Extended Conway-Maxwell-Poisson distribution and its properties and applications

A new three parameter natural extension of the Conway-Maxwell-Poisson (COM-Poisson) distribution is proposed. This distribution includes the recently proposed COM-Poisson type negative binomial (COM-NB) distribution [Chakraborty, S. and Ong, S. H. (2014): A COM-type Generalization of the Negative Binomial Distribution, Accepted in Communications in Statistics-Theory and Methods] and the generalized COM-Poisson (GCOMP) distribution [Imoto, T. :(2014) A generalized Conway-Maxwell-Poisson distribution which includes the negative binomial distribution, Applied Mathematics and Computation, 247, 824-834]. The proposed distribution is derived from a queuing system with state dependent arrival and service rates and also from an exponential combination of negative binomial and COM-Poisson distribution. Some distributional, reliability and stochastic ordering properties are investigated. Computational asymptotic approximations, different characterizations, parameter estimation and data fitting example also discussed.

The distribution is defined in the parameter space . When  is a positive integer, ) ;  Imoto (2014) proposed another generalization where a random variable X is said to follow the GCOM-Poisson distribution of Imoto (2014) with parameters ) , ,

Generalized COM-Poisson distribution
if its pmf is given by The distribution is defined in the parameter space In the present article a natural three parameter generalization of the COM-Poisson distribution which includes COM-NB and GCOM-Poisson distributions is proposed. Some important distributional properties of the proposed distribution are presented in the section 2. Reliability and stochastic ordering results are discussed in section3. Concluding remarks is made in the final section.

A hypergeometric type series
We introduce the series  is the Pochhammer's notation (see Johnson et al., 2005). For  , and m all positive integers, it reduces to ) ; , , , , a particular generalized hypergeometric function. With this notation we have  [Chakraborty and Ong, 2014] ii. [Imoto, 2014] iii. ) , [Conway and Maxwell,1962] iv.
Some important limiting cases of ) positive.

Extended COM-Poisson (ECOMP) distribution
Here we introduce a new distribution that unifies both the COM-NB and GCOMP distributions.

Definition1.
A random variable X is said to follow the extended COM-Poisson distribution with parameters ) , , , ] if its pmf is given by The distribution is defined in the parameter space Remark1. Unlike COM-NB where the parameter 1   and in GCOMP the parameter 1   , in case of ECOMP these two parameters can be either positive or negative with the restriction of    . Remark2. The pmf in (6) can alternatively expressed as

Particular cases
Following discrete distributions are particular cases of ECOMP ) , , , The distribution in (7) is log-convex as will be seen in section 2.8.

Approximation of the normalizing constant
The normalizing constant ) distribution is not expressed in a closed form and includes the summation of infinite series. Therefore, we need approximations of this constant to compute the pmf and moments of the distribution numerically.
A simply approximation is to truncate the series that is where m is an integer chosen such that Then the relative error about the pmf is give by where ) (k P is given by the r.h.s. of equation (6) in section 2 and ) (k P m is given by the r.h.s. of (6) with ) The upper bound of the relative truncation error is then found to be This formula reduces to asymptotic formula by Minka Table 1 gives the percentage errors ) Since the ECOMP distribution is a member of exponential family, the mean is given by differentiating the logarithm of the normalizing constant with respect to p . Here we consider the differentiation of the logarithm of the function (9), or )) . This function approximates the mean of the ECOMP distribution for large p and small | |    , where it is difficult to compute the approximation by truncation. Table 2 gives the percentage errors of the approximated mean.

Recurrence relation for probabilities
The ECOMP ) , , , pmf has a simple recurrence relation given by . This will be useful for the computation of the probabilities.

Exponential family
The pmf in (6) can also be expressed as ] when  is a nuisance parameter or its value is given.

