Conway–Maxwell–Poisson distribution

The CMP distribution was originally proposed by Conway and Maxwell in 1962{{Citation|last1=Conway| first1=R. W.| last2=Maxwell| first2= W. L.| year=1962| title=A queuing model with state dependent service rates|journal= Journal of Industrial Engineering| volume=12| pages=132–136}} as a solution to handling queueing systems with state-dependent service rates. The CMP distribution was introduced into the statistics literature by Boatwright et al. 2003 Boatwright, P., Borle, S. and Kadane, J.B. "A model of the joint distribution of purchase quantity and timing." Journal of the American Statistical Association 98 (2003): 564–572. and Shmueli et al. (2005).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.[https://dx.doi.org/10.1111/j.1467-9876.2005.00474.x] The first detailed investigation into the

probabilistic and statistical properties of the distribution was published by Shmueli et al. (2005). Some theoretical probability results of COM-Poisson distribution is studied and reviewed by Li et al. (2019),Li B., Zhang H., Jiao H. "Some Characterizations and Properties of COM-Poisson Random Variables." Communications in Statistics - Theory and Methods, (2019).[https://doi.org/10.1080/03610926.2018.1563164] especially the characterizations of COM-Poisson distribution.

The CMP distribution is defined to be the distribution with probability mass function

:$$

P(X = x) = f(x; \lambda, \nu) = \frac{\lambda^x}{(x!)^\nu}\frac{1}{Z(\lambda,\nu)}.

where :

:$$

Z(\lambda,\nu) = \sum_{j=0}^\infty \frac{\lambda^j}{(j!)^\nu}.

The function $Z(\backslash lambda,\backslash nu)$ serves as a normalization constant so the probability mass function sums to one. Note that $Z(\backslash lambda,\backslash nu)$ does not have a closed form.

The domain of admissible parameters is $\backslash lambda,\backslash nu>0$, and , $\backslash nu=0$.

The additional parameter $\backslash nu$ which does not appear in the Poisson distribution allows for adjustment of the rate of decay. This rate of decay is a non-linear decrease in ratios of successive probabilities, specifically

:$$

\frac{P(X = x-1)}{P(X = x)} = \frac{x^\nu}{\lambda}.

When $\backslash nu\; =\; 1$, the CMP distribution becomes the standard Poisson distribution and as $\backslash nu\; \backslash to\; \backslash infty$, the distribution approaches a Bernoulli distribution with parameter $\backslash lambda/(1+\backslash lambda)$. When $\backslash nu=0$ the CMP distribution reduces to a geometric distribution with probability of success $1-\backslash lambda$ provided .

For the CMP distribution, moments can be found through the recursive formula

:$$

\operatorname{E}[X^{r+1}] = \begin{cases}

\lambda \, \operatorname{E}[X+1]^{1-\nu} & \text{if } r = 0 \\

\lambda \, \frac{d}{d\lambda}\operatorname{E}[X^r] + \operatorname{E}[X]\operatorname{E}[X^r] & \text{if } r > 0. \\

\end{cases}

For general $\backslash nu$, there does not exist a closed form formula for the cumulative distribution function of $X\backslash sim\backslash mathrm\{CMP\}(\backslash lambda,\backslash nu)$. If $\backslash nu\backslash geq1$ is an integer, we can, however, obtain the following formula in terms of the generalized hypergeometric function:Nadarajah, S. "Useful moment and CDF formulations for the COM–Poisson distribution." Statistical Papers 50 (2009): 617–622.

:$$

F(n)=P(X\leq n)=1-\frac{ _1F_{\nu-1}(;n+2,\ldots,n+2;\lambda)}{{\{(n+1)!\}^{\nu-1}} _0F_{\nu-1}(;1,\ldots,1;\lambda)}.

Many important summary statistics, such as moments and cumulants, of the CMP distribution can be expressed in terms of the normalizing constant $Z(\backslash lambda,\backslash nu)$. Indeed, The probability generating function is $\backslash operatorname\{E\}s^X=Z(s\backslash lambda,\backslash nu)/Z(\backslash lambda,\backslash nu)$, and the mean and variance are given by

:$$

\operatorname{E}X=\lambda\frac{d}{d\lambda}\big\{\ln(Z(\lambda,\nu))\big\},

:$$

\operatorname{var}(X)=\lambda\frac{d}{d\lambda}\operatorname{E}X.

