On Discrete Distributions Generated through Mittag-Leffler Function and their Properties

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1 On Discrete Distributions Generated through Mittag-Leffler Function and their Properties Mariamma Antony Department of Statistics Little Flower College, Guruvayur Kerala, India Abstract The Poisson distribution is a popular model for count data. However, its use is restricted by the equality of its mean and variance (equi-dispersion). Many models with the ability to represent under, equi and over dispersed data have been proposed in the literature to overcome this restriction. One of this is the hyper Poisson(HP) distribution of Bardwell and Crow(1964). Charaborty and Ong(2017) introduced a generalization of HP distribution using Mittag-Leffler function. The hyper Poisson, displaced Poisson, Poisson and geometric distribution are particular cases of this distribution. This Mittag- Leffler function distribution(mlfd) belongs to the generalized hyper geometric and generalized power series family and also arises as weighted Poisson distributions. MLFD is a flexible distribution with varying shape and can be modeled for under, equi or over dispersed data. Charaborty and Ong(2017) studied various distributional properties of MLFD such as recurrence relations for pmf, generating functions, formulae for different types of moments, reliability properties and stochastic ordering. They showed that the MLFD is suitable in empirical modeling of differently dispersed count data when compared with hyper Poisson distribution. Pillai and Jayaumar (1995) introduced a discrete distribution as a discrete analogue of Mittag- Leffler distribution. They established its properties such as infinite divisibility, membership in discrete class L, etc. In this paper, we review and study various discrete distributions developed using Mittag-Leffler function and discuss their applications. Keywords - Discrete Class L; Hyper Poisson; Infinite Divisibility; Mittag-Leffler Function; Power Series Family I. INTRODUCTION Mittag-Leffler functions and distributions have attracted the attention of various researchers recently. Pillai(1990) introduced the Mittag-Leffler distribution in terms of the Mittag-Leffler function.the Mittag-Leffler function was introduced by Swedish Mathematician Gosta Mittag- Leffler in 1903 (Mittag-Leffler, 1903) in connection with his method of summations of some divergent series. The function E α (u) = =0, u (0, ) u (1+ α) is nown as Mittag-Leffler function. It arises as the solution of certain boundary value problems involving fractional differential equation. During various developments of fractional calculus in the last two decades, this function has gained importance and popularity on account of its vast applications in the field of Science and Technology (see, Mathai (2010) and Pillai(1990)). Feller(1971) showed that the Laplace transform of E α ( x α ) for 0 α 1 is α 1 1+ α, 0. But E α ( x α ) is not a probability distribution. Pillai(1990) showed that F α (x) = 1- E α ( x α ) is a distribution function. Hence he named F α (x) as Mittag-Leffler distribution. ( 1) 1 x α We have F α (x) = =1, 0< α 1, x 0 (1+α) and the corresponding density function is f α (x) = ( 1) 1 x α 1 =1. Huillet(2016) discussed various (α) properties of Mittag-Leffler distribution and related stochastic processes. Jayaumar and Pillai(1996) characterized Mittag-Leffler distribution using spectrum function corresponding to an infinetely divisible distribution. 155