Dispersion level of with respect to the Poisson distribution
The pmf in (6) can be seen as a weighted Poisson (p) distribution with weight function and under dispersed for  (Conway and Maxwell, 1962), ECOMP ) , , (   p can also be derived as the probability of the system being in the k th state for a queuing system with state dependent service and arrival rate. Consider a single server queuing system with state dependent (that is dependent on the system state, k th state means k number of units in the system) arrival rate where,  / 1 and  / 1 are respectively the normal mean service and mean arrival time for a unit when that unit is the only one in the system;  and  are the pressure coefficients, reflecting the degree to which the service and arrival rates of the system are affected by the system state. This set up implies that while the arrival rate and the service rate increases exponentially as queue lengthens (i.e. n increases).
Following Conway and Maxwell (1962), the system differential difference equations are given by and . Then from (12) and (13) we get Assuming a steady state (i.e. 0 ) ( Since we have assumed a steady state (i.e. 0 ) ( for all k) ) (t P k can be replaced by k P .

ECOMP
as exponential combination formulation The general form of the exponential combination of two pmfs say ) ; is given by (Atkinson, 1970) This combining of the pmf was suggested by Cox (1961Cox ( , 1962 for combining the two hypotheses ( 1   i.e. the distribution is 1 f and 0   that is the distribution is 2 f ) in a general model of which they would both be special cases. The inferences about  made in the usual way and testing the hypo-thesis that the value of  is zero or one is equivalent to testing for departures from one model in the direction of the other. Now the probability function resulting from the exponential combination of NB (   , ) and COM-Poisson (  , ) is given by can be regarded as a natural extension of COM-Poisson of Conway and Maxwell (1962), COM-NB of Chakraborty and Ong (2014), as well as the GCOMP distribution of Imoto (2014).

Log-concavity and modality
> 0, for all t.
has a log-convex probability mass function for } ,

Momemts
The r th factorial moment

Reliability characteristics and stochastic ordering 3.1 Survival and Failure rate functions
The survival function is given by while the failure rate function is where the second expressions in terms of hypergeometric function is for the case when  , are positive integers.

Stochastic orderings
This is clearly increasing in n as 1   (Shaked andShanthikumar, 2007 andGupta et al., 2014). Hence the result is proved.
This is clearly increasing in n as 1   (Shaked andShanthikumar, 2007 andGupta et al., 2014). Hence the result is proved.

A numerical example
To fit the proposed distribution, we have to estimate the parameters ) , , , (    p in (6). The maximum likelihood (ML) estimation is often used for fitting to real data, but the log likelihood function of the proposed distribution where i f is the observed frequency of i events, , k is the highest observed value, has some local maximum points for some datasets, or the likelihood equations do not always have unique solution.
Here we have used profile likelihood method to fit the proposed distribution to the data of Corbet (1942) on Malayan butteries with zeros, which also has been used by Blumer (1974)  Corbet caught altogether 620 species, but he also estimated that the total buttery fauna of the area contained 924 species, so that 304 species were missing from the collection and treated as count zero. We also fit the Poisson-lognormal (P-Log), COMNB ((2) of section 1) and GCOMP ((3) of Section 1) distributions to the same data for comparing the fitting with the proposed distribution. The ML estimates and fitting results of the distributions are given in Table 3. rejects the negative binomial distribution (p-value is 0.001).
The Poisson-lognormal distribution was derived for long-tailed count data and fitted to the same data as in this section. Comparing the distributions, we see that the ECOMP distribution gives better fitting in the sense of AIC and 2  goodness of fit, and especially gives very good fittings for the count 0 and the tail part 25+. We have tried fitting the ECOMP distribution to various count data and seen that the distribution gives good fittings for the count data with many zeros or with long-tail.

Concluding Remarks
In this paper a new discrete distribution that extends the Conway-Maxwell-Poisson distribution and also unifies the COM-NB (Chakraborty and Ong, 2014) and GCOMP (Imoto, 2014) is proposed and its important distributional properties investigated. This distribution which arises from queuing theory set up and also as exponential combination has many desirable properties with potential applications in modeling varieties of count data.