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Conway–Maxwell–binomial distribution

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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

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]

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

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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

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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

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

where . Note that in general, unless .

Generating functions

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Let

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

Moments

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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

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Let and define

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

Stein characterisation

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

Approximation by the Conway–Maxwell–Poisson distribution

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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

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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

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  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.