Extended ConwayMaxwellPoisson distribution and its properties and applications
 Subrata Chakraborty^{1}Email author and
 Tomoaki Imoto^{2}
DOI: 10.1186/s4048801600441
© Chakraborty and Imoto. 2016
Received: 10 October 2015
Accepted: 9 February 2016
Published: 25 February 2016
Abstract
A new four parameter extended ConwayMaxwellPoisson (ECOMP) distribution which unifies the recently proposed COMPoisson type negative binomial (COMNB) distribution [Chakraborty, S. and Ong, S. H. (2014): A COMtype Generalization of the Negative Binomial Distribution, Accepted in Communications in StatisticsTheory and Methods] and the generalized COMPoisson (GCOMP) distribution [Imoto, T. :(2014) A generalized ConwayMaxwellPoisson distribution which includes the negative binomial distribution, Applied Mathematics and Computation, 247, 824–834] is proposed. The additional parameter allows this distribution to have longer (shorter) tail compared to COMNB and GCOMP. The proposed distribution can be formulated as an exponential combination of negative binomial and COMPoisson distribution and also arises from a queuing system with state dependent arrival and service rates and belongs to exponential family when one of the parameter is considered as nuisance. Important distributional, reliability and stochastic ordering properties along with asymptotic approximations for the normalizing constant and the mean of this distribution is investigated. Method of parameter estimation and three comparative data fitting applications are also discussed.
Keyword
COMPoisson COMNegative binomial Generalized COMPoisson State dependent service and arrival rate Queues Laplace methodMathematics Subject Classification (2010)
62E15 60K25 62N051 Introduction
Recently, two new generalizations of the well known COMPoisson (Conway and Maxwell 1962) was proposed. One by Chakraborty and Ong (2014) known as the COMNegative binomial distribution and the other by Imoto (2014) referred to as the generalized COMPoisson Distribution. In this section we briefly introduce these two distributions along with a hypergeometric type series which is used in the sequel.
COMPoisson type negative binomial distribution: Chakraborty and Ong (2014) proposed a new COMPoisson type generalization of negative binomial distribution that includes some wellknown distributions including COMPoisson, Negative Binomial (page 208–250, Chapter 5, Johnson et al. 2005), as particular case and Bernoulli (page 108, Chapter 3, Johnson et al. 2005), COMPoisson as limiting cases among others. This distribution is logconcave and flexible enough to model under, equi and over dispersed count data.
When α is a positive integer, _{1} H _{ α − 1}(ν; 1; p) can be expressed as a particular case of generalized hypergeometric series \( {}_mF_n\left({a}_1,{a}_2,\cdots, {a}_m;{b}_1,{b}_2,\cdots, {b}_m;z\right)={\displaystyle \sum_{k=0}^{\infty}\frac{{\left({a}_1\right)}_k{\left({a}_2\right)}_k\cdots {\left({a}_m\right)}_k}{\;{\left({b}_1\right)}_k{\left({b}_2\right)}_k\cdots {\left({b}_n\right)}_k}\;\frac{z^k}{k\;!}} \) as _{1} F _{ α − 1}(ν; 1, 1, ⋯, 1; p).
 i.
\( {}_1S_{\alpha 1}^1\left(\nu;\;1;p\right)={}_1H_{\alpha 1}\left(\nu;\;1;p\right) \) [Chakraborty and Ong, 2014]
 ii.
\( {}_1S_{{}_0}^{\beta}\left(\nu;\;1;p\right)=C\left(\beta,\;\nu,\;p\right)/{\left(\Gamma \nu \right)}^{\beta } \) [Imoto 2014]
 iii.
\( {}_1S_0^1\left(\nu;\;1;p\right)={\left(1p\right)}^{\nu } \) [geometric series]
 iv.
\( {}_1S_{{}_{\alpha 1}}^{\beta}\left(1;\;1;p\right)=Z\left(p,\alpha \beta \right) \) [Conway and Maxwell 1962]
 v.$$ {}_1S_{\gamma}^{\gamma}\left(1;\;1;p\right)= \exp (p) $$
Some important limiting cases of \( {}_1S_{\alpha 1}^{\beta}\left(\nu;\;1;p\right) \) are
 vi.$$ \underset{\alpha \to \infty }{ \lim}\kern0.24em {}_1S_{\alpha 1}^{\beta}\left(\nu;\;1;p\right)=1+{\nu}^{\beta }p. $$
 vii.
