Mittag  Leffler function distribution  a new generalization of hyperPoisson distribution
 Subrata Chakraborty^{1} and
 S. H. Ong^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s4048801700609
© The Author(s). 2017
Received: 18 February 2016
Accepted: 29 May 2017
Published: 13 July 2017
Abstract
In this paper a new generalization of the hyperPoisson distribution is proposed using the MittagLeffler function. The hyperPoisson, displaced Poisson, Poisson and geometric distributions among others are seen as particular cases. This MittagLeffler function distribution (MLFD) belongs to the generalized hypergeometric and generalized power series families and also arises as weighted Poisson distributions. MLFD is a flexible distribution with varying shapes and has a unique mode at zero or it is unimodal with one/two nonzero modes. It can be under, equi or over dispersed. Various distributional properties like recurrence relation for probability mass function, cumulative distribution function, generating functions, formulas for different type of moments, their recurrence relations, index of dispersion and its classification, logconcavity, reliability properties like survival, increasing failure rate, unimodality, and stochastic ordering with respect to hyperPoisson distribution are discussed. A particular case of the distribution is shown to arise as the steady state probability of a queuing system under state dependent service rate. The distribution has been found to fare well when compared with the hyperPoisson and COMPoisson type negative binomial distributions in its suitability in empirical modeling of differently dispersed count data. It is therefore expected that the proposed MLFD with its interesting features and flexibility will be a useful addition as a model for count data.
Keywords
Index of dispersion Reliability Logconcavity Unimodality Stochastic ordering Generalized power series Hypergeometric family Empirical modelingMathematics Subject Classification (2000)
62E15 · 62 F03 · 62 N05Introduction
The Poisson distribution is a popular model for count data. However its use is restricted by the equality of its mean and variance (equidispersion). Many models with the ability to represent under, equi and over dispersion have been proposed in the research literature to overcome this restriction. Notable among these distributions are the hyperPoisson (HP) of Bardwell and Crow (1964), generalized Poisson of Consul (1989), doublePoisson of Efron (1986), Poisson polynomial of Cameron and Johansson (1997), weighted Poisson of Castillo and PérezCasany (2005) and COMPoisson of Conway and Maxwell (1962) (see also Shmueli et al., 2005).
where \( \varphi \left(1,\beta; \lambda \right)={\displaystyle \sum_{j=0}^{\infty}\frac{(1)_j}{{\left(\beta \right)}_j}\frac{\lambda^j}{j!}}, \) is the confluent hypergeometric function and (β)_{ j } = β(β + 1) ⋯ (β + j − 1).
Staff (1964) studied a displaced Poisson distribution which is the HP distribution with the parameter β restricted to be a positive integer. The case when β is negative was investigated later by Staff (1967). The HP distribution attracted the attention of many researchers of late. Kemp (2002) dealt with a qanalogue of the distribution and Ahmad (2007) proposed a ConwayMaxwellHP distribution. Roohi and Ahmad (2003a, 2003b) investigated moments of the HP distribution. Kumar and Nair (2011, 2012, 2013, 2015) studied various extensions and alternatives of the HP distribution. SáezCastillo and CondeSánchez (2013) studied a HP regression model for overdispersed and underdispersed count data. Best (2001) and Antic et al. (2006) considered the HP distribution in word length and text length research. Khazraee et al. (2015) investigated the application of the HP generalized linear model for analyzing motor vehicle crashes.
Consequently the proposed distribution is called the MittagLeffler function distribution (MLFD). In general adding an extra parameter increases the complexity. However, for the MLFD, the extra parameter α adds flexibility but retains computational tractability since computation of E _{ α, β }(λ) does not pose a problem due to many software packages (example MATLAB) which offered routines for its quick computation. When α = β = 1, the MLFD is the Poisson distribution which is equidispersed. However for α, β ≠ 1, the MLFD still can be equidispersed and this characteristic allows more flexibility in modeling of equidispersion by a nonPoisson multiparameter model compared to the single parameter Poisson model. The MLFD is shown to be logconcave and this confers a number of attractive properties for modeling and inference; see Walther (2009) for a good review of statistical modeling and inference with logconcave distributions. The proposed MLFD should not be confused with a class of discrete MittagLeffler distributions proposed by Pillai and Jayakumar (1995). It is pertinent to give a brief review of some developments in statistical models involving the MittagLeffler function.
Since for α = 1 this distribution reduces to the exponential distribution with mean 1, it can be treated as a generalization of the exponential distribution. Pillai (1990) studied different properties of this distribution. Jose and Pillai (1986), Jayakumar and Pillai (1993), Lin (1998), Jayakumar (2003), Jose et al. (2010) studied different aspects of this distribution.
Pillai and Jayakumar (1995) proposed a class of discrete MittagLeffler (DML) distributions having pgf P(z) = E(z ^{ X }) = 1/[1 + c(1 − z)^{ α }]. The DML distribution arises as a mixture of the Poisson distribution with parameter θλ, where θ is a constant and λ follows the MittagLeffler distribution in (3). They have studied different properties of the DML distribution, gave a probabilistic derivation and an application in a first order autoregressive discrete process. The DML is also a particular case of the discrete Linnik distribution (Devroye, 1990).
In this article we have taken a completely different route to propose a discrete distribution based on the MittagLeffler function. The proposed distribution, which under certain conditions also arises from a queuing theory setup, is simple and extremely flexible in its shape and modality and it can model under, equi and over dispersed count data. Section 2 defines the MLFD and basic structural properties are given. MLFD as a distribution in a queuing system is given in Section 3. Reliability and stochastic ordering properties are discussed in Section 4. Section 5 deals with parameter estimation and examples of applications of MLFD. The conclusion is given in Section 6.
MittagLeffler Function Distribution: Definition and Properties
In this section we define the proposed MLFD and investigate its main distributional, reliability and ordering properties.
where \( {E}_{\alpha, \beta}\left(\lambda \right)={\displaystyle \sum_{j=0}^{\infty }{\lambda}^j/\varGamma \left(\alpha j+\beta \right)} \) is the generalized MittagLeffler function. The distribution henceforth will be denoted by MLFD (λ, α, β).