The cumulant generating function is

:$$

g(t)=\ln(\operatorname{E}[e^{tX}])=\ln(Z(\lambda e^{t},\nu))-\ln(Z(\lambda,\nu)),

and the cumulants are given by

:$$

\kappa_n=g^{(n)}(0)=\frac{\partial^n}{\partial t^n}\ln(Z(\lambda e^t,\nu)) \bigg|_{t=0}, \quad n\geq1.

Whilst the normalizing constant $Z(\backslash lambda,\backslash nu)=\backslash sum\_\{i=0\}^\backslash infty\backslash frac\{\backslash lambda^i\}\{(i!)^\backslash nu\}$ does not in general have a closed form, there are some noteworthy special cases:

• $Z(\backslash lambda,1)=\backslash mathrm\{e\}^\{\backslash lambda\}$
• $Z(\backslash lambda,0)=(1-\backslash lambda)^\{-1\}$
• $\backslash lim\_\{\backslash nu\backslash rightarrow\backslash infty\}Z(\backslash lambda,\backslash nu)=1+\backslash lambda$
• $Z(\backslash lambda,2)=I\_0(2\backslash sqrt\{\backslash lambda\})$, where $I\_0(x)=\backslash sum\_\{k=0\}^\backslash infty\backslash frac\{1\}\{(k!)^2\}\backslash big(\backslash frac\{x\}\{2\}\backslash big)^\{2k\}$ is a modified Bessel function of the first kind.
• For integer $\backslash nu$, the normalizing constant can expressed as a generalized hypergeometric function: $Z(\backslash lambda,\backslash nu)=\_0F\_\{\backslash nu-1\}(;1,\backslash ldots,1;\backslash lambda)$.

Because the normalizing constant does not in general have a closed form, the following asymptotic expansion is of interest. Fix $\backslash nu>0$. Then, as $\backslash lambda\backslash rightarrow\backslash infty$,Gaunt, R.E., Iyengar, S., Olde Daalhuis, A.B. and Simsek, B. "An asymptotic expansion for the normalizing constant of the Conway–Maxwell–Poisson distribution." To appear in Annals of the Institute of Statistical Mathematics (2017+) DOI 10.1007/s10463-017-0629-6

:$$

Z(\lambda,\nu)=\frac{\exp\left\{\nu\lambda^{1/\nu}\right\}}{\lambda^{(\nu-1)/2\nu}(2\pi)^{(\nu-1)/2}\sqrt{\nu}}\sum_{k=0}^\infty c_k\big(\nu\lambda^{1/\nu}\big)^{-k},

where the $c\_j$ are uniquely determined by the expansion

:$$

\left(\Gamma(t+1)\right)^{-\nu}=\frac{\nu^{\nu (t+1/2)}}{\left(2\pi\right)^{(\nu-1)/2}}\sum_{j=0}^\infty\frac{c_j}{\Gamma(\nu t+(1+\nu)/2+j)}.

In particular, $c\_0=1$, $c\_1=\backslash frac\{\backslash nu^2-1\}\{24\}$, $c\_2=\backslash frac\{\backslash nu^2-1\}\{1152\}\backslash left(\backslash nu^2+23\backslash right)$. Further coefficients are given in.

For general values of $\backslash nu$, there does not exist closed form formulas for the mean, variance and moments of the CMP distribution. We do, however, have the following neat formula. Let $(j)\_r=j(j-1)\backslash cdots(j-r+1)$ denote the falling factorial. Let $X\backslash sim\backslash mathrm\{CMP\}(\backslash lambda,\backslash nu)$, $\backslash lambda,\backslash nu>0$. Then

:$$

\operatorname{E}[((X)_r)^\nu]=\lambda^r,

for $r\backslash in\backslash mathbb\{N\}$.