\( \underset{\alpha \to \infty }{ \lim}\kern0.24em {}_1S_{\alpha 1}^{\beta}\left(\nu;\;1;p\right)={\displaystyle \sum_{k=0}^{\infty }{\lambda}^k/{\left(k!\right)}^{\alpha }}=Z\left(\lambda, \alpha \right) \), where ν ^{ β } p = λ is finite positive.
In the present article we propose a natural four parameter extension of the COMPoisson distribution which includes the recently introduced COMNB and GCOMPoisson distributions as special cases. This new distribution with additional parameters is more flexible in terms of tail length and dispersion index. The definition of the proposed distribution along with some of its important distributional properties are presented in the Section 2. Reliability and stochastic ordering results are discussed in Section 3. In Section 4 we presented applications of the proposed distribution by considering three real life data sets. Concluding remarks is provided in the Section 5 which if followed by an appendix containing the proofs of the results and propositions in the article.
2 Extended COMPoisson (ECOMP) distribution
Here we introduce a new distribution that unifies both the COMNB and GCOMP distributions.
It may be noted that unlike in the COMNB distribution where the parameter α ≥ 1 and in the GCOMP distribution where the parameter β ≤ 1, in the ECOMP distribution these two parameters can be either positive or negative with the restriction of α ≥ β.
For 0 < ν ≤ 1, the distribution in (7) is logconvex as will be seen in proposition 4 in the Section 2.7.
2.1 Shape of the pmf
2.2 Approximations of the normalizing constant
2.2.1 Approximation using truncation of the series
The normalizing constant \( {}_1S_{\alpha 1}^{\beta}\left(\nu; 1;p\right) \) of the ECOMP(v, p, α, β) 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.
For α − β ≥ 1, this truncated approximation is good because ε _{ m } = O(1/m) and thus, the truncation point m is not large. However, for 0 < α − β < 1 and p > 1, the truncation point become too large to compute the approximation. For example, when ν = 1.5, p = 3, α = 3.1, β = 3, m has to be over 50,000. This is not practicable. To avoid this difficulty it is useful to make a restriction for the parameter p such that p < 1 when α − β → 0. For example, with the restriction p < 10^{ α − β }, we see the relative truncation error R _{50}(1.5, 3, 3.1, 3) < 0.001.
2.2.2 Asymptotic approximation of the normalizing constant using the Laplace’s method
This formula reduces to the asymptotic formula by Minka et al. (2003) when ν = 1 or β = 0 and that by Imoto (2014) when α = 1. The proof and numerical investigation about the formula (9) are given in Appendix A.1.
2.3 Recurrence relation for probabilities
2.4 Exponential family
Which immediately implies that the ECOMP (ν, p, α, β) distribution belongs to the exponential family with parameters ( log p, α, β) when v, is a nuisance parameter or when its value is given.
2.5 Index of dispersion
Hence, ECOMP (ν, p, α, β) is over dispersed (i) if α < 1, β ≥ 0 for all v (ii) if {α ≥ 1, β > 0} or {α < 1, β < 0} when {0 < ν ≤ 1, β ≤ α ≤ β + 1} or {ν > 1, α ≤ 1} and under dispersed (i) if α ≥ 1, β < 0 for all v (ii) for {α ≥ 1, β > 0} or {α < 1, β < 0} if {0 < ν ≤ 1, α ≥ β + 1} or {ν > 1, α ≥ 1}.
As a particular cases of the above result, when β = 1, we can see that the COMNB (ν, p, α) distribution always over dispersed for {0 < ν ≤ 1, 1 ≤ α < 2} or {ν > 1, α = 1} and under dispersed compared to COMP distribution for {0 < ν ≤ 1, α ≥ 2}. Similarly when α = 1, the GCOMP(v, p, β) distribution is seen to be is over dispersed for 0 < β ≤ 1 and under dispersed for β < 0. When ν = 1, we derive that COMP (p, α − β) is over dispersed for α − β > 1 under dispersed for α − β < 1 and equidispersed when α − β = 1. Finally, the new generalized NB distribution with pmf (7) is over dispersed when γ = 1 (which is when it reduces to Negative binomial) and under dispersed if γ > 1.
It can also be checked that ECOMP (v, p, α, β) is over (under) dispersed for α ≥ β > (≤)0 w.r.t. COMNB (v, p, α) and w.r.t. GCOMPoisson (v, p, β) it is over (under) dispersed for β ≤ α < 1 (1 < β ≤ α).