Remark 1: The MLFD pmf (5) may be obtained by replacing k ! in the Poisson pmf e ^{− λ } λ ^{ k }/k ! with Γ(αk + β), and the normalization constant e ^{− λ } is now 1/E _{ α, β }(λ).
Recurrence relation between probabilities
with P(X = 0) = 1/{Γ(β) E _{ α, β }(λ)}.
When α is a positive integer, (6) can be expressed as (αk + β)_{ α } P(X = k + 1) = λP(X = k).
The distribution exhibits long tailedness for 0 < α < 1 as the ratio of successive probabilities varies slowly (this corresponds to overdispersion) as k tends to infinity while for α ≥ 1 this ratio tends to zero faster implying presence of a Poissontype tail.
The recurrence relation in (6) facilitates easy computation of the probabilities. The computation of the normalizing constant E _{ α, β }(λ) is only required for P(X = 0).
Note that the recurrence relation (or the difference equation) in (6) reduces to that of HP (λ, β) distribution for α = 1 and displaced Poisson distribution when α = 1 and β is an integer (further discussed in Section 2.5.1).
Computation of the generalized MittagLeffler function E_{αβ}(λ)
For other values of λ, asymptotic series and integral representation (see equations (2.3), (2.4) and (2.7) of Seybold and Hilfer, 2008) are employed. Error estimates are also given for these cases. The computation of the MittagLeffler function is given by many software packages like Matlab (MLF (alpha, Z, P)) and Mathematica (MittagLefflerE [a, b, z]). (See also Gorenflo et al., 2002.) See also Garrappa (2015) for a recent contribution towards numerical evaluation of MittagLeffler function.
Shapes of pmf
The pmf of MLFD (λ, α, β) is plotted for a number of combinations of parameters to study the different shapes of the distribution.
From the plots of the pmf it is seen that the distribution can be unimodal with nonzero mode (see Fig. 1(a)) or it can have nonzero modes at two points (see Fig. 1(h)) or nonincreasing with the mode at 0 (see Fig. 1(g)). See Section 4.3 item (i) for further discussion on the modes.
Cumulative distribution function and generating functions
by using the known relation \( {\lambda}^r{E}_{\alpha, \beta + r\;\alpha}\left(\lambda \right)={E}_{\alpha, \beta}\left(\lambda \right){\displaystyle \sum_{j=0}^{r1}{\lambda}^j/\varGamma \left(\alpha j+\beta \right)} \) (Haubold et al., 2011).
Related distributions and connections with other families of distributions
Particular cases of MLFD (λ, α, β)
 (i)
When α = β = 1, MLFD (λ, α, β) reduces to the Poisson distribution with parameter λ.
 (ii)
When α = 0, β (≥0) MLFD (λ, α, β) becomes the geometric distribution with parameter λ provided 0 < λ < 1. Since for α → 0^{+}, \( \underset{\alpha \to {0}^{+}}{ \lim }{E}_{\alpha, \beta}\left(\lambda \right)\to {\displaystyle \sum_{j=0}^{\infty}\frac{\lambda^j}{\varGamma \left(\beta \right)}}=\frac{1}{\left(1\lambda \right)\varGamma \left(\beta \right)},0<\lambda <1 \) (Hanneken et al., 2009).
 (iii)
When α = 1, β (≥0) MLFD (λ, α, β) reduces to the HP (λ, β) distribution (Bardwell and Crow, 1964; Johnson et al., 2005, p. 200).
Proof: \( P\left( X= k\right)=\frac{\lambda^k}{\varGamma \left( k+\beta \right){E}_{1,\beta}\left(\lambda \right)}, k=0,1,2,\cdots; \lambda >0 \) where \( {E}_{1,\beta}\left(\lambda \right)={\displaystyle \sum_{j=0}^{\infty }{\lambda}^j/\varGamma \left( j+\beta \right)=}{\displaystyle \sum_{j=0}^{\infty }{\lambda}^j/\left\{{\left(\beta \right)}_j\varGamma \left(\beta \right)\right\}} \) \( =\frac{1}{\varGamma \left(\beta \right)}{\displaystyle \sum_{j=0}^{\infty}\frac{(1)_j}{{\left(\beta \right)}_j}\frac{\lambda^j}{j!}}=\frac{\varphi \left(1,\beta; \lambda \right)}{\varGamma \left(\beta \right)} \).
Hence P(X = k) reduces to the pmf of HP (λ, β) distribution given in Eq. (1).
 (v)When α = 2 and β = 2, MLFD (λ, α, β) reduces to a new discrete distribution with parameter λ and pmf$$ P\left( X= k\right)=\frac{\lambda^{k+\left(1/2\right)}}{\varGamma \left(2\left( k+1\right)\right)}\frac{\kern0.24em 1}{ \sinh \left(\sqrt{\lambda}\right)}=\frac{{\left(\sqrt{\lambda}\right)}^{2 k+1}}{\left(2 k+1\right)!}\frac{\kern0.24em 1}{ \sinh \left(\sqrt{\lambda}\right)}, k=0,1,2,\cdots; \lambda >0, $$
 (vi)When α = 2 and β = 1, MLFD (λ, α, β) reduces to a new distribution with parameter λ and pmf$$ P\left( X= k\right)=\frac{\lambda^k}{\varGamma \left(2 k+1\right)}\frac{\kern0.24em 1}{ \cosh \left(\sqrt{\lambda}\right)}=\frac{{\left(\sqrt{\lambda}\right)}^{2 k}}{\left(2 k\right)!}\frac{\kern0.24em 1}{ \cosh \left(\sqrt{\lambda}\right)}, k=0,1,2,\cdots; \lambda >0 $$
 (vii)When α = 1/2 and β = 1, MLFD (λ, α, β) reduces to a new distribution with parameter λ and pmf$$ P\left( X= k\right)=\frac{ \exp \left({\lambda}^2\right)\kern0.24em {\lambda}^k}{\left( k/2\right)!\kern0.36em erfc\;\left(\lambda \right)}, k=0,1,2,\cdots; \lambda >0 $$
since \( {E}_{1/2,1}\left(\sqrt{\lambda}\right)= \exp \left(\lambda \right)\; erfc\;\left(\sqrt{\lambda}\right) \) (Haubold et al., 2011) where erfc (λ) is the
complementary error function defined as \( erfc\;\left(\lambda \right)=1 e r f\left(\lambda \right)=1\frac{2}{\sqrt{\pi}}{\displaystyle \underset{0}{\overset{\lambda}{\int }} \exp \left({t}^2\right)\; dt} \).
