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 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. A shifted form of the distribution has been called the beta-Pascal distribution.
If parameters of the beta distribution are α and β, and if
where
then the marginal distribution of X is a beta negative binomial distribution:
In the above, NB is the negative binomial distribution and B is the beta distribution.

Definition

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

PMF expressed with Gamma

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

PMF expressed with the rising Pochammer symbol

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

Properties

Non-identifiable

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.

Relation to other distributions

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

Heavy tailed

By Stirling's approximation to the beta function, it can be easily shown that
which implies that the beta negative binomial distribution is heavy tailed and that moments less than or equal to do not exist.