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Concept
Beta negative binomial distribution
Summary
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
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 k-th factorial moment of a beta negative binomial random variable X is defined for and in this case is equal to
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.
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