2.6 Different formulations of ECOMP (v, p, α, β)
Two different formulations of the proposed distribution are presented in this section.
2.6.1 ECOMP (v, p, α, β) as a distribution from a queuing set up
Like the COMPoisson distribution, the ECOMP (v, p, α, β) distribution 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 λ _{ k } = (ν + k)^{ β } λ, and state dependent service rate μ _{ k } = k ^{ α } μ , where, 1/μ and 1/λ are respectively the normal mean service and mean arrival time for a unit when that unit is the only one in the system; α and v are the pressure coefficients, reflecting the degree to which the service and arrival rates of the system are affected by the system state.
Proposition 1. Under the above set up where the arrival rate and the service rate increases exponentially as queue lengthens (i.e. as k increases) the probability of the system being in the k ^{th} state is ECOMP (v, p, α, β).
Proof: See Appendix B.1
2.6.2 ECOMP (v, p, α, β) as exponential combination formulation
This combining of the pmf was suggested by Cox (1961, 1962) for combining the two hypotheses (β = 1, i.e. the distribution is f _{1} and β = 0 that is the distribution is f _{2}) in a general model of which they would both be special cases. The inferences about β made in the usual way and testing the hypothesis that the value of β is zero or one is equivalent to testing for departures from one model in the direction of the other.
Proposition 2. ECOMP (ν, p, α, β) distribution is an exponential combination NB (v, λ) and COMPoisson (μ, θ) distributions, with λ ^{ β } μ ^{1 − β } = p and α = θ(1 − β) + β.
Proof: See Appendix B.2.
From the above formulations it is clear that for ECOMP (v, p, α, β), β close to zero will indicate departure from COMPoisson towards NB, while β close to one will indicate the reverse. Thus ECOMP (v, p, α, β) can also be regarded as a natural extension of COMPoisson, and negative binomial distributions.
2.7 Logconcavity and modality
Proposition 3. The ECOMP (v, p, α, β) has a logconcave pmf when {ν > 1, p > 0, α ≥ β}
Proof: See Appendix B.3.
From the above result the corresponding results of COMNB (v, p, α) and GCOMP (v, p, β) can be obtained as particular cases. That is COMNB (v, p, α) is logconcave when {ν > 1, p > 0, α ≥ 1} and GCOMP (v, p, β) is logconcave when {ν > 1, p > 0, β ≤ 1}.
Following two important results follows as a consequence of logconcavity:

➢ a strongly unimodal distribution

➢ has an increasing failure rate function
 (i)
a non increasing pmf with a unique mode at X = 0 if ν ^{ β } p < 1,
e.g. ν = 2, α = 3, β = 2, p should be less than 0.25 to have unique mode at X = 0.
 (ii)
a unique mode at X = k if k ^{ α }/(ν + k − 1)^{ β } < p < (k + 1)^{ α }/(ν + k)^{ β }
e.g. ν = 2, α = 3, β = 2, p should be between 1.6875 and 2.560 to have unique mode at X = 3.
 (iii)
two modes at X = k and X = k − 1 if (ν + k − 1)^{ β } p = k ^{ α }. In particular the two modes are at X = 0 and X = 1 if ν ^{ β } p = 1.
e.g. ν = 2, α = 3, β = 2, p should be equal to 4.408 to have two modes at X = 5 and X = 6.
Graphical illustrations of the above three examples are presented in the first plots of Fig. 1. It is interesting to note that the distribution may be bimodal with one of the mode always at zero as shown the last two plots in Fig. 1.
Proposition 4. ECOMP (v, p, α, β) has a logconvex pmf for {0 < ν ≤ 1, α = β}
Proof. See Appendix B.4.
Following important results follows as a consequence of logconvexity:

➢ is Infinitely divisible (see Warde and Katti 1971) distribution, hence Discrete Compound Poisson distribution. (see page 409 of GómezDéniz et al. 2011)

➢ has an decreasing failure rate function, hence increasing mean residual life function

➢ has an upper bound for variance as p ν ^{ β } (using result of page 410 of GómezDéniz et al. 2011)
2.8 Moments
This function approximates the mean of the ECOMP (v, p, α, β) distribution for large p and small α − β, where it is difficult to compute the approximation by truncation. A numerical illustration of this asymptotic approximation is presented in the Appendix A.2.
