 Conway–Maxwell–Poisson distribution

Conway–Maxwell–Poisson parameters: support: pmf: cdf: mean: median: No closed form mode: Not listed variance: skewness: Not listed ex.kurtosis: Not listed entropy: Not listed mgf: Not listed cf: Not listed In probability theory and statistics, the Conway–Maxwell–Poisson (CMP or COMPoisson) 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. It is a member of the exponential family, has the Poisson distribution and geometric distribution as special cases and the Bernoulli distribution as a limiting case.
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Conway–Maxwell–Poisson distribution
The COMPoisson distribution was originally proposed by Conway and Maxwell in 1962 ^{[1]} as a solution to handling queueing systems with statedependent service rates. The probabilistic and statistical properties of the distribution were published by Shmueli et al. (2005).^{[2]}
The COMPoisson is defined to be the distribution with probability mass function
for x = 0,1,2,... , λ > 0 and ν ≥ 0, where
The function Z(λ,ν) serves as a normalization constant so the probability mass function sums to one. Note that Z(λ,ν) does not have a closed form.
The additional parameter ν which does not appear in the Poisson distribution allows for adjustment of the rate of decay. This rate of decay is a nonlinear decrease in ratios of successive probabilities, specifically
When ν = 1, the COMPoisson distribution becomes the standard Poisson distribution and as , the distribution approaches a Bernoulli distribution with parameter λ / (1 + λ). When ν = 0 the CoMPoisson distribution reduces to a geometric distribution with probability of success 1 − λ provided λ < 1.
For the COMPoisson distribution, moments can be found through the recursive formula
Parameter estimation
There are a few methods of estimating the parameters of the CMP distribution from the data. Two methods will be discussed, the "quick and crude method" and the "accurate and intensive method".
Quick and crude method: weighted least squares
The "quick and crude method" 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. Following the use of this method, an alternative method should be employed to compute more accurate estimates of the parameters if the model is deemed appropriate.
This method uses the relationship of successive probabilities as discussed above. By taking logarithms of both sides of this equation, the following linear relationship arises
where p_{x} denotes . When estimating the parameters, the probabilities can be replaced by the relative frequencies of x and x − 1. To determine if the CMP distribution is an appropriate model, these values should be plotted against log x for all ratios without zero counts. If the data appear to be linear, then the model is likely to be a good fit.
Once the appropriateness of the model is determined, the parameters can be estimated by fitting a regression of on log x. 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 onestep covariances on the first offdiagonal, both given below.
Accurate and intensive method: maximum likelihood
The COMPoisson likelihood function is
where and . 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. In each iteration, the expectations, variances, and covariance of X and log X! are approximated by using the estimates for λ and ν from the previous iteration in the expression
This is continued until convergence of and .
Generalized linear model
The basic COMPoisson distribution discussed above has also been used as the basis for a generalized linear model (GLM) using a Bayesian formulation. A duallink GLM based on the CMP distribution has been developed,^{[3]} and this model has been used to evaluate traffic accident data.^{[4]}^{[5]} The CMP GLM developed by Guikema and Coffelt (2008) is based on a reformulation of the CMP distribution above, replacing λ with μ = λ^{1 / ν}. The integral part of μ is then the mode of the distribution. A full Bayesian estimation approach has been used with MCMC sampling implemented in WinBugs with noninformative priors for the regression parameters.^{[3]}^{[4]} 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.
A classical GLM formulation for a COMPoisson regression has been developed which generalizes Poisson regression and logistic regression.^{[6]} This takes advantage of the exponential family properties of the COMPoisson distribution to obtain elegant model estimation (via maximum likelihood), inference, diagnostics, and interpretation. This approach requires substantially less computational time than the Bayesian approach, at the cost of not allowing expert knowledge to be incorporated into the model.^{[6]} 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. Code for fitting a COMPoisson regression, testing for dispersion, and evaluating fit is available.^{[7]}
The two GLM frameworks developed for the COMPoisson distribution significantly extend the usefulness of this distribution for data analysis problems.
References
 ^ Conway, R. W.; Maxwell, W. L. (1962), "A queuing model with state dependent service rates", Journal of Industrial Engineering 12: 132–136
 ^ 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]
 ^ ^{a} ^{b} Guikema, S.D. and J.P. Coffelt (2008) "A Flexible Count Data Regression Model for Risk Analysis", Risk Analysis, 28 (1), 213–223. doi:10.1111/j.15396924.2008.01014.x
 ^ ^{a} ^{b} Lord, D., S.D. Guikema, and S.R. Geedipally (2008) "Application of the Conway–Maxwell–Poisson Generalized Linear Model for Analyzing Motor Vehicle Crashes," Accident Analysis & Prevention, 40 (3), 1123–1134. doi:10.1016/j.aap.2007.12.003
 ^ Lord, D., S.R. Geedipally, and S.D. Guikema (2010) "Extension of the Application of ConwayMaxwellPoisson Models: Analyzing Traffic Crash Data Exhibiting UnderDispersion," Risk Analysis, 30 (8), 12681276. doi:10.1111/j.15396924.2010.01417.x
 ^ ^{a} ^{b} Sellers, K. S. and Shmueli, G. (2010), "A Flexible Regression Model for Count Data", Annals of Applied Statistics, 4 (2), 943961
 ^ Code for COM_Poisson modelling, Georgetown Univ.
External links
 Conway–Maxwell–Poisson distribution package for R (compoisson) by Jeffrey Dunn, part of Comprehensive R Archive Network (CRAN)
 Conway–Maxwell–Poisson distribution package for R (compoisson) by Tom Minka, third party package
Categories: Discrete distributions
 Poisson processes
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