r, \alpha$ and $\beta$ 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 $\alpha$ red balls (the stopping color) and $\beta$ 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 $r$ red colored balls are drawn. The random variable $X$ of observed draws of blue balls are distributed according to a $\mathrm(r, \alpha, \beta)$. Note, at the end of the experiment, the urn always contains the fixed number $r+\alpha$ of red balls while containing the random number $X+\beta$ blue balls.

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

See also

* Negative binomial distribution
* Dirichlet negative multinomial distribution

Notes

References

*Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) ''Univariate Discrete Distributions'', 2nd edition, Wiley (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

External links

* Interactive graphic
Univariate Distribution Relationships

{{ProbDistributions, discrete-infinite

Discrete distributions
Compound probability distributions
Factorial and binomial topics