Since in general closed form formulas are not available for moments and cumulants of the CMP distribution, the following asymptotic formulas are of interest. Let $X\backslash sim\; \backslash mathrm\{CMP\}(\backslash lambda,\backslash nu)$, where $\backslash nu>0$. Denote the skewness $\backslash gamma\_1=\backslash frac\{\backslash kappa\_3\}\{\backslash sigma^3\}$ and excess kurtosis $\backslash gamma\_2=\backslash frac\{\backslash kappa\_4\}\{\backslash sigma^4\}$, where $\backslash sigma^2=\backslash mathrm\{Var\}(X)$. Then, as $\backslash lambda\backslash rightarrow\backslash infty$,

:$$

\operatorname{E}X= \lambda^{1/\nu}\left(1-\frac{\nu-1}{2\nu} \lambda^{-1/\nu}-\frac{\nu^2-1}{24\nu^2}\lambda^{-2/\nu}-\frac{\nu^2-1}{24\nu^3}\lambda^{-3/\nu}+\mathcal{O}(\lambda^{-4/\nu}) \right),

:$$

\mathrm{Var}(X)= \frac{\lambda^{1/\nu}}{\nu}\bigg(1+\frac{\nu^2-1}{24\nu^2}\lambda^{-2/\nu}+\frac{\nu^2-1}{12\nu^3}\lambda^{-3/\nu}+\mathcal{O}(\lambda^{-4/\nu})\bigg),

:$$

\kappa_n= \frac{\lambda^{1/\nu}}{\nu^{n-1}}\bigg(1+\frac{(-1)^n(\nu^2-1)}{24\nu^2}\lambda^{-2/\nu}+\frac{(-2)^n(\nu^2-1)}{48\nu^3}\lambda^{-3/\nu}+\mathcal{O}(\lambda^{-4/\nu})\bigg),

:$$

\gamma_1= \frac{\lambda^{-1/2\nu}}{\sqrt{\nu}}\bigg(1-\frac{5(\nu^2-1)}{48\nu^2}\lambda^{-2/\nu}-\frac{7(\nu^2-1)}{24\nu^3}\lambda^{-3/\nu}+\mathcal{O}(\lambda^{-4/\nu})\bigg),

:$$

\gamma_2= \frac{\lambda^{-1/\nu}}{\nu}\bigg(1-\frac{(\nu^2-1)}{24\nu^2}\lambda^{-2/\nu}+\frac{(\nu^2-1)}{6\nu^3}\lambda^{-3/\nu}+\mathcal{O}(\lambda^{-4/\nu})\bigg),

:$$

\operatorname{E}[X^n]= \lambda^{n/\nu}\bigg(1+\frac{n(n-\nu)}{2\nu}\lambda^{-1/\nu}+a_2\lambda^{-2/\nu}+\mathcal{O}(\lambda^{-3/\nu})\bigg),

where

:$$

a_2=-\frac{n(\nu-1)(6n\nu^2-3n\nu-15n+4\nu+10)}{24\nu^2}+\frac{1}{\nu^2}\bigg\{\binom{n}{3}+3\binom{n}{4}\bigg\}.

The asymptotic series for $\backslash kappa\_n$ holds for all $n\backslash geq2$, and $\backslash kappa\_1=\backslash operatorname\{E\}X$.

When $\backslash nu$ is an integer explicit formulas for moments can be obtained. The case $\backslash nu=1$ corresponds to the Poisson distribution. Suppose now that $\backslash nu=2$. For $m\backslash in\backslash mathbb\{N\}$,

:$$

\operatorname{E}[(X)_m]=\frac{\lambda^{m/2}I_m(2\sqrt{\lambda})}{I_0(2\sqrt{\lambda})},

where $I\_r(x)$ is the modified Bessel function of the first kind.

Using the connecting formula for moments and factorial moments gives

:$$

\operatorname{E}X^m=\sum_{k=1}^m\left\{ {m \atop k} \right\}\frac{\lambda^{k/2}I_k(2\sqrt{\lambda})}{I_0(2\sqrt{\lambda})}.

In particular, the mean of $X$ is given by

:$$

\operatorname{E}X=\frac{\sqrt{\lambda}I_1(2\sqrt{\lambda})}{I_0(2\sqrt{\lambda})}.

Also, since $\backslash operatorname\{E\}X^2=\backslash lambda$, the variance is given by

:$$

\mathrm{Var}(X)=\lambda\left(1-\frac{I_1(2\sqrt{\lambda})^2}{I_0(2\sqrt{\lambda})^2}\right).

Suppose now that $\backslash nu\backslash geq1$ is an integer. Then

:$$

\operatorname{E}[(X)_m]=\frac{{\lambda^m}}{{(m!)^{\nu-1}}} \frac{_0F_{\nu-1}(;m+1,\ldots,m+1;\lambda)}{_0F_{\nu-1}(;1,\ldots,1;\lambda)}.