3 Reliability characteristics and stochastic ordering
3.1 Survival and failure rate functions
3.2 Stochastic orderings
An rv X with pmf P(X = n) is said to be smaller than another rv Y pmf P(Y = n) in the likelihood ratio order that is X ≤ _{ lr } Y if P(Y = n)/P(X = n) increases in n over the union of the supports of X and Y. Again X ≤ _{ lr } Y implies X is smaller than Y in the hazard rate order and subsequently in the mean residual (MRL) life order (see Gupta et al. 2014).
Theorem 1. X ~ ECOMP (ν, p, α, β) is smaller than Y ~ COMNB (ν, p, α) in the likelihood ratio order i.e. X ≤ _{ lr } Y when β < 1.
This is clearly increasing in n as β < 1 (Definition 1.C.1 of Chapter 1, Shaked and Shanthikumar 2007 and Gupta et al. 2014). Hence the result is proved.
As an implication of theorem 1, we get X ≤ _{ hr } Y ⇒ X ≤ _{ MRL } Y, for β < 1.
Theorem 2. X ~ ECOMP (v, p, α, β) is smaller than Y ~ GCOMP (v, p, β) in the likelihood ratio order i.e. X ≤_{ lr } Y when α > 1.
This is clearly increasing in n as α > 1 (Definition 1.C.1 of Chapter 1, Shaked and Shanthikumar 2007 and Gupta et al. 2014). Hence the result is proved.
As an implication of theorem 2, we get X ≤ _{ hr } Y ⇒ X ≤ _{ MRL } Y, for α > 1.
4 Numerical examples
By using this method, we fit the proposed distribution to three datasets and compare with NB (r, p), COMP (θ, p) COMNB (v, p, α) and GCOMP (v, p, β). Simultaneously, we fit Delaporte distribution, which is derived from the convolution of a NB (r, p) and Poisson (λ) rv, and some mixed Poisson distributions; mixing with generalized gamma distribution of Agarwal and Kalla (1996) with parameters (δ, m, α, n), mixing with generalized inverse Gaussian gamma distribution of Jorgensen (1982) with parameters (χ, η, ω, λ), mixing with generalized exponential distribution of Ong and Lee (1986) with parameters (v, a, d, β). These distributions are derived as the generalized negative binomial distributions and used for longtailed count data. The detailed studies are given in Gupta and Ong (2005). Here we show only the best fitting distribution among these distributions in Gupta and Ong (2005). The performances of various distributions are compared using the χ ^{2} goodness of fit and the Akaike Information Criterion (AIC). Following Burnham and Anderson (2004) we look at the difference Δ_{ i } = AIC_{ i } − AIC_{min} where AIC_{min} is the minimum of the AIC values of the all the fitted model and AIC_{ i } is that of the i ^{th} model. According to Burnham and Anderson (2004), models having Δ_{ i } ≤ 2 had substantial support (evidence) and those in which 4 ≤ Δ_{ i } have considerably less support. For computing the χ ^{2} goodness of fit statistics we group the cells whose expected number is less than 5 such that the expected number of grouped cell is not less than 5.
4.1 The spots in southern pine beetle
Distribution of Corbet’s Malayan Buttery with zeros (Corbet 1942)
Count  Observed  NB  Delaporte  COMP  COMNB  GCOMP  ECOMP 

0  304  315.36  303.10  104.93  315.36  315.36  304.97 
1  118  94.24  123.28  93.03  94.24  94.24  117.12 
2  74  59.76  62.83  82.47  59.76  59.76  67.25 
3  44  44.58  43.29  73.11  44.58  44.58  45.92 
4  24  35.74  33.73  64.81  35.74  35.74  34.51 
5  29  29.85  27.77  57.45  29.85  29.85  27.57 
6  22  25.60  23.61  50.93  25.60  25.60  22.94 
7  20  22.37  10.51  45.14  22.37  22.37  19.66 
8  19  19.81  18.10  40.02  19.81  19.81  17.23 
9  20  17.73  16.16  35.47  17.73  17.73  15.35 
