The Conway-Maxwell-Poisson distribution:
Distributional theory and approximation

Daly, Fraser; Gaunt, Robert E.

doi:10.30757/alea.v13-25

Cited by 32 publications

(25 citation statements)

References 32 publications

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“…Here we want to give an alternative proof, see Appendix for the proof. By using the Stein identity lemma above and the Stein-Chen method, Daly and Gaunt (2016) have shown some convergence results and approximations, including a bound on the total variation distance between a COM-binomial r.v. and the corresponding COM-Poisson r.v..…”

Section: Functional Operator Characterizationmentioning

confidence: 99%

Some characterizations and properties of COM-Poisson random variables

Zhang

2019

Communications in Statistics - Theory and Methods

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This paper introduces some new characterizations of COM-Poisson random variable. First, it extends Moran-Chatterji characterization, and generalizes Rao-Rubin characterization of Poisson distribution to COM-Poisson distribution. Then, it defines the COM-type discrete r.v. X ν of the discrete random variable X. The probability mass function of X ν has a link to the Rényi entropy and Tsallis entropy of order ν of X. And then we can get the characterization of Stam inequality for COM-type discrete version Fisher information. By using the recurrence formula, the property that COM-Poisson random variables (ν = 1) is not closed under addition are obtained. Finally, under the property of "not closed under addition" of COM-Poisson random variables, a new characterization of Poisson distribution is found.

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Section: Functional Operator Characterizationmentioning

confidence: 99%

Some characterizations and properties of COM-Poisson random variables

Zhang

2019

Communications in Statistics - Theory and Methods

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“…As noted in Shmueli et al (), the moments can be represented recursively as

normalE false(X^{r + 1} false) = ({, \begin{matrix} array λ {[E (X + 1)]}^{1 - ν}, & array r = 0 \\ array λ \frac{\partial}{\partial λ} E (X^{r}) + E (X) E (X^{r}), & array r > 0 . \end{matrix})

In particular, the expected value and variance can be written in the form and approximated respectively as

normalE false(X false) = \frac{\partial \ln ζ false(λ, ν false)}{\partial \ln λ} \approx λ^{1 false/ ν} - \frac{ν - 1}{2 ν}, and

Var false(X false) = \frac{\partial normalE false(X false)}{\partial \ln λ} \approx \frac{1}{ν} λ^{1 false/ ν},

where the approximations are especially good for

ν \leq 1

λ > 1 0^{ν}

(Shmueli et al ). More broadly, Daly and Gaunt () show that

E ((), {false({false(X false)}_{k} false)}^{ν}) = λ^{k}

, where

{false(j false)}_{k} = j false(j - 1 false) \dots false(j - k + 1 false)

denotes the descending factorial for some

j

.…”

Section: Motivating Distributionsmentioning

confidence: 99%

“…The pgf and mgf of

Y

have the form,

E ((), t^{Y}) = \frac{τ ((), \frac{t p}{1 - p}, ν, m)}{τ ((), \frac{p}{1 - p}, ν, m)} and M_{Y} false(t false) = \frac{τ ((), \frac{p e^{t}}{1 - p}, ν, m)}{τ ((), \frac{p}{1 - p}, ν, m)},

respectively, where

τ false(θ, ν, m false) = \sum_{y = 0}^{m} {((), \binom{m}{y})}^{ν} θ^{y}

for some

θ

. Daly and Gaunt () show that

E false({false({false(Y false)}_{k} false)}^{ν} false) = \frac{χ false(p, ν, m - k false)}{χ false(p, ν, m false)}

{false({false(m false)}_{k} false)}^{ν} p^{k}

, where

{false(j false)}_{k} = j false(j - 1 false) \dots false(j - k + 1 false)

denotes the descending factorial for some

j

.…”

