Occupancy Distributions Arising in Sampling from Gibbs-Poisson Abundance Models
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Occupancy Distributions Arising in Sampling from Gibbs-Poisson Abundance Models Thierry Huillet · Servet Martínez
Received: 5 March 2013 / Accepted: 3 October 2013 / Published online: 16 October 2013 © Springer Science+Business Media New York 2013
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 nspecies 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 Pn,k , the number of distinct sampled species. It is shown that it always makes sense to study these occupancy