10  15  16.0  14.57  31.44  16.00  16.00  13.86 
11  12  14.53  13.23  27.87  14.53  14.53  12.64 
12  14  13.27  12.09  24.70  13.27  13.27  11.63 
13  6  12.18  11.10  21.90  12.18  12.18  10.77 
14  12  11.23  10.24  19.41  11.23  11.23  10.04 
15  6  10.38  9.49  17.20  10.38  10.38  9.39 
16  9  9.63  8.82  15.25  9.63  9.63  8.83 
17  9  8.96  8.21  13.51  8.96  8.96  8.32 
18  6  8.35  7.68  11.98  8.35  8.35  7.86 
19  10  7.80  7.19  10.62  7.80  7.80  7.45 
20  10  7.30  6.74  9.41  7.30  7.30  7.07 
21  11  6.84  6.34  8.34  6.84  6.84  6.73 
22  5  6.42  5.97  7.39  6.42  6.42  6.40 
23  3  6.04  5.63  6.55  6.04  6.04  6.10 
24  3  5.69  5.32  5.81  5.69  5.69  5.82 
25+  119  114.36  119.00  45.27  114.36  114.36  118.57 
Total  924  924  924  924  924  924  924 
AIC  4516.22  4508.40  45021.16  4518.22  4518.22  4510.02  
χ ^{2}  28.50  19.46  660.91  28.50  28.50  18.57  
pvalue  0.20  0.62  0.00  0.15  0.15  0.61  
MLEs  \( \widehat{\nu}=0.31 \)  \( \widehat{r}=0.26 \)    \( \widehat{\nu}=0.31 \)  \( \widehat{\nu}=0.31 \)  \( \widehat{\nu}=2.86 \)  
\( \widehat{p}=0.97 \)  \( \widehat{p}=0.97 \)  \( \widehat{p}=0.01 \)  \( \widehat{p}=0.97 \)  \( \widehat{p}=0.97 \)  \( \widehat{p}=1.26 \)  
\( \widehat{\lambda}=0.15 \)  \( \widehat{\alpha}=0.89 \)  \( \widehat{\alpha}=1.00 \)    \( \widehat{\alpha}=1.07 \)  
\( \widehat{\beta}=1.00 \)  \( \widehat{\beta}=1.13 \) 
More over for it can be observed, the ML estimate \( \widehat{\alpha} \) of the COMNB distribution and ML estimate \( \widehat{\beta} \) of the GCOMP distribution show these two distributions reduce to the negative binomial distribution, while the proposed ECOMP distribution does not seem to reduce to the negative binomial distribution. Actually, the likelihood ratio test for H _{0}: Negative binomial distribution (α = β = 1) Vs H _{1}: ECOMP distribution (α ≠ 1 or β ≠ 1) rejects the negative binomial distribution (pvalue is 0.001). So the ECOMP distribution brings in substantial improvement in fitting this data set over both COMNB and GCOMP distributions.
4.2 The spots in southern pine beetle
The number of spots in southern pine beetle (Lin 1985)
Count  Observed  NB  PoiGE  COMP  COMNB  GCOMP  ECOMP 

0  1169  1165.32  1168.98  922.62  1168.56  1168.71  1169.00 
1  144  169.40  148.01  373.44  152.87  151.88  147.59 
2  92  78.91  82.91  151.15  80.17  80.69  84.21 
3  54  45.18  51.87  61.18  49.76  50.13  52.00 
4  29  28.28  33.51  24.76  32.60  32.75  33.17 
5  18  18.61  21.93  10.02  21.80  21.82  21.54 
6  10  12.65  14.44  4.06  14.68  14.64  14.14 
7  12  8.78  9.55  1.64  9.89  9.84  9.36 
8  6  6.20  6.32  0.66  6.65  6.60  6.23 
9  9  4.44  4.19  0.27  4.45  4.41  4.16 
10  3  3.20  2.78  0.11  2.96  2.93  2.79 
11  2  2.33  1.85  0.04  1.96  1.94  1.88 
12  0  1.71  1.23  0.02  1.29  1.28  1.27 
13  0  1.26  0.81  0.01  0.84  0.84  0.85 
14  1  0.93  0.54  0.00  0.55  0.55  0.58 
15  0  0.69  0.36  0.00  0.35  0.35  0.39 
16  0  0.52  0.24  0.00  0.23  0.23  0.27 
17  0  0.39  0.16  0.00  0.15  0.15  0.18 
18  0  0.29  0.11  0.00  0.09  0.09  0.12 
19  1  0.92  0.22  0.00  0.15  0.16  0.28 
Total  1550  1550.00  1550.00  1550.00  1550.00  1550.00  1550.00 
AIC  3117.28  3116.33  3518.85  3113.24  3112.99  3113.84  
χ ^{2}  14.79  8.56  669.08  8.99  8.65  8.18  
pvalue  0.06  0.13  0.00  0.25  0.28  0.23  
