On Geometric Infinite Divisibility and Stability

Aly, Emad‐Eldin A. A.; Bouzar, Nadjib

doi:10.1023/a:1017589613321

Cited by 15 publications

(18 citation statements)

References 11 publications

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“…More recently, VAN HARN and STEUTEL (1993) studied stability equations for Z þ and R þ -valued processes with stationary independent increments and used Poisson mixtures to deduce the results in the R þ -case from their counterparts in the Z þ -case (see also PAKES 1995). ALY and BOUZAR (2000) used the same approach in their study of geometric infinite divisibility and geometric stability of distributions on Z þ and R þ .…”

Section: Proof: By Proposition 1 (Applied Withmentioning

confidence: 99%

Mixture representations for discrete Linnik laws

Bouzar

2002

Statistica Neerlandica

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In this paper we give mixture representations for the discrete Mittag-Leffler and Linnik laws. Our results form the discrete analogues of the mixture representations for the generalized symmetric Linnik distribution obtained by PAKES (1998). As an application, we derive the infinite divisibility of mixtures of Mittag-Leffler distributions. Alternative mixture representations in terms of discrete stable distributions are also presented.

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Section: Proof: By Proposition 1 (Applied Withmentioning

confidence: 99%

Mixture representations for discrete Linnik laws

Bouzar

2002

Statistica Neerlandica

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“…For the applications of g.i.d. distribution and distribution with c.m.d., see Klebanov et al (1985)and Aly and Bouzar (2000). Figure 1 gives a comparison between pdf of HP distribution and pdf of Pareto distribution of third kind for different values of λ.…”

Section: Introductionmentioning

confidence: 99%

Heavy Tailed Pareto Distribution: Properties and Applications

Jayakumar¹,

Kuttykrishnan²,

Krishnan³

2021

Journal of Data Science

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In this article, we introduce a class of distributions that have heavy tails as compared to Pareto distribution of third kind, which we termed as Heavy Tailed Pareto (HP) distribution. Various structural properties of the new distribution are derived. It is shown that HP distribution is in the domain of attraction of minimum of Weibull distribution. A representation of HP distribution in terms of Weibull random variable is obtained. Two characterizations of HP distribution are obtained. The method of maximum likelihood is used for estimation of model parameters and simulation results are presented to assess the performance of new model. Marshall-Olkin Heavy Tailed Pareto (MOHP) distribution is also introduced and some of its properties are studied. It is shown that MOHP distribution is geometric extreme stable. An autoregressive time series model with the new model as marginal distribution is developed and its properties are studied.

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“…It is clear in this connection that the distribution given by ψ is infinitely divisible (ID), with the ChF ψ u corresponding to the random variable X(Z(u)), with X(t) as before and {Z(t), t ≥ 0} being the gamma Lévy process (so that Z(1) is exponential). Special cases of random variables with ChF ψ u include generalized Mittag-Leffler (see, e.g., [2]), generalized Linnik (see, e.g., [32,152]), and generalized Laplace (or Bessel function) distributions (see, e.g., [77]).…”

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confidence: 99%

“…The concept of geometric stability as defined in (3) needs a modification in order to be applicable to discrete distributions. Such modification appeared in [2,14], where the multiplication by a p was replaced by the binomial thinning operator , introduced in [188] in connection with extending the concepts of self-decomposability and stability to this setting. Recall that for any discrete random variable N supported on the set of non-negative integers Z + = {0, 1, .…”

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confidence: 99%

“…In this setting, a Z + -valued random variable X is discrete strictly geometric stable (DSGS) if for each p ∈ (0, 1) there exist an a p ∈ (0, 1) such that X d = a p S p , where S p is the sum (1) whose terms are IID copies of X, independent of the geometric number of terms N p . As shown in [2], there is a correspondence between DSGS distributions and discrete stable laws (see [24,188]), which can be stated as…”

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confidence: 99%

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Geometric infinite divisibility, stability, and self-similarity: an overview

Kozubowski¹

2010

Banach Center Publications

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The concepts of geometric infinite divisibility and stability extend the classical properties of infinite divisibility and stability to geometric convolutions. In this setting, a random variable X is geometrically infinitely divisible if it can be expressed as a random sum of Np components for each p ∈ (0, 1), where Np is a geometric random variable with mean 1/p, independent of the components. If the components have the same distribution as that of a rescaled X, then X is (strictly) geometric stable. This leads to broad classes of probability distributions closely connected with their classical counterparts. We review fundamental properties of these distributions and discuss further extensions connected with geometric sums, including multivariate and operator geometric stability, discrete analogs, and geometric self-similarity.

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On Geometric Infinite Divisibility and Stability

Cited by 15 publications

References 11 publications

Mixture representations for discrete Linnik laws

Mixture representations for discrete Linnik laws

Heavy Tailed Pareto Distribution: Properties and Applications

Geometric infinite divisibility, stability, and self-similarity: an overview

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