 (viii)
MLFD (λ, α, β) degenerates with mass only at zero for either α → ∞ or β → ∞ or both and also when λ → 0^{+}.
Remark 2. For 0 ≤ α ≤ 1, the MLFD (λ, α, β) can be viewed as a continuous bridge between the geometric (α = 0) and HP (α = 1) distributions in the range of the parameter α; in particular, the MLFD (λ, α, 1) can be viewed as a continuous bridge between the geometric (α = 0) and Poisson (α = 1) distributions in the range of the parameter α, a property also shared by the COMPoisson distribution.
MLFD as weighted Poisson distribution
then for integer α and β it can be shown that weighted distribution with weight function 1/(k + 1)_{(α − 1)k + β − 1} gives the pmf of MLFD (λ, α, β). Since the weight function 1/(k + 1)_{(α − 1)k + β − 1} is monotonically decreasing in k for α, β > 1, MLFD (λ, α, β) is stochastically smaller than the Poisson distribution when α, β > 1. (See Patil et al. 1986; Ross, 1983; Castillo and PérezCasany, 2005)
MLFD as member of some families of discrete distributions
 i.
MLFD (λ, α, β) is a member of the generalized hypergeometric family (Kemp 1968a, b). This can be checked by comparing the recurrence relation in Eq. (6) with that of the generalized hypergeometric distributions (see equation (2.63) in page 91 of Johnson et al., 2005).
 ii.
MLFD (λ, α, β) is a member of the generalized power series distribution (Patil 1962, 1964) when λ is the primary parameter.
 iii.
For fixed values of the parameters α and β, the MLFD (λ, α, β) is also a member of the exponential family of distributions.
Moments and related results
we can derive the following formulas:
\( {\mu}_2=\frac{1}{\alpha^2}\left\{\frac{E_{\alpha, \beta 2}\left(\lambda \right)}{\;{E}_{\alpha, \beta}\left(\lambda \right)}{\left(\frac{E_{\alpha, \beta 1}\left(\lambda \right)}{\;{E}_{\alpha, \beta}\left(\lambda \right)}\right)}^2+\frac{E_{\alpha, \beta 1}\left(\lambda \right)}{\;{E}_{\alpha, \beta}\left(\lambda \right)}\right\} \), provided α > 0 and β > 2.
The above results can alternatively be derived easily by first deriving E(αX + β)_{[r]}, r = 1, 2 and then simplifying.
(see Erdelyi 1955; Hanneken et al., 2009).
where \( \rho (n)={\displaystyle \sum_{i=1}^n\frac{1}{i}} \) and γ = − Γ ^{/}(1) is the Euler’s constant.
Recurrence relations of moments
 (i)$$ {\mu}_{r+1}^{/}=\lambda\;\frac{d}{ d\lambda}\;{\mu}_r^{/}+{\mu}_r^{/}\kern0.24em {\mu}_1^{/} $$
 (ii)$$ {\mu}_{r+1}=\lambda\;\frac{d}{ d\lambda}\;{\mu}_r+ r{\mu}_{r1}\;{\mu}_2 $$
 (iii)$$ {\mu}_{{}^{\left[ r+1\right]}}=\lambda\;\frac{d}{ d\lambda}\;{\mu}_{{}^{\left[ r\right]}}+\left({\mu}_1^{/} r\;\right){\mu}_{{}^{\left[ r\right]}} $$
 (iv)$$ E{\left(\alpha \left( X1\right)+\beta \right)}_{\alpha}=\lambda +{\left(\beta \alpha \right)}_{\alpha} P\left( X=0\right) $$
The relations (i) to (iii) can be proved by using the general relations for GPSD or by direct manipulation while (iv) follows from the difference equation in (6).
Since μ _{2} > 0 and λ ≠ 0, \( {\mu}_2=\lambda\;\frac{d}{ d\lambda}\;{\mu}_1^{/}>0 \) implies \( \frac{d}{ d\lambda}\;{\mu}_1^{/}>0 \). Hence \( {\mu}_1^{/} \) is a monotonically increasing function of λ.
Alternative formulae for moments
An alternative formula for moments is given by
\( E\left[{\left( X+1\right)}_r\right]= r!{E}_{\alpha, \beta}^{r+1}\left(\lambda \right)/{E}_{\alpha, \beta}\left(\lambda \right) \),
where \( {E}_{\alpha, \beta}^{\rho}\left(\lambda \right)={\displaystyle \sum_{k=0}^{\infty}\frac{{\left(\rho \right)}_k}{k!}\frac{\lambda^k}{\varGamma \left(\alpha k+\beta \right)}} \) is the generalized MittagLeffler function (Prabhakar, 1971).
since k ! (k + 1)_{ r } = r ! (r + 1)_{ k }.