In particular,

: $$

\operatorname{E}[X]=\lambda \frac{_0F_{\nu-1}(;2,\ldots,2;\lambda)}{_0F_{\nu-1}(;1,\ldots,1;\lambda)},

and

\mathrm{Var}(X)=\frac{{\lambda^2}}{{2^{\nu-1}}} \frac{_0F_{\nu-1}(;3,\ldots,3;\lambda)}{_0F_{\nu-1}(;1,\ldots,1;\lambda)}+\operatorname{E}[X]-(\operatorname{E}[X])^2.

Let $X\backslash sim\backslash mathrm\{CMP\}(\backslash lambda,\backslash nu)$. Then the mode of $X$ is $\backslash lfloor\backslash lambda^\{1/\backslash nu\}\backslash rfloor$ if

The mean deviation of $X^\backslash nu$ about its mean $\backslash lambda$ is given by Daly, F. and Gaunt, R.E. " The Conway–Maxwell–Poisson distribution: distributional theory and approximation." ALEA Latin American Journal of Probability and Mathematical Statistics 13 (2016): 635–658.

:$$

\operatorname{E}|X^\nu-\lambda| = 2Z(\lambda,\nu)^{-1} \frac{\lambda^{\lfloor\lambda^{1/\nu}\rfloor+1}}{\lfloor\lambda^{1/\nu}\rfloor!}.

No explicit formula is known for the median of $X$, but the following asymptotic result is available. Let $m$ be the median of $X\backslash sim\backslash mbox\{CMP\}(\backslash lambda,\backslash nu)$. Then

:$$

m=\lambda^{1/\nu}+\mathcal{O}\left(\lambda^{1/2\nu}\right),

as $\backslash lambda\backslash rightarrow\backslash infty$.

Let $X\backslash sim\backslash mbox\{CMP\}(\backslash lambda,\backslash nu)$, and suppose that $f:\backslash mathbb\{Z\}^+\backslash mapsto\backslash mathbb\{R\}$ is such that and . Then

:$$

\operatorname{E}[\lambda f(X+1)-X^\nu f(X)]=0.

Conversely, suppose now that $W$ is a real-valued random variable supported on $\backslash mathbb\{Z\}^+$ such that $\backslash operatorname\{E\}[\backslash lambda\; f(W+1)-W^\backslash nu\; f(W)]=0$ for all bounded $f:\backslash mathbb\{Z\}^+\backslash mapsto\backslash mathbb\{R\}$. Then $W\backslash sim\; \backslash mbox\{CMP\}(\backslash lambda,\backslash nu)$.

Let $Y\_n$ have the Conway–Maxwell–binomial distribution with parameters $n$, $p=\backslash lambda/n^\backslash nu$ and $\backslash nu$. Fix $\backslash lambda>0$ and $\backslash nu>0$. Then, $Y\_n$ converges in distribution to the $\backslash mathrm\{CMP\}(\backslash lambda,\backslash nu)$ distribution as $n\backslash rightarrow\backslash infty$. This result generalises the classical Poisson approximation of the binomial distribution. More generally, the CMP distribution arises as a limiting distribution of Conway–Maxwell–Poisson binomial distribution. Apart from the fact that COM-binomial approximates to COM-Poisson, Zhang et al. (2018)Zhang H., Tan K., Li B. "COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data." Frontiers of Mathematics in China, 2018, 13(4): 967–998.[https://doi.org/10.1007/s11464-018-0714-z] illustrates that COM-negative binomial distribution with probability mass function

:$$

\mathrm{P}(X = k) =

\frac {{{{(\frac{{\Gamma (r + k)}}{{k! \Gamma (r)}})}^\nu }{p^k}{{(1 - p)}^r}}}

{{\sum\limits_{i = 0}^\infty {{{(\frac{{\Gamma (r + i)}}{{i! \Gamma (r)}})}^\nu }} {p^i}{{(1 - p)}^r}}}

={{\left(\frac{{\Gamma (r + k)}}{{k! \Gamma (r)}}\right)}^\nu }

{{p^k}{{(1 - p)}^r}}

\frac{1}{{C(r,\nu ,p)}},\quad (k = 0,1,2, \ldots ),

convergents to a limiting distribution which is the COM-Poisson, as $\{r\; \backslash to\; +\; \backslash infty\; \}$ .

