Parameters |
shape (real) shape (real) — number of successes until the experiment is stopped (integer but can be extended to real) | ||
---|---|---|---|

Support | |||

PMF | |||

Mean | |||

Variance | |||

Skewness | |||

MGF | does not exist | ||

CF | where is the gamma function and is the hypergeometric function. |

In probability theory, a **beta negative binomial distribution** is the probability distribution of a discrete random variable equal to the number of failures needed to get successes in a sequence of independent Bernoulli trials. The probability 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**^{[1]} or simply abbreviated as the **BNB** distribution. A shifted form of the distribution has been called the **beta-Pascal distribution**.^{[1]}

If parameters of the beta distribution are and , and if

where

then the marginal distribution of is a beta negative binomial distribution:

In the above, is the negative binomial distribution and is the beta distribution.

Denoting the densities of the negative binomial and beta distributions respectively, we obtain the PMF of the BNB distribution by marginalization:

Noting that the integral evaluates to:

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

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

- .

More generally, the PMF can be written

or

- .

Using the properties of the Beta function, the PMF with integer can be rewritten as:

- .

More generally, the PMF can be written as

- .

The PMF is often also presented in terms of the Pochammer symbol for integer

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

The beta negative binomial distribution contains the beta geometric distribution as a special case when either or . It can therefore approximate the geometric distribution arbitrarily well. It also approximates the negative binomial distribution arbitrary well for large . It can therefore approximate the Poisson distribution arbitrarily well for large , and .

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

which implies that the beta negative binomial distribution is heavy tailed and that moments less than or equal to do not exist.

The beta geometric distribution is an important special case of the beta negative binomial distribution occurring for . In this case the pmf simplifies to

- .

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

Further, when 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 then .

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

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