Approximation of the mean and variance for large values of λ
Using the result that for large values of λ, E _{ α,β }(λ) → {λ ^{(1 − β)/α } exp(λ ^{1/α })}/α (see Gerhold, 2012) we can derive approximations for the mean and variance of MLFD (λ, α, β) as (1 − β + λ ^{1/α })/α and λ ^{1/α }/α ^{2} respectively. The expression for the mean follows from either \( {\mu}_1^{/}=\frac{E_{\alpha, \beta 1}\left(\lambda \right)}{\;\alpha\;{E}_{\alpha, \beta}\left(\lambda \right)}+\frac{1\beta}{\alpha} \) or directly from \( {\mu}_1^{/}=\lambda \frac{d}{ d\lambda} \log \left[{E}_{\alpha, \beta}\left(\lambda \right)\right] \). Then variance can be obtained by using the relation \( {\mu}_2=\lambda\;\frac{d}{ d\lambda}\;{\mu}_1^{/} \).
In particular for MLFD (λ, α, 1) the approximate mean and variance will be λ ^{1/α }/α and λ ^{1/α }/α ^{2}. These approximations are good when α ∈ (0, 2] (see Simon, 2013) and may be useful in a regression formulation where the covariates are linked through the mean and variance.
Index of dispersion
From Fig. 2 (a) and (b), it is obvious that the MLFD (λ, α, β) is very flexible with respect to the ID and is able to accommodate under, equi and over dispersion in count data. Interestingly, this family includes a nonPoisson distribution with equidispersion when λ is kept fixed. Some such pairs of values for (α, β) can be easily taken from line of equidispersion in the contour plots in Fig. 2(a) when λ = 0.25 and from Fig. 2(b) when λ = 5.
Using results of the Section 2.6.3 for large values of λ it can be stated that the ID of MLFD (λ, α, β) is approximately given by λ ^{1/α }/α((1 − β) + λ ^{1/α }) which reduces to 1/α for MLFD. (λ, α, 1). Thus for large λ MLFD (λ, α, 1) expected to be over (under) dispersed depending on α < (>)1, while MLFD (λ, α, β) will be underdispersed in the region α > 1, β < 1.
MLFD (z, α, 1) as a distribution in a queuing system
MLFD (z, α, 1), like the COMPoisson distribution, can be derived as the probability of the system being in the kth state for a queuing system with state dependent service rate.
Consider a queuing system with Poisson inter arrival times with parameter λ, firstcome firstserved policy, and exponential service times that depend on the system state (nth state means n number of units in the system). The mean service time in the nth state is μ _{ n } = μ (n α)_{[α]}, n ≥ 1 and μ _{ n } = μ for α = 0, where, 1/μ is the normal mean service time for a unit when that unit is the only one in the system and α is the pressure coefficient, a constant reflecting the degree to which the service rate of the system is affected by the system. For the sake of completeness, the proof that the probability is the pmf of MLFD (z, α, 1) where z = λ/μ is given as follows:
From (7) we get P _{0}(t + Δ) − P _{0}(t) = − λ ΔP _{0}(t) + μ α ! ΔP _{1}(t) since μ _{1} = μ(α)_{ α } = μα !. This implies, \( \underset{\varDelta \to 0}{ \lim}\frac{P_0\left( t+\varDelta \right){P}_0(t)}{\varDelta}=\lambda\;{P}_0(t)+\mu\;\alpha !{P}_1(t) \)
or \( {P}_0^{/}(t)=\lambda\;{P}_0(t)+\mu\;\alpha !{P}_1(t) \).
Assuming a steady state (i.e. \( {P}_n^{/}(t)=0 \) for all n), we get P _{1}(t) = zP _{0}(t)/α ! where λ/μ = z. Similarly, from (8), we get
\( \underset{\varDelta \to 0}{ \lim}\frac{P_n\left( t+\varDelta \right){P}_n(t)}{\varDelta}=\left(\lambda +{\left( n\alpha \right)}_{\left[\alpha \right]}\mu\;\right){P}_n(t)+\lambda \kern0.24em {P}_{n1}(t)+{\left(\left( n+1\right)\alpha \right)}_{\left[\alpha \right]}\mu\;{P}_{n+1}(t) \).
because \( {P}_n^{/}(t)=0 \) for all n and λ/μ = z.
This implies that (z + (nα)_{[α]} )P _{ n }(t) = z P _{ n − 1}(t) + ((n + 1)α)_{[α]} P _{ n + 1}(t) since μ ≠ 0.
Putting n = 1 we get
(z + α !)P _{1}(t) = zP _{0}(t) + (2α)_{[α]} P _{2}(t),
P _{2}(t) = {z ^{2}/(2α) ! }P _{0}(t). Since α ! (2α)_{[α]} = (2α) !
P _{2}(t) = {z ^{3}/(3α) ! }P _{0}(t),
since (3α ! )(3α)_{[α]} = (3α) !
In general, P _{ n }(t) = {z ^{ n }/(nα) ! }P _{0}(t),
where \( {P}_0(t)=1/{\displaystyle \sum_{n=0}^{\infty}\left\{{z}^n/\left( n\alpha \right)!\right\}} \). This is the pmf of MLFD (z, α, 1).
In the case when α is not an integer one can use α ! = Γ(α + 1).
Reliability, stochastic ordering and log concavity
Discrete life time models have lately been a favorite subject of many studies since in many situations the life of a system may be observed as counts, and even when the life is measured in a continuous scale the actual observations may be recorded in a way making a discrete model more appropriate. It is therefore important to study the reliability properties of the proposed discrete distribution. Stochastic ordering is a closely related important area that has found applications in many diverse areas such as economics, reliability, survival analysis, insurance, finance, actuarial and management sciences (see Shaked and Shanthikumar, 2007). In this section we study the reliability properties and stochastic ordering of the MLFD (λ, α, β) distribution.
Survival and failure rate function
Stochastic ordering with HP
The following result stochastically compares the MLFD (λ, α, β) with the HP (λ, β) by using the likelihood ratio order.
Definitions: Let X and Y be two discrete random variables with pmfs f(x) and g(x). Then X is said to be smaller than Y in the likelihood ratio order denoted by X ≤ _{ lr } Y if g(x)/f(x) increases in x over the union of the supports of X and Y; X is smaller than Y in the hazard rate order X ≤ _{ hr } Y if r _{ X }(t) ≥ r _{ Y }(t) for all t; X is smaller than Y in the mean residual life order X ≤ _{ MRL } Y if μ _{ X }(t) ≤ μ _{ Y }(t) for all t, where r _{ X }(.) and μ _{ X }(.) are respectively the hazard rate and mean residual life (MRL) functions of X.