• $X\backslash sim\backslash operatorname\{CMP\}(\backslash lambda,1)$, then $X$ follows the Poisson distribution with parameter $\backslash lambda$.
• Suppose . Then if $X\backslash sim\backslash mathrm\{CMP\}(\backslash lambda,0)$, we have that $X$ follows the geometric distribution with probability mass function $P(X=k)=\backslash lambda^k(1-\backslash lambda)$, $k\backslash geq0$.
• The sequence of random variable $X\_\backslash nu\backslash sim\backslash mathrm\{CMP\}(\backslash lambda,\backslash nu)$ converges in distribution as $\backslash nu\backslash rightarrow\backslash infty$ to the Bernoulli distribution with mean $\backslash lambda(1+\backslash lambda)^\{-1\}$.

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. 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

:$$

\log \frac{p_{x-1}}{p_x} = - \log \lambda + \nu \log x

where $p\_x$ denotes $\backslash Pr(X\; =\; x)$. 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 $\backslash log\; x$ 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 $\backslash log\; (\backslash hat\; p\_\{x-1\}\; /\; \backslash hat\; p\_x)$ on $\backslash log\; x$. 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 one-step covariances on the first off-diagonal, both given below.

:$$

\operatorname{var}\left[\log \frac{\hat p_{x-1}}{\hat p_x}\right] \approx \frac{1}{np_x} + \frac{1}{np_{x-1}}

:$$

\text{cov}\left(\log \frac{\hat p_{x-1}}{\hat p_x}, \log \frac{\hat p_x}{\hat p_{x+1}} \right) \approx - \frac{1}{np_x}

The CMP likelihood function is

:$$

\mathcal{L}(\lambda,\nu\mid x_1,\dots,x_n) = \lambda^{S_1} \exp(-\nu S_2) Z^{-n}(\lambda, \nu)

where $S\_1\; =\; \backslash sum\_\{i=1\}^n\; x\_i$ and $S\_2\; =\; \backslash sum\_\{i=1\}^n\; \backslash log\; x\_i!$. Maximizing the likelihood yields the following two equations

:$$

\operatorname{E}[X] = \bar X

:$$

\operatorname{E}[\log X!] = \overline{\log X!}

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 $\backslash log\; X!$ are approximated by using the estimates for $\backslash lambda$ and $\backslash nu$ from the previous iteration in the expression

:$$

\operatorname{E}[f(x)] = \sum_{j=0}^\infty f(j) \frac{\lambda^j}{(j!)^\nu Z(\lambda, \nu)}.

This is continued until convergence of $\backslash hat\backslash lambda$ and $\backslash hat\backslash nu$.

The basic CMP distribution discussed above has also been used as the basis for a generalized linear model (GLM) using a Bayesian formulation. A dual-link GLM based on the CMP distribution has been developed,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.1539-6924.2008.01014.x}}

and this model has been used to evaluate traffic accident data.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 Conway–Maxwell–Poisson Models: Analyzing Traffic Crash Data Exhibiting Under-Dispersion," Risk Analysis, 30 (8), 1268–1276. {{doi|10.1111/j.1539-6924.2010.01417.x}} The CMP GLM developed by Guikema and Coffelt (2008) is based on a reformulation of the CMP distribution above, replacing $\backslash lambda$ with $\backslash mu=\backslash lambda^\{1/\backslash nu\}$. The integral part of $\backslash mu$ is then the mode of the distribution. A full Bayesian estimation approach has been used with MCMC sampling implemented in WinBugs with non-informative priors for the regression parameters. 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 CMP regression has been developed which generalizes Poisson regression and logistic regression.Sellers, K. S. and Shmueli, G. (2010), [http://projecteuclid.org/euclid.aoas/1280842147 "A Flexible Regression Model for Count Data"], Annals of Applied Statistics, 4 (2), 943–961 This takes advantage of the exponential family properties of the CMP 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. 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 CMP regression, testing for dispersion, and evaluating fit is available.[http://www9.georgetown.edu/faculty/kfs7/research.html Code for COM_Poisson modelling], Georgetown Univ.

The two GLM frameworks developed for the CMP distribution significantly extend the usefulness of this distribution for data analysis problems.

• [https://cran.r-project.org/web/packages/compoisson/index.html Conway–Maxwell–Poisson distribution package for R (compoisson) by Jeffrey Dunn, part of Comprehensive R Archive Network (CRAN)]
• [http://alumni.media.mit.edu/~tpminka/software/compoisson Conway–Maxwell–Poisson distribution package for R (compoisson) by Tom Minka, third party package]

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