Discrete stable random variables

Christoph, Gerd; Schreiber, Karina

doi:10.1016/s0167-7152(97)00123-5

Cited by 37 publications

(23 citation statements)

References 3 publications

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“…There exists a discrete version of the notion of selfdecomposability [66]. Some accounts on the sub-class of discrete-stable random variables may also be found here and in [12].…”

Section: Thierry E Huilletmentioning

confidence: 98%

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On Mittag-Leffler distributions and related stochastic processes

Huillet

2016

Journal of Computational and Applied Mathematics

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International audienceRandom variables with Mittag-Leffler distribution can take values either in the set of non-negative integers or in the positive real line. There can be of two different types, one (type-1) heavy-tailed with index α ∈ (0, 1), the other (type-2) possessing all its moments. We investigate various stochastic processes where they play a key role, among which: the discrete space/time Neveu branching process, the discrete-space continuous-time Neveu branching process, the continuous space/time Neveu branching process (CSBP) and renewal processes with rare events. Its relation to (discrete or continuous) self-decomposability and branching processes with immigration is emphasized. Special attention will be paid to the Neveu CSBP for its connection with the Bolthausen-Sznitman coalescent. In this context, and following a recent work of Möhle [49], a type-2 Mittag-Leffler process turns out to be the Siegmund dual to Neveu's CSBP block-counting process arising in sampling from P D e −t , 0. Further combinatorial developments of this model are investigated. 1. Sibuya random variables (rvs) and related branching processes We first investigate a class of integral-valued rvs that will show important for our general purpose. 1.1. Sibuya rvs and related ones. We start with their definition and main properties. • One parameter Sibuya(α) rv. Let X α ≥ 1 be an integer-valued random variable with support N = {1, 2, ....} defined as follows: X α = inf (l ≥ 1 : B α (l) = 1) , where (B α (l)) l≥1 is a sequence of independent Bernoulli rvs obeying P (B α (l) = 1) = α/l where α ∈ (0, 1). It is thus the first epoch of a success in a Bernoulli trial when the probability of success is inversely proportional to the number of the trial. X α is called a Sibuya(α) rv. Then P (X α = k) = (−1

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“…There exists a discrete version of the notion of selfdecomposability [66]. Some accounts on the sub-class of discrete-stable random variables may also be found here and in [12].…”

Section: Thierry E Huilletmentioning

confidence: 98%

“…We also have [51], s (i) = s (∞) (1 − P (τ i,0 < ∞)) , together with τ i,j = inf (n ≥ 1 : S n = j | S 0 = i) , j > i and (12) P…”

Section: Thierry E Huilletmentioning

confidence: 99%

On Mittag-Leffler distributions and related stochastic processes

Huillet

2016

Journal of Computational and Applied Mathematics

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“…We begin with a few examples to illustrate the situation often encountered when new distributions are created. The distribution was introduced by Christoph and Schreiber [1].…”

Section: New Distribution Created Using Probability Generating Functionsmentioning

confidence: 99%

“…The recursive formula for its mass function has been obtained; see expression (8) given by Christoph and Schreiber [1].…”

Section: New Distribution Created Using Probability Generating Functionsmentioning

confidence: 99%

Simulated Minimum Hellinger Distance Inference Methods for Count Data

Luong¹,

Bilodeau²,

Blier-Wong³

2018

OJS

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In this paper, we consider simulated minimum Hellinger distance (SMHD) inferences for count data. We consider grouped and ungrouped data and emphasize SMHD methods. The approaches extend the methods based on the deterministic version of Hellinger distance for count data. The methods are general, it only requires that random samples from the discrete parametric family can be drawn and can be used as alternative methods to estimation using probability generating function (pgf) or methods based matching moments. Whereas this paper focuses on count data, goodness of fit tests based on simulated Hellinger distance can also be applied for testing goodness of fit for continuous distributions when continuous observations are grouped into intervals like in the case of the traditional Pearson's statistics. Asymptotic properties of the SMHD methods are studied and the methods appear to preserve the properties of having good efficiency and robustness of the deterministic version.

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“…See Steutel and van Harn (1979) for a necessary and sufficient condition to have discrete self decomposability. Downloaded by [University of Sussex Library] at 21:10 07 June 2016 (f) Probabilities Expanding the pgf P X z of the discrete stable random variable X in a power series, Christoph and Schreiber (1998) obtain for p k the series…”

Section: (B) Compound Poisson Distributionmentioning

confidence: 99%

Some Simple Method of Estimation for the Parameters of the Discrete Stable Distribution with the Probability Generating Function

Doray

Jiang

Luong

2009

Communications in Statistics - Simulation and Computation

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In this article, we develop a method to estimate the two parameters of the discrete stable distribution. By minimizing the quadratic distance between transforms of the empirical and theoretical probability generating functions, we obtain estimators simple to calculate, asymptotically unbiased, and normally distributed. We also derive the expression for their variance-covariance matrix. We simulate several samples of discrete stable distributed datasets with different parameters, to analyze the effect of tuncation on the right tail of the distribution.

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Discrete stable random variables

Cited by 37 publications

References 3 publications

On Mittag-Leffler distributions and related stochastic processes

On Mittag-Leffler distributions and related stochastic processes

Simulated Minimum Hellinger Distance Inference Methods for Count Data

Some Simple Method of Estimation for the Parameters of the Discrete Stable Distribution with the Probability Generating Function

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