Theorem 1. For α > 1, X ~ MLFD (λ, α, β) is smaller than Y ~ HP (λ, β) distribution in the likelihood ratio order i.e. X ≤ _{ lr } Y, while for 0 < α < 1, HP (λ, β) is greater than MLFD (λ, α, β) distribution in the likelihood ratio order i.e. Y ≤ _{ lr } X.
For α > 1 this ratio is clearly increasing in n (see Shaked and Shanthikumar, 2007 and Gupta et al., 2014). Hence X ≤ _{ lr } Y is proved. While for 0 < α < 1, the ratio is decreasing in n which proves that Y ≤ _{ lr } X.
Corollary 1. For α > 1, X ~ MLFD (λ, α, β) is smaller than Y ~ HP (λ, β) distribution in the MRL life order that is X ≤ _{ MRL } Y.
Proof. The result follows since X ≤ _{ lr } Y ⇒ X ≤ _{ hr } Y ⇒ X ≤ _{ MRL } Y. (see Gupta et al., 2014).
Logconcavity
The logconcavity of any probability distribution has important implications on its reliability function, failure rate function, tail probabilities and moments. The MLFD (λ, α, β) has a logconcave pmf since for this distribution (Gupta et al., 1997)
\( \varDelta\;\eta (t)=\frac{P\left( t+1\right)}{P(t)}\frac{P\left( t+2\right)}{P\left( t+1\right)}=\lambda \frac{\varGamma \left(\alpha\; t+\beta \right)\varGamma \left(\alpha\; t+2\;\alpha +\beta \right){\left\{\varGamma \left(\alpha\; t+\alpha +\beta \right)\right\}}^2}{\varGamma \left(\alpha\; t+\alpha +\beta \right)\varGamma \left(\alpha\; t+2\;\alpha +\beta \right)} \) > 0.
 i.MLFD (λ, α, β) is a strongly unimodal distribution due to the logconcavity of its pmf (see Steutel 1985).

➢ MLFD (λ, α, β) has a unique mode at X = k if

$$ \varGamma \left(\alpha k+\beta \right)/\varGamma \left(\alpha k\alpha +\beta \right)<\lambda <\varGamma \left(\alpha k+\alpha +\beta \right)/\varGamma \left(\alpha k+\beta \right) $$

Proof: This follows easily from the probability recurrence relation given in (5).

➢ MLFD (λ, α, β) has a non increasing pmf with a unique mode at X = 0 if λ < Γ(α + β)/Γ(β) (See the pmf plots in Fig. 1(g) for some choices of (λ, α, β) satisfying the condition.)

➢ MLFD (λ, α, β) has two modes at X = k and X = k + 1 if

$$ \lambda =\varGamma \left(\alpha k+\alpha +\beta \right)/\varGamma \left(\alpha k+\beta \right) $$

(See the pmf plot in Fig. 1(h) for some (λ, α, β) satisfying the condition.)

 ii.
MLFD (λ, α, β) has non decreasing failure rate function.
 iii.
MLFD (λ, α, β) has at most an exponential tail.
 iv.
MLFD (λ, α, β) remains logconcave if truncated.
 v.
\( \frac{P\left( X= i+ k\right)}{P\left( X= i\right)}\ge \frac{P\left( X= j+ k\right)}{P\left( X= j\right)}\;\mathrm{f}\mathrm{o}\mathrm{r}\kern0.24em i< j \).
Data Fitting
Parameter estimation
Suppose that we have a sample of size n from MLFD (λ, α, β) reported as grouped frequencies in k classes, like (X, f) = {(x _{1}, f _{1}), (x _{2}, f _{2,}), . . . , (x _{ k }, f _{ k })}, where f _{ i } is the frequency of ith observed value x _{ i } and n = ∑f _{ i } is the sample size. Then the loglikelihood function is given by
\( \log L\left(\left.{x}_1,{x}_2,\cdots, {x}_k\right\;\lambda, \alpha, \beta \right)=\left({\displaystyle \sum_{i=1}^k{f}_i\;{x}_i}\right) \log \lambda {\displaystyle \sum_{i=1}^k{f}_i \log \varGamma \left(\alpha\;{x}_i+\beta \right)} n \log {E}_{\alpha, \beta}\left(\lambda \right) \).
Numerical optimization method is used to obtain the maximum likelihood estimates (MLE) of the parameters required for the data fitting and the likelihood ratio test.
