The Generalized Discrete Linnik Distributions

Christoph, Gerd; Schreiber, Karina

doi:10.1007/978-1-4612-2234-7_1

Cited by 17 publications

(11 citation statements)

References 11 publications

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“…We also observe that h (z) i , as a solution to the functional equation (11), is the pgf of N ∞ started at i, the total limiting number of cumulated individuals which appeared over time in the population (possibly infinite on the set of explosion). It can be solved by the Lagrange inversion formula, [14].…”

Section: Thierry E Huilletmentioning

confidence: 96%

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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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Section: Thierry E Huilletmentioning

confidence: 96%

“…This is consistent with the following result: let τ i,0 = inf (n ≥ 1 : S n = 0 | S 0 = i) . Then, by first-step analysis ( [56], p. 92), E (z τ i,0 ) = h (z) i , where h (z) solves the functional equation (11) h (z) = zφ α,λ (h (z)) .…”

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 recall that this model could also be derived by assuming that, under the biological motivation of the BCH model, the number of tumor cells M follows the generalized discrete Linnik distribution with characteristic exponent λ ∈(0,1], scale parameter θ >0 and form parameter

γ ⩾ 0

(symb. M ∼ D L ( λ , θ , γ ); ). Then, the probability generating function of M will be given by

φ (z) = \{\begin{matrix} {(1 + θ γ {(1 - z)}^{λ})}^{- 1 / γ}, & γ > 0 \\ \exp ((), - θ {(1 - z)}^{λ}), & γ = 0 \end{matrix}

resulting, in view of , to model again.…”

Section: The Modelmentioning

confidence: 99%

“…Specifically, assume that M (the number of tumor cells) follows a Poisson distribution with random parameter Ξ= Y 1/ λ V , where Y , V are independent random variables with Y following a Gamma distribution with scale and shape parameter equal to γ , and V being a positive random variable with Laplace transform

e^{- θ z^{λ}}

. Then, the distribution of M is the generalized Linnik distribution and hence, the cure rate model generated under this setup is again the one given by .…”

Section: Introductionmentioning

confidence: 99%

“…A family of discrete distributions that includes as special cases many of the classical probability models (e.g., Poisson, negative binomial, and geometric) is the class of generalized Linnik distributions ( [29]). Taking into account that the probability generating function of the generalized Linnik distribution is given by (z) = (1+ (1−z) ) −1∕ , with ⩾ 0, one can easily verify that, the population survival function S P (t) obtained by (4) coincides to the respective survival function described by the generalized subject-specific frailty model given by (3).…”

Section: Introductionmentioning

confidence: 99%

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A flexible family of transformation cure rate models

Koutras

Milienos

2017

Statistics in Medicine

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In this paper, we introduce a flexible family of cure rate models, mainly motivated by the biological derivation of the classical promotion time cure rate model and assuming that a metastasis-competent tumor cell produces a detectable-tumor mass only when a specific number of distinct biological factors affect the cell. Special cases of the new model are, among others, the promotion time (proportional hazards), the geometric (proportional odds), and the negative binomial cure rate model. In addition, our model generalizes specific families of transformation cure rate models and some well-studied destructive cure rate models. Exact likelihood inference is carried out by the aid of the expectationŰmaximization algorithm; a profile likelihood approach is exploited for estimating the parameters of the model while model discrimination problem is analyzed by the aid of the likelihood ratio test. A simulation study demonstrates the accuracy of the proposed inferential method. Finally, as an illustration, we fit the proposed model to a cutaneous melanoma data-set. Copyright © 2017 John Wiley & Sons, Ltd.

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2015

Fundamental Aspects of Operational Risk and Insurance Analytics

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The Generalized Discrete Linnik Distributions

Cited by 17 publications

References 11 publications

On Mittag-Leffler distributions and related stochastic processes

On Mittag-Leffler distributions and related stochastic processes

A flexible family of transformation cure rate models

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