1. Probability mass function

2. Relation to Conway–Maxwell–Poisson distribution

3. Sum of possibly associated Bernoulli random variables

4. Generating functions

5. Moments

6. Mode

7. Stein characterisation

8. Approximation by the Conway–Maxwell–Poisson distribution

9. Conway–Maxwell–Poisson binomial distribution

10. References

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

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

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

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

where . Note that in general, unless .

Generating functions

Let

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

Moments

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

Let and define

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

Stein characterisation

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

Approximation by the Conway–Maxwell–Poisson distribution

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

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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