Numerical examples
Observed and expected frequencies of trips made by Dutch households owning at least one car during a particular survey week in 1989(van Ophem, 2000)
x  Observed  Expected Frequencies  

HP (λ, β)  MLFD (λ, α, 1)  MLFD (λ, α, β)  COMNB (λ, α, β)  
0  75  93.208  99.150  74.175  89.965 
1  312  269.133  273.608  308.769  284.076 
2  384  403.029  399.3458  420.586  411.107 
3  421  407.420  399.724  389.683  397.963 
4  307  310.848  305.795  286.233  296.794 
5  183  190.456  189.821  178.113  183.213 
6  77  97.491  99.311  97.428  97.687 
7  47  42.853  44.958  47.962  46.258 
8  15  16.504  17.9533  21.598  19.836 
9  9  5.656  6.4181  9.003  7.813 
10  5  1.746  2.078  3.506  2.858 
11  0  0.490  0.615  1.285  0.979 
12  0  0.126  0.166  0.445  0.316 
13  1  0.030  0.042  0.147  0.097 
14  2  0.007  0.010  0.046  0.028 
15  0  0.001  0.002  0.014  0.008 
16  0  0.000  0.000  0.004  0.002 
17  1  0.000  0.000  0.002  0.001 
Total  1839  1839  1839  1839  1839 
\( \widehat{\lambda} \)  3.110 (0.121)  2.697 (0.182)  1.000 (0.130)  3.288 (4.25)  
\( \widehat{\alpha} \)    0.943 (0.032)  0.576 (0.063)  0.960 (1.22)  
\( \widehat{\beta} \)  1.080(0.121)    0.183 (0.058)  1.509 (0.40)  
χ ^{2}, df  37.034, 7  32.564, 8  17.716, 8  20.676, 8  
pval  <0.01  <0.01  0.023  <0.01  
AIC  7234.620  7231.970  7213.917  7218.610 
Frequency distribution of the α particles emitted by a radioactive substance in 2608 periods (Rutherford and Geiger, 1910)
x  Observed  Expected Frequencies  

HP (λ, β)  MLFD (λ, α, 1)  MLFD (λ, α, β)  COMNB (λ, α, β)  
0  57  49.916  48.766  56.731  40.352 
1  203  206.489  202.884  198.608  202.736 
2  383  408.368  406.600  393.766  420.265 
3  525  530.655  533.254  527.594  543.646 
4  532  513.471  518.076  523.971  515.429 
5  408  395.776  398.948  408.328  389.504 
6  273  253.493  254.119  259.549  246.804 
7  139  138.885  137.888  138.392  135.597 
8  45  66.481  65.122  63.227  66.139 
9  27  28.253  27.212  25.167  29.143 
10  10  10.796  10.192  8.846  11.756 
11  4  3.748  3.457  2.776  4.387 
12  0  1.192  1.071  0.785  1.527 
13  1  0.350  0.306  0.202  0.499 
14  1  0.127  0.106  0.059  0.217 
Total  2608  2608  2608  2608  2608 
\( \widehat{\lambda} \)  3.789(0.101)  4.228(0.286)  10.166 (7.06)  5.284(16.70)  
\( \widehat{\alpha} \)    1.037(0.028)  1.281 (0.182)  0.951(3.121)  
\( \widehat{\beta} \)  0.916(0.096)    2.170 (1.044)  1.527(0.921)  
χ ^{2}, df  18.382, 11  11.367, 10  8.179, 9  22.326, 9  
pval  0.073  0.330  0.516  0.008  
AIC  10707.500  10706.412  10705.400  10718.180 
The MLFD (λ, α, β) model is a generalization/extension of the HP (λ, β) and MLFD (λ, α, 1). Therefore the MLFD (λ, α, β) model has been fitted and compared with the HP (λ, β), MLFD (λ, α, 1) to ascertain the benefits accrued through the proposed generalization. In addition, we have also considered a recently introduced threeparameter distribution namely the COMPoisson type negative binomial [COMNB (λ, α, β)] distribution having pmf P(X = k) = (λ)_{ k } α ^{ k }/{(k !)^{ β } S _{ β,λ }(α)}, k = 0, 1, 2, ⋯ where S _{ β,λ }(α) is the normalizing constant (Chakraborty and Ong, 2016) for comparative data fitting. The performances of various distributions are compared using the AIC (Akaike Information Criterion) defined as AIC = −2 log L + 2 k, where k is the number of parameter(s) and log L is the maximum of loglikelihood for a given data set (Burnham and Anderson, 2004). We also provide chisquare goodness of fit statistics with pvalues. In Tables 1 and 2 the degrees of freedoms for χ ^{2} are given alongside its value and the standard errors for parameter estimates are given within parentheses.
From Table 1 it can be seen that the MLFD (λ, α, β) gives the best fit since it has the lowest χ ^{2} value and is also the first choice in model selection with lowest AIC value. Moreover MLFD (λ, α, β) is the only distribution with good fit at 1% since its pvalue is more than 0.01 while for the rest the pvalues are less than 0.01.
From Table 2 considering the χ ^{2} values together with pvalues it can be seen that all the distributions except the COMNB (λ, α, β) give adequate fits. But among them the MLFD (λ, α, β) gives the best fit since it has the lowest value of χ ^{2} with the highest pvalue. It is also the selected model because it has the lowest AIC value.
Conclusion
A new generalization of the HP distribution which is a continuous bridge between the geometric and HP is derived using the generalized MittagLeffler function. Some known and new distributions are seen as particular cases of this distribution. This new generalization belongs to the generalized power series, generalized hypergeometric families and also arises as weighted Poisson distributions. Like the HP, COMPoisson and generalized Poisson distributions, this distribution is also able to cater for under, equi and over dispersion. Although the new generalization of the HP distribution has an extra parameter, it is computationally not more complicated than the HP since it retains the twoterm probability recurrence formula and the normalizing constant, in terms of the generalized MittagLeffler function, is readily computed. It has many interesting probabilistic and reliability properties and is found to be a better empirical model than the HP distribution.
Declarations
Acknowledgements
The first author wishes to acknowledge the hospitality of Institute of Mathematical Sciences, University of Malaya during his visit to the institute in the summer of 2014. S.H. Ong wishes to acknowledge support in parts from Ministry of Education FRGS grant FP0452015A and University of Malaya’s UMRGS grant RP009A13AFR. The authors also wish to acknowledge the significant comments and suggestions of the Associate Editor and reviewers in earlier versions of the paper which have helped to improve the presentation tremendously.