2 DEFINITION 2.1 Pillai and Jayaumar(1995) proposed a class of discrete Mittag-Leffler distribution(dml) having probability generating function(pgf) P(z)=1/[1+c(1-z) α ], as a discrete analogue of the Mittag-Leffler distribution. Charaborty and Ong(2017) introduced a generalization of hyper Poisson distribution using Mittag-Leffler function called Mittag- Leffler function distribution. This paper is organised as follows. Section 2 deals with discrete Mittag- Leffler distribution its mathematical origin and some properties are discussed. In section 3, we study Mittag-Leffler function distribution. The applications of Mittag- Leffler distribution in various areas are discussed in section 4. II. DISCRETE MITTAG-LEFFLER DISTRIBUTION A discrete version of the Mittag Leffler distribution was introduced by Pillai and Jayaumar(1995). The discrete Mittag- Leffler (DML) distribution is a generalisation of geometric distribution. A.Mathematical origin of DML distribution Consider a sequence of independent Bernoulli trials in which the th trial has probability of success α/, 0 < α<1, =1,2,3. Let N be the trial number in which the first success occurs. Then Probability that N=r is given by P r=(1- α)(1- α/2).. (1- α/(r-1) ) α /r =(-1) r-1 α(α-1)( α-2) (α-r+1)/r!. The probability generating function(pgf) of N is given by G(z)=1-(1-z) α. Let X 1, X 2, X 3,, X n be independent and identically distributed as N. Let M be geometric with parameter p. That is, P(M=) = q p, =0,1,2,, 0<p<1, q=1-p. Then X 1+X 2+X 3+ + X M has the pgf P(z)=p/{1-q[1-(1- z) α ]} = 1/1+c(1-z) α with p=1/(1+c). A random variable X on {0,1,2, } is said to follow DML distribution if its pgf is P(z)= 1/(1+c(1-z) α ) with 0<α 1, c >0. The DML distribution can be viewed as the distribution of geometric sum of independent and identically distributed Sibuya random variables. The Sibuya distribution was introduced first by Sibuya(1979) while studying the generalized hypergeometric, digamma and trigamma distributions. Devroye(1993) studied some properties of Sibuya distribution. Sibuya distribution plays an important role in the mathematical origin of DML random variables. DEFINITION 2.2 In a sequence of independent Bernoulli trials let α/ be the probability of success in th trial, 0<α1. Then the number of trials required to obtain the first success has Sibuya distribution. Note that the pgf of Sibuya disribution is G(z)=1- (1-z) α. B.Some Properties of discrete Mittag- Leffler distribution (a) DML (α) is geometrically infinitely divisible and hence infinitely divisible. (b) DML (α) is normally attracted to stable(α). (c) ML(α) is obtained as a sequence of DML(α). (d) For α=1, the distribution is geometric. (e) DML(α) is in discrete class L. (f) DML (α) is directly attracted to discrete stable(α). III. MITTAG-LEFFLER FUNCTION DISTRIBUTION Charaborty and Ong(2017) proposed a new generalisation of the HP distribution using the Mittag-Leffler function called Mittag- Leffler function distribution (MLFD). The distribution with above said pgf is called DML distribution with parameter α. The Poisson distribution is a popular model for count data. However, its use is restricted by the equality of its mean and variance (equi-dispersion). Many models with the ability to represent under, equi and over dispersed data have been proposed in the 156

3 literature to overcome this restriction. One of this is The probability recurrence relation for the HP distribution of Bardwell and Crow(1964). MLFD pmf in (3) is The HP distribution proposed by Bardwell and Crow(1964) can handle both over and under dispersion. The probability mass function of the HP distribution is given by PX 1 PX, 1 (4) P ( ) ( ) (1,, ) X, 0,1,2, where (1,, ) 0 (1), ( )! (1) is the confluent hypergeometric function. A new generalization of the HP distribution, which is a continuous bridge between geometric and HP, is derived by Charaborty and Ong(2017) using the generalized Mittag-Leffler function. The generalization of the HP distribution is done by replacing (+) in (1) with (α+), α > 0 and the normalization constant becomes Eα,(λ) which is the generalized Mittag-Leffler function defined by E, DEFINITION 3.1 z ( z) 0 ( ) A discrete random variable follow MFLD if its pmf is given by where E P X (2) X is said to ( ), E, =0, 1, 2, ; λ, α, > 0, (3) z z, is the generalized 0 Mittag-Leffler function. The distribution with the above pmf is denoted as MLFD(λ,α,). with P X 0 1 ( ) ( ). E, A. Related Distributions and Connection with Other Families of Distributions The MLFD(λ,α,) includes a number of wellnown distribution as particular cases: (1) When α==1, MLFD(λ,α,) reduces to the Poisson distribution with parameter λ. (2) When α=0,( 0), MLFD(λ,α,) becomes geometric distribution with parameter λ provided lλl<1. (3) When α=1,( 0), MLFD(λ,α,) reduces to the HP(λ,) distribution described in Bardwell and Crow(1964). (4) When α=1 and (=t+1) is a positive integer, REMARK 3.1 MLFD(λ,α,) reduces to displaced Poisson distribution with parameters λ and t. (a)for 0 α 1, the MLFD(λ,α,) can be viewed as a continuous bridge between geometric (α=0) and HP (α=1) distributions in the range of the parameter α. (b)for 0 α 1, the MLFD(λ,α,1) can be viewed as a continuous bridge between geometric (α=0) and Poisson (α=1) distributions in the range of the parameter α. This property is also shared by the Com- Poisson distribution. B. MLFD as Member of Some Families of Discrete Distributions (1) MLFD(λ,α,) is a member of the generalized hypergeometric family. 157