MLEs  \( \overset{\frown }{r}=0.18 \)  \( \overset{\frown }{v}=0.08 \)  \( \overset{\frown }{\theta }=0.00 \)  \( \overset{\frown }{v}=0.13 \)  \( \overset{\frown }{v}=0.10 \)  \( \overset{\frown }{v}=0.002 \)  
\( \overset{\frown }{p}=0.79 \)  \( \overset{\frown }{p}=0.59 \)  \( \overset{\frown }{p}=0.40 \)  \( \overset{\frown }{p}=1.03 \)  \( \overset{\frown }{p}=0.98 \)  \( \overset{\frown }{p}=0.69 \)  
\( \overset{\frown }{\beta }=2.03 \)    \( \overset{\frown }{\alpha }=1.15 \)    \( \overset{\frown }{\alpha }=0.28 \)  
\( \overset{\frown }{d}=0.68 \)  \( \overset{\frown }{\beta }=0.87 \)  \( \overset{\frown }{\beta }=0.28 \)  
\( \overset{\frown }{l}=1.05 \) 
4.3 Borrowing library books
The number of books that were borrowed k times (Burrell and Cane 1982)
Count  Observed  NB  PoiGE  COMP  COMNB  GCOMP  ECOMP 

1  9647  9522.59  9557.15  9364.75  9604.38  9576.21  9648.57 
2  4351  4595.67  4538.89  4706.62  4401.32  4458.54  4341.37 
3  2275  2296.39  2296.51  2365.50  2314.68  2311.93  2291.26 
4  1250  1167.09  1179.55  1188.87  1231.23  1213.94  1244.90 
5  663  599.12  608.51  597.51  643.23  630.96  663.27 
6  355  309.61  313.75  300.31  327.22  322.40  339.49 
7  154  160.75  161.73  150.93  161.73  161.61  165.46 
8  72  83.76  83.27  75.86  77.67  79.45  76.51 
9  37  43.76  42.82  38.12  36.28  38.33  33.54 
10  14  22.91  22.00  19.16  16.50  18.16  13.95 
11  6  12.02  11.28  9.63  7.32  8.46  5.51 
12  2  6.31  5.78  4.84  3.17  3.87  2.07 
13  0  3.32  2.96  2.43  1.34  1.75  0.74 
14  1  3.89  3.28  2.46  0.93  1.37  0.56 
Total  18,827  18827.00  18827.00  18827.00  18827.00  18827.00  18827.00 
AIC  52461.94  52453.14  52472.22  52414.70  52422.75  52411.67  
χ ^{2}  51.96  38.33  66.83  7.20  14.75  2.37  
pvalue  0.00  0.00  0.00  0.51  0.06  0.94  
MLEs  \( \overset{\frown }{r}=0.81 \)  \( \overset{\frown }{v}=0.27 \)  \( \overset{\frown }{\theta }=0.00 \)  \( \overset{\frown }{v}=0.01 \)  \( \overset{\frown }{v}=0.01 \)  \( \overset{\frown }{v}=2.30 \)  
\( \overset{\frown }{p}=0.53 \)  \( \overset{\frown }{p}=0.38 \)  \( \overset{\frown }{p}=0.50 \)  \( \overset{\frown }{p}=1.18 \)  \( \overset{\frown }{p}=0.93 \)  \( \overset{\frown }{p}=6.40 \)  
\( \overset{\frown }{\beta }=1.78 \)    \( \overset{\frown }{\alpha }=1.37 \)    \( \overset{\frown }{\alpha }=2.98 \)  
\( \overset{\frown }{d}=1.16 \)  \( \overset{\frown }{\beta }=0.74 \)  \( \overset{\frown }{\beta }=3.95 \)  
\( \overset{\frown }{l}=1.03 \) 
5 Concluding remarks
Extended ConwayMaxwellPoisson distribution proposed here unifies the COMNB and GCOMP which were recently introduced to add more flexibility to the COMPoisson distribution. The proposed distribution with additional parameter has more flexibility in terms of its tail behavior and dispersion level. Further it also arises from queuing theory set up and as exponential combination of negative binomial and COMPoisson distribution and has many interesting properties. It is therefore envisaged that ECOMP distribution has the potential in modeling varieties of count data.
Declarations
Acknowledgments
The corresponding author Prof. Subrata Chakraborty would like to thank the Editors –inChief Prof. Felix Famoye andProf. Carl Lee, for the invitation to write a paper for this esteemed Journal. Both the authors acknowledge the comments and suggestions of the editor and both the reviewers which lead to substantial improvement in the presentation of the work.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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