Correction on ‘Population Genetics Models with Skewed Fertilities: A Forward and Backward Analysis’

doi:10.1080/15326349.2012.700799

Stochastic Models

2012

DOI: 10.1080/15326349.2012.700799

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Correction on ‘Population Genetics Models with Skewed Fertilities: A Forward and Backward Analysis’

Abstract: The article 'Population genetics models with skewed fertilities: a forward and backward analysis ', Stochastic Models 27, 521-554 (2011)

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Occupancy Distributions Arising in Sampling from Gibbs-Poisson Abundance Models

Huillet¹,

Martı́nez²

2013

J Stat Phys

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Abstract. Estimating the number n of unseen species from a k−sample displaying only p ≤ k distinct sampled species has received attention for long. It requires a model of species abundance together with a sampling model. We start with a discrete model of iid stochastic species abundances, each with Gibbs-Poisson distribution. A k−sample drawn from the n−species abundances vector is the one obtained while conditioning it on summing to k. We discuss the sampling formulae (species occupancy distributions, frequency of frequencies) in this context. We then develop some aspects of the estimation of n problem from the size k of the sample and the observed value of P n,k , the number of distinct sampled species.It is shown that it always makes sense to study these occupancy problems from a Gibbs-Poisson abundance model in the context of a population with infinitely many species. From this extension, a parameter γ naturally appears, which is a measure of richness or diversity of species. We rederive the sampling formulae for a population with infinitely many species, together with the distribution of the number P k of distinct sampled species. We investigate the estimation of γ problem from the sample size k and the observed value of P k .We then exhibit a large special class of Gibbs-Poisson distributions having the property that sampling from a discrete abundance model may equivalently be viewed as a sampling problem from a random partition of unity, now in the continuum. When n is finite, this partition may be built upon normalizing n infinitely divisible iid positive random variables by its partial sum. It is shown that the sampling process in the continuum should generically be biased on the total length appearing in the latter normalization. A construction with sizebiased sampling from the ranked normalized jumps of a subordinator is also supplied, would the problem under study present infinitely many species. We illustrate our point of view with many examples, some of which being new ones.

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Occupancy Distributions Arising in Sampling from Gibbs-Poisson Abundance Models

Huillet¹,

Martı́nez²

2013

J Stat Phys

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Asymptotics of Symmetric Compound Poisson Population Models

Huillet¹,

Möhle²

2014

Combinator. Probab. Comp.

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Compound Poisson population models are particular conditional branching process models. A formula for the transition probabilities of the backward process for general compound Poisson models is verified. Symmetric compound Poisson models are defined in terms of a parameter θ ∈ (0, ∞) and a power series φ with positive radius r of convergence. It is shown that the asymptotic behaviour of symmetric compound Poisson models is mainly determined by the characteristic value θrφ (r−). If θrφ (r−) 1, then the model is in the domain of attraction of the Kingman coalescent. If θrφ (r−) < 1, then under mild regularity conditions a condensation phenomenon occurs which forces the model to be in the domain of attraction of a discrete-time Dirac coalescent. The proofs are partly based on the analytic saddle point method. They draw heavily from local limit theorems and from results of S. Janson on simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Several examples of compound Poisson models are provided and analysed.2010 Mathematics subject classification: Primary 60F05, 60G09, 60J10, 60J80 Secondary 60K35, 92D10, 92D25

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Asymptotic genealogies for a class of generalized Wright–Fisher models

Huillet¹,

Möhle²

2021

Modern Stochastics: Theory and Applications

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A class of Cannings models is studied, with population size N having a mixed multinomial offspring distribution with random success probabilities ${W_{1}},\dots ,{W_{N}}$ induced by independent and identically distributed positive random variables ${X_{1}},{X_{2}},\dots $ via ${W_{i}}:={X_{i}}/{S_{N}}$, $i\in \{1,\dots ,N\}$, where ${S_{N}}:={X_{1}}+\cdots +{X_{N}}$. The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into N subintervals of lengths ${W_{1}},\dots ,{W_{N}}$. Convergence results for the genealogy of these Cannings models are provided under assumptions that the tail distribution of ${X_{1}}$ is regularly varying. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by Huillet [J. Math. Biol. 68 (2014), 727–761] for the case when ${X_{1}}$ is Pareto distributed and complement those obtained by Schweinsberg [Stoch. Process. Appl. 106 (2003), 107–139] for models where sampling is performed without replacement from a supercritical branching process.

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Correction on ‘Population Genetics Models with Skewed Fertilities: A Forward and Backward Analysis’

Abstract: The article 'Population genetics models with skewed fertilities: a forward and backward analysis ', Stochastic Models 27, 521-554 (2011)

Cited by 3 publications

References 2 publications

Occupancy Distributions Arising in Sampling from Gibbs-Poisson Abundance Models

Occupancy Distributions Arising in Sampling from Gibbs-Poisson Abundance Models

Asymptotics of Symmetric Compound Poisson Population Models

Asymptotic genealogies for a class of generalized Wright–Fisher models

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