Beta Negative Binomial
— number of successes until the experiment is stopped (integer but can be extended to real)
does not exist|
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 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 and , and if
then the marginal distribution of is a beta negative binomial distribution:
In the above, is the negative binomial distribution and is the beta distribution.
Definition and derivation
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
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
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.
Relation to other distributions
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.
Beta geometric distribution
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 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 .