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.

Beta Negative Binomial
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.

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.

Definition and derivation edit

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

 .

PMF expressed with Gamma edit

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 edit

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

 

Properties edit

Factorial Moments edit

The k-th factorial moment of a beta negative binomial random variable X is defined for   and in this case is equal to

 

Non-identifiable edit

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 edit

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  .

Heavy tailed edit

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 edit

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  .

Beta negative binomial as a Pólya urn model edit

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.


See also edit

Notes edit

  1. ^ a b Johnson et al. (1993)

References edit

  • Johnson, N.L.; Kotz, S.; Kemp, A.W. (1993) Univariate Discrete Distributions, 2nd edition, Wiley ISBN 0-471-54897-9 (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 doi:10.1016/j.jspi.2010.09.020

External links edit

  • Interactive graphic: Univariate Distribution Relationships