4 (2) MLFD(λ,α,) is a member of the generalized is suitable in empirical modeling of differently dispersed count data when compared with HP power series distribution when λ is the primary distribution. MLFD is a flexible distribution with parameter. varying shape and can be modeled for under, equi or (3) For fixed value of the parameter α and the over dispersed data. MLFD(λ,α,) is a member of the exponential family of distribution. Charaborty and Ong(2017) studied various distribution properties of MLFD such as recurrence relations for pmf, generating functions, formulae for different types of moments, reliability properties and stochastic ordering. 4. APPLICATIONS During the last two decades, the Mittag- Leffler distribution was applied in diverse areas lie Physics, Reliability, Financial Modelling, Time series Analysis, Queueing Theory, etc.(see, Mariamma(2017)). Kozubowsi and Rachev(1994) discussed the applications of Mittag-Leffler distribution in modeling financial data. Semi Mittag- Leffler distribution, which is a generalization of Mittag-Leffler distribution, arises as the stationary solution of a first order autoregressive equation (see, Jayaumar and Pillai,1993). Bunge(1996) used Mittag-Leffler distribution in the study of random stability. Lin(1998) studied some distributional properties of Mittag-Leffler distribution. Kozubowsi and Rachev(1999) developed certain representations of the Mittag-Leffler distribution. Jayaumar and Pillai(1993) developed autoregressive Mittag-Leffler process as a generalization of the first order autoregressive exponential model and obtained the stationary solution of a first order autoregressive equation. Jayaumar(2003) used the Mittag-Leffler distribution to model the rate of flow of water in Kallada river, Kerala, India. Seetha Lashmi and Jose(2002) developed geometric Mittag-Leffler tailed autoregressive processes and studied its properties. The DML distribution proposed by Pillai and Jayaumar(1995) arises as a mixture of Poisson and Mittag-Leffler distribution. They have studied different properties of DML distribution and gave a probabilistic derivation and application in a first order auto regressive integer-valued process. Charaborty and Ong(2017) showed that the MLFD Acnowledgement The author would lie to than UGC for the financial support under Minor Research Project No:MRP 14-15/KLCA033/UGC-SWRO. References [1] Bardwell,G.E. and Crow,E.L.(1964) A two parameter family of hyper-poisson distribution. Journal of the American Statistical Association, 59, [2] Bunge,J.(1996) Composition semigroups and random stability. Annals of Probability, 24, [3] Charaborty, S. and Ong, S.H.(2017) Mittag-Leffler distribution- A new generalization of hyper-poisson distribution. Journal of Statistical Distribution and Applications, 4, [4] Devroye, L.(1993) A trypitch of discrete distributions related to the stable law. Statistics and Probability Letters, 18, [5] Feller,W.(1971) An Introduction to Probability Theory and Its Applications Vol. II. Wiley, New Yor. [6] Huillet,T.E.(2016) On Mittag-Leffler distributions and related stochastic processes. Journal of Computational and Applied Mathematics, 296, [7] Jayaumar,K.(2003) Mittag Leffler Processes. Mathematical and Computer Modeling, 37, [8] Jayaumar,K and Pillai,R.N.(1993) The first order autoregressive Mittag- Leffler Process. Journal of Applied Probability, 30, [9] Jayaumar, K. and Pillai, R. N.(1996) Characterizations of Mittag-Leffler distribution. Journal of Applied Statistical Science, 4, [10] Kozubowsi,T.J. and Rachev,S.T.(1994) The theory of geometric stable laws and its use in modeling financial data. European Journal of Operations Research, 74, [11] Kozubowsi,T.J. and Rachev,S.T.(1999) Univariate geometric stable distributions. Journal of Computational Analysis and Applications, 1, [12] Lin,G.D.(1998) On the Mittag-Leffler distributions. Journal of Statistical Planning and Inference, 74, 1-9. [13] Mariamma, A.(2017) A study on Mittag-Leffler distribution. VIRJ for Pure and Applied Science, 3, [14] Mathai, A.M. (2010) Some properties of Mittag-Leffler functions and matrix-variate analogues: A statistical perspective. Fractional Calculus and Applied Analysis, 13, [15] Mittag-Leffler, G.M.(1903) Sur la nouvelle function E(x). C.R. Acad.Sci., Paris (Ser.II), 137, [16] Pillai,R.N.(1990) On Mittag-Leffler functions and related distributions. Annals of the Institute of Statistical Mathematics, 42, [17] Pillai,R.N. and Jayaumar, K.(1995) Discrete Mittag-Leffler distributions. Statistics and Probability Letters, 23, [18] SeethaLashmi,V. and Jose,K.K.(2002) Geometric Mittag Leffler tailed autoregressive processes. Far East Journal of Theoretical Statistics, 6,

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