In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COM–Poisson) distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion.
[2] The CMP distribution was originally proposed by Conway and Maxwell in 1962[3] as a solution to handling queueing systems with state-dependent service rates.
[2] The first detailed investigation into the probabilistic and statistical properties of the distribution was published by Shmueli et al.
serves as a normalization constant so the probability mass function sums to one.
This rate of decay is a non-linear decrease in ratios of successive probabilities, specifically When
[2] For the CMP distribution, moments can be found through the recursive formula [2] For general
, there does not exist a closed form formula for the cumulative distribution function of
is an integer, we can, however, obtain the following formula in terms of the generalized hypergeometric function:[6] Many important summary statistics, such as moments and cumulants, of the CMP distribution can be expressed in terms of the normalizing constant
does not in general have a closed form, there are some noteworthy special cases: Because the normalizing constant does not in general have a closed form, the following asymptotic expansion is of interest.
, there does not exist closed form formulas for the mean, variance and moments of the CMP distribution.
[7] This result generalises the classical Poisson approximation of the binomial distribution.
[7] Apart from the fact that COM-binomial approximates to COM-Poisson, Zhang et al. (2018)[9] illustrates that COM-negative binomial distribution with probability mass function convergents to a limiting distribution which is the COM-Poisson, as
There are a few methods of estimating the parameters of the CMP distribution from the data.
Two methods will be discussed: weighted least squares and maximum likelihood.
The weighted least squares approach is simple and efficient but lacks precision.
Maximum likelihood, on the other hand, is precise, but is more complex and computationally intensive.
The weighted least squares provides a simple, efficient method to derive rough estimates of the parameters of the CMP distribution and determine if the distribution would be an appropriate model.
By taking logarithms of both sides of this equation, the following linear relationship arises where
To determine if the CMP distribution is an appropriate model, these values should be plotted against
Once the appropriateness of the model is determined, the parameters can be estimated by fitting a regression of
However, the basic assumption of homoscedasticity is violated, so a weighted least squares regression must be used.
The inverse weight matrix will have the variances of each ratio on the diagonal with the one-step covariances on the first off-diagonal, both given below.
Maximizing the likelihood yields the following two equations which do not have an analytic solution.
Instead, the maximum likelihood estimates are approximated numerically by the Newton–Raphson method.
A dual-link GLM based on the CMP distribution has been developed,[10] and this model has been used to evaluate traffic accident data.
A full Bayesian estimation approach has been used with MCMC sampling implemented in WinBugs with non-informative priors for the regression parameters.
[10][11] This approach is computationally expensive, but it yields the full posterior distributions for the regression parameters and allows expert knowledge to be incorporated through the use of informative priors.
[13] This takes advantage of the exponential family properties of the CMP distribution to obtain elegant model estimation (via maximum likelihood), inference, diagnostics, and interpretation.
[13] In addition it yields standard errors for the regression parameters (via the Fisher Information matrix) compared to the full posterior distributions obtainable via the Bayesian formulation.
It also provides a statistical test for the level of dispersion compared to a Poisson model.