Authors’ contributions
Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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References
 Ahmad, M.: A short note on ConwayMaxwellhyper Poisson distribution. Pak J Stat 23, 135–137 (2007)MathSciNetMATHGoogle Scholar
 Antic, G., Stadlober, E., Grzybek, P., Kelih, E.: Word length and frequency distributions in different text genres. In: Spiliopoulou, M., Kruse, R., Nürnberger, A., Borgelt, C., Gaul, W. (eds.) From Data and Information Analysis to Knowledge Engineering, pp. 310–317. Springer, Heidelberg (2006)View ArticleGoogle Scholar
 Bardwell, G.E., Crow, E.L.: A two parameter family of hyperPoisson distributions. J Am Stat Assoc 59, 133–141 (1964)MathSciNetView ArticleMATHGoogle Scholar
 Best, K.: Kommentierte Bibliographie zum Göttinger Projekt. In: Best, K.H. (ed.) Häufigkeitsverteilungen in Texten, pp. 248–310. Pest & Gutschmidt, Göttingen (2001)Google Scholar
 Burnham, K.P., Anderson, D.R.: Multimodel Inference, Understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304 (2004)MathSciNetView ArticleGoogle Scholar
 Cameron, A.C., Johansson, P.: Count Data Regression Using series expansions: with applications. J Appl Econ 12(3), 203–223 (1997)View ArticleGoogle Scholar
 Castillo, J.D., PérezCasany, M.: Overdispersed and underdispersed Poisson generalizations. Journal of Statistical Planning and Inference 134, 486–500 (2005)MathSciNetView ArticleMATHGoogle Scholar
 Chakraborty, S., Ong, S.H.: A COMPoissontype generalization of the negative binomial distribution. Communications in Statistics Theory and Methods 45(14), 4117–4135 (2016)MathSciNetView ArticleMATHGoogle Scholar
 Consul, P.C.: Generalized Poisson distributions: properties and applications. Marcel Dekker Inc, New York/Basel (1989)MATHGoogle Scholar
 Conway, R.W., Maxwell, W.L.: A queuing model with state dependent service rates. J Ind Eng 12, 132–136 (1962)Google Scholar
 Crow, E.L., Bardwell, G.E.: Estimation of the parameters of the hyperPoisson distributions. In: Patil, G.P. (ed.) Classical and Contagious Discrete Distributions, pp. 127–140. Pergamon, Oxford (1965)Google Scholar
 Devroye, L.: A note on Linnik’s distribution. Statistics and Probability Letters 9, 305–306 (1990)MathSciNetView ArticleMATHGoogle Scholar
 Efron, B.: Double ExponentialFamilies and their use in Generalized LinearRegression. J Am Stat Assoc 81(395), 709–721 (1986)MathSciNetView ArticleMATHGoogle Scholar
 Erdelyi, A.: Higher transcendental functions, vol. 3. McGrawHill, New York (1955)MATHGoogle Scholar
 Fisher, B., Kilicman, A.: Some results on the gamma function for negative integers. Appl Math Inf Sci 6(2), 173–176 (2012)MathSciNetGoogle Scholar
 Garrappa, R.: Numerical evaluation of two and three parameter MittagLeffler functions. Siam J Numer Anal 53(3), 1350–1369 (2015)MathSciNetView ArticleMATHGoogle Scholar
 Gerhold, S.: Asymptotics for a variant of the MittagLeffler function. Integral Transforms and Special functions 23(6), 397–403 (2012)MathSciNetView ArticleMATHGoogle Scholar
 Gorenflo, R., Joulia, L., Luchko, Y.: Computation of the MittagLeffler function Eα, β(z) and its derivatives. Fractional Calculus & Applied Analysis 5(4), 491–518 (2002)MathSciNetMATHGoogle Scholar
 Gupta, P.L., Gupta, R.C., Tripathi, R.C.: On the monotonic properties of discrete failure Rates. Journal of Statistical Planning and Inference 65, 255–268 (1997)MathSciNetView ArticleMATHGoogle Scholar
 Gupta, R.C., Sim, S.Z., Ong, S.H.: Analysis of discrete data by ConwayMaxwell Poisson distribution. AStA Advances in Data Analysis 98, 327–343 (2014)MathSciNetView ArticleGoogle Scholar
 Hanneken, J.W., Achar, B.N.N., Puzio, R., Vaught, D.M.: Properties of the Mittag–Leffler function for negative alpha. Phys Scr. T136, 014037, 1–5 (2009). doi:10.1088/00318949/2009/T136/014037
 Haubold, H.J., Mathai, A.M., Saxena, R.K.: MittagLeffler functions and their applications. J Applied Math, Article ID 298628, p. 51 (2011). doi:10.1155/2011/298628
 Jayakumar, K.: On MittagLeffler process. Math Comput Model 37, 1427–1434 (2003)MathSciNetView ArticleMATHGoogle Scholar
 Jayakumar, K., Pillai, R.N.: The first order autoregressive MittagLeffler process. J Appl Probab 30, 462–466 (1993)MathSciNetMATHGoogle Scholar
 Johnson, N.L., Kemp, A.W., Kotz, S.: Univariate Discrete Distributions. Wiley, New York (2005)View ArticleMATHGoogle Scholar
 Jose, K.K., Abraham, B.: A count data model based on MittagLeffler inter arrival times. Statistica 71(4), 501–514 (2011)Google Scholar
 Jose, K.K., Pillai, R.N.: In: Yagen Thomas, P. (ed.) Generalized autoregressive time series models in MittagLeffler variables. Recent Advances in Statistics, pp. 96–103. (1986)Google Scholar
 Jose, K.K., Uma, P., Seethalekshmi, V., Haubold, H.J.: Generalized MittagLeffler processes for applications in astrophysics and time series modelling. In: Haubold, H.J., Mathai, A.M. (eds.) Proceedings of the third UN/ESA/NASA workshop on the International Heliophysical Year 2007 and Basic sciences, Astrophysics and Space Science Proceedings, pp. 79–92. (2010). doi:10.1007/9783642033254_9 Google Scholar
 Kemp, A.W.: Studies in Univariate discrete distribution theory based on the generalized hypergeometric function and associated differential equations. PhD Thesis. The Queen’s University of Belfast, Belfast (1968a)Google Scholar
