Abstract
In this paper a new generalization of the hyper-Poisson distribution is proposed using the Mittag-Leffler function. The hyper-Poisson, displaced Poisson, Poisson and geometric distributions among others are seen as particular cases. This Mittag-Leffler 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 non-zero 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, log-concavity, reliability properties like survival, increasing failure rate, unimodality, and stochastic ordering with respect to hyper-Poisson 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 hyper-Poisson and COM-Poisson 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.
Highlights
The Poisson distribution is a popular model for count data
In this paper we propose a new generalization of the HP distribution by replacing Γ(k + β) in (1) with Γ(α k + β), α > 0 and the normalization constant becomes Eα, β(λ) which is the generalized Mittag-Leffler function defined by
The discrete Mittag-Leffler (DML) distribution arises as a mixture of the Poisson distribution with parameter θλ, where θ is a constant and λ follows the Mittag-Leffler distribution in (3)
Summary
The Poisson distribution is a popular model for count data. its use is restricted by the equality of its mean and variance (equi-dispersion). The DML distribution arises as a mixture of the Poisson distribution with parameter θλ, where θ is a constant and λ follows the Mittag-Leffler 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. This family includes a non-Poisson distribution with equidispersion when λ is kept fixed. Consider a queuing system with Poisson inter arrival times with parameter λ, firstcome- first-served policy, and exponential service times that depend on the system a
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