Section: Motivating Distributionsmentioning

confidence: 99%

A Flexible Univariate Autoregressive Time‐Series Model for Dispersed Count Data

Sellers

Peng

Arab

2019

Journal Time Series Analysis

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Integer‐valued time series data have an ever‐increasing presence in various applications (e.g., the number of purchases made in response to a marketing strategy, or the number of employees at a business) and need to be analyzed properly. While a Poisson autoregressive (PAR) model would seem like a natural choice to model such data, it is constrained by the equi‐dispersion assumption (i.e., that the variance and the mean equal). Hence, data that are over‐ or under‐dispersed (i.e., have the variance greater or less than the mean respectively) are improperly modeled, resulting in biased estimates and inaccurate forecasts. This work instead develops a flexible integer‐valued autoregressive model for count data that contain over‐ or under‐dispersion. Using the Conway–Maxwell–Poisson (CMP) distribution and related distributions as motivation, we develop a first‐order sum‐of‐CMP's autoregressive (SCMPAR(1)) model that will instead offer a generalizable construct that captures the PAR, and versions of what we refer to as a negative binomial AR model, and binomial AR model respectively as special cases, and serve as an overarching representation connecting these three special cases through the dispersion parameter. We illustrate the SCMPAR model's flexibility and ability to effectively model count time series data containing data dispersion through simulated and real data examples.

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“…Some recent applications of the COM-Poisson distribution include Lord et al (2010) for the analysis of traffic crash data, Sellers and Shmueli (2010) for the modelling of airfreight breakage and book purchases, Huang (2017) on the analysis of attendance data, takeover bids and cotton boll counts, and Chatla and Shmueli (2018) to model counts from bike sharing systems. Theoretical results and approximations derived for this distribution are discussed by Shmueli et al (2005), Daly and Gaunt (2016) and Gaunt et al (2017). The main disadvantage of the COM-Poisson regression model as presented in Sellers and Shmueli (2010) is that its location parameter does not correspond to the expectation of the distribution.…”

Section: Introductionmentioning

confidence: 99%

Reparametrization of COM–Poisson regression models with applications in the analysis of experimental data

Ribeiro

Zeviani

Bonat

et al. 2019

Statistical Modelling

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In the analysis of count data often the equidispersion assumption is not suitable, hence the Poisson regression model is inappropriate. As a generalization of the Poisson distribution the COM-Poisson distribution can deal with under-, equi-and overdispersed count data. It is a member of the exponential family of distributions and has the Poisson and geometric distributions as special cases, as well as the Bernoulli distribution as a limiting case. In spite of the nice properties of the COM-Poisson distribution, its location parameter does not correspond to the expectation, which complicates the interpretation of regression models specified using this distribution. In this paper, we propose a straightforward reparametrization of the COM-Poisson distribution based on an approximation to the expectation of this distribution. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the count response variable, as usual in the context of generalized linear models. Furthermore, the estimation and inference for the new COM-Poisson regression model can be done based on the likelihood paradigm. We carried out simulation studies to verify the finite sample properties of the maximum likelihood estimators. The results from our simulation study show that the maximum likeli-hood estimators are unbiased and consistent for both regression and dispersion parameters. We observed that the empirical correlation between the regression and dispersion parameter estimators is close to zero, which suggests that these parameters are orthogonal. We illustrate the application of the proposed model through the analysis of three data sets with over-, under-and equidispersed count data. The study of distribution properties through a consideration of dispersion, zero-inflated and heavy tail indices, together with the results of data analysis show the flexibility over standard approaches. Therefore, we encourage the application of the new parametrization for the analysis of count data in the context of COM-Poisson regression models. The com-arXiv:1801.09795v1 [stat.AP] 29 Jan 2018 2 Ribeiro Jr et al. putational routines for fitting the original and new version of the COM-Poisson regression model and the analyzed data sets are available in the supplementary material.

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The Conway-Maxwell-Poisson distribution: Distributional theory and approximation

Cited by 32 publications

References 32 publications

Some characterizations and properties of COM-Poisson random variables

Some characterizations and properties of COM-Poisson random variables

A Flexible Univariate Autoregressive Time‐Series Model for Dispersed Count Data

Reparametrization of COM–Poisson regression models with applications in the analysis of experimental data

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