 Kemp, A.W.: A wide class of discrete distributions and the associated differential equations. Sankhya, Series A 30, 401–410 (1968b)MATHGoogle Scholar
 Kemp, C.D.: qanalogues of the hyperPoisson distribution. Journal of Statistical Planning and Inference 101, 179–183 (2002)MathSciNetView ArticleMATHGoogle Scholar
 Khazraee, S.H., SáezCastillo, A.J., Geedipally, S.R.: Lord D: Application of the hyperPoisson generalized linear model for analyzing motor vehicle crashes. Risk Anal 35(5), 919–930 (2015)View ArticleGoogle Scholar
 Kumar, C.S., Nair, B.U.: A modified version of hyperPoisson distribution and its applications. J Stat Appl 6, 23–34 (2011)Google Scholar
 Kumar, C.S., Nair, B.U.: An alternative hyperPoisson distribution. Statistica 3, 357–369 (2012)Google Scholar
 Kumar, C.S., Nair, B.U.: Modified alternative hyper Poisson distribution. In: Kumar, C.S., Chacko, M., Sathar, E.I.A. (eds.) Collection of recent statistical methods and applications, Department of Statistics, pp. 97–109. University of Kerala publication, Trivandrum (2013)Google Scholar
 Kumar, C.S., Nair, B.U.: On a class of hyperPoisson and alternative hyperPoisson distributions. OPSEARCH (Operational Research Society of India) 52(1), 86–100 (2015). doi:10.1007/s1259701301697 MathSciNetMATHGoogle Scholar
 Lee, P.A., Ong, S.H., Srivastava, H.M.: Some integrals of the products of Laguerre polynomials. Internat J Comput Math 78, 303–321 (2001)MathSciNetView ArticleMATHGoogle Scholar
 Lin, G.D.: On MittagLeffler distribution. Journal of Statistical Planning and Inference 74, 1–9 (1998)MathSciNetView ArticleMATHGoogle Scholar
 Mark, Y.A.: Logconcave probability distributions: Theory and statistical testing. Working paper, Published by Center for Labour Market and Social Research, University of Aarhus and the Aarhus School of Business, WorkingPaper 96–01, (1996)Google Scholar
 MittagLeffler, G.: Sur la nouvelle fonction E_{α}(x). Comptes Rendus Acad Sci Paris 137, 554–558 (1903)MATHGoogle Scholar
 MittagLeffler, G.: Sur la représentation analytique d’une branche uniforme d’une fonction monogene. Acta Math 29, 101–181 (1905)MathSciNetView ArticleMATHGoogle Scholar
 Patil, G.P.: Estimation by two moments method for generalized power series distribution and certain applications. Sankhya, B 24(3&4), 201–214 (1962)MathSciNetGoogle Scholar
 Patil, G.P.: In: Rao, C.R. (ed.) Estimation for the generalized power series distribution with two parameters and its application to binomial distribution. Contributions to Statistics, pp. 335–344. Calcutta: Statistical Publishing Society; Oxford, Pergamon (1964)Google Scholar
 Patil, G.P., Rao, C.R., Ratnaparkhi, M.V.: On discrete weighted distributions and their use in model for observed data. Communications in StatisticsTheory and Methods 15(3), 907–918 (1986)MathSciNetView ArticleMATHGoogle Scholar
 Pillai, R.N.: On MittagLeffler functions and related distributions. Ann Inst Statist Math 42, 157–161 (1990)MathSciNetView ArticleMATHGoogle Scholar
 Pillai, R.N., Jayakumar, K.: Discrete MittagLeffler distributions. Statistics & Probability Letters 23, 271–274 (1995)MathSciNetView ArticleMATHGoogle Scholar
 Prabhakar, T.R.: A singular integral equation with a generalized MittagLeffler function in the Kernel. Yokohama Math J 19, 7–15 (1971)MathSciNetMATHGoogle Scholar
 Roohi, A., Ahmad, M.: Estimation of the parameter of hyperPoisson distribution using negative moments. Pak J Stat 19, 99–105 (2003a)MathSciNetMATHGoogle Scholar
 Roohi, A., Ahmad, M.: Inverse ascending factorial moments of the hyperPoisson probability distribution. Pak J Stat 19, 273–280 (2003b)MathSciNetMATHGoogle Scholar
 Ross, S.M.: Stochastic Processes. Wiley, New York (1983)MATHGoogle Scholar
 Rutherford, E., Geiger, H.: The probability variations in the distribution of α particles. Philos Mag 20, 698–704 (1910)View ArticleGoogle Scholar
 SáezCastillo, A.J., CondeSánchez, A.: A hyperPoisson regression model for overdispersed and underdispersed count data. Computational Statistics and Data Analysis 61, 148–157 (2013)MathSciNetView ArticleMATHGoogle Scholar
 Seybold, H., Hilfer, R.: Numerical algorithm for calculating the generalized Mittag Leffler function. SIAM J Numer Anal 47(1), 69–88 (2008)MathSciNetView ArticleMATHGoogle Scholar
 Shaked, M., Shanthikumar, J.G.: Stochastic Orders. Springer Verlag (2007)Google Scholar
 Shmueli, G., Minka, T.P., Kadane, J.B., Borle, S., Boatwright, S.: A useful distribution for fitting discrete datarevival of the ConwayMaxwell Poisson distribution. Appl Stat 54, 127–142 (2005)MathSciNetMATHGoogle Scholar
 Simon, T.: Mittag  Leffler functions and complete monotonicity. arXiv: 1312. 4513v [math.CA] (2013)Google Scholar
 Staff, P.J.: The displaced Poisson distribution. Australian Journal of Statistics 6, 12–20 (1964)MathSciNetView ArticleMATHGoogle Scholar
 Staff, P.J.: The displaced Poisson distribution. Region B. Journal of American Statistical Association 62(318), 643–654 (1967)MathSciNetMATHGoogle Scholar
 Steutel, F.W.: In: Kotz, S., Johnson, N.L., Read, C.B. (eds.) Logconcave and logconvex distributions. Volume 5, Encyclopedia of Statistical Sciences. (1985)Google Scholar
 van Ophem, H.: Modeling selectivity in countdata models. J Bus Econ Stat 18, 503–511 (2000)View ArticleGoogle Scholar
 Walther, G.: Inference and modeling with logconcave distributions. Stat Sci 24(3), 319–327 (2009)MathSciNetView ArticleMATHGoogle Scholar