Conway–Maxwell–binomial distribution

In probability theory and statistics, the Conway–Maxwell–binomial (CMB) distribution is a three parameter discrete probability distribution that generalises the binomial distribution in an analogous manner to the way that the Conway–Maxwell–Poisson distribution generalises the Poisson distribution. The CMB distribution can be used to model both positive and negative association among the Bernoulli summands,.[1][2]

Conway–Maxwell–binomial
Parameters
Support
PMF
CDF
Mean Not listed
Median No closed form
Mode See text
Variance Not listed
Skewness Not listed
Excess kurtosis Not listed
Entropy Not listed
MGF See text
CF See text

The distribution was introduced by Shumeli et al. (2005),[1] and the name Conway–Maxwell–binomial distribution was introduced independently by Kadane (2016) [2] and Daly and Gaunt (2016).[3]

Probability mass function

edit

The Conway–Maxwell–binomial (CMB) distribution has probability mass function

 

where  ,   and  . The normalizing constant   is defined by

 

If a random variable   has the above mass function, then we write  .

The case   is the usual binomial distribution  .

Relation to Conway–Maxwell–Poisson distribution

edit

The following relationship between Conway–Maxwell–Poisson (CMP) and CMB random variables [1] generalises a well-known result concerning Poisson and binomial random variables. If   and   are independent, then  .

Sum of possibly associated Bernoulli random variables

edit

The random variable   may be written [1] as a sum of exchangeable Bernoulli random variables   satisfying

 

where  . Note that   in general, unless  .

Generating functions

edit

Let

 

Then, the probability generating function, moment generating function and characteristic function are given, respectively, by:[2]

 
 
 

Moments

edit

For general  , there do not exist closed form expressions for the moments of the CMB distribution. The following neat formula is available, however.[3] Let   denote the falling factorial. Let  , where  . Then

 

for  .

Mode

edit

Let   and define

 

Then the mode of   is   if   is not an integer. Otherwise, the modes of   are   and  .[3]

Stein characterisation

edit

Let  , and suppose that   is such that   and  . Then [3]

 

Approximation by the Conway–Maxwell–Poisson distribution

edit

Fix   and   and let   Then   converges in distribution to the   distribution as  .[3] This result generalises the classical Poisson approximation of the binomial distribution.

Conway–Maxwell–Poisson binomial distribution

edit

Let   be Bernoulli random variables with joint distribution given by

 

where   and the normalizing constant   is given by

 

where

 

Let  . Then   has mass function

 

for  . This distribution generalises the Poisson binomial distribution in a way analogous to the CMP and CMB generalisations of the Poisson and binomial distributions. Such a random variable is therefore said [3] to follow the Conway–Maxwell–Poisson binomial (CMPB) distribution. This should not be confused with the rather unfortunate terminology Conway–Maxwell–Poisson–binomial that was used by [1] for the CMB distribution.

The case   is the usual Poisson binomial distribution and the case   is the   distribution.

References

edit
  1. ^ a b c d e Shmueli G., Minka T., Kadane J.B., Borle S., and Boatwright, P.B. "A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution." Journal of the Royal Statistical Society: Series C (Applied Statistics) 54.1 (2005): 127–142.[1]
  2. ^ a b c Kadane, J.B. " Sums of Possibly Associated Bernoulli Variables: The Conway–Maxwell–Binomial Distribution." Bayesian Analysis 11 (2016): 403–420.
  3. ^ a b c d e f Daly, F. and Gaunt, R.E. " The Conway–Maxwell–Poisson distribution: distributional theory and approximation." ALEA Latin American Journal of Probabability and Mathematical Statistics 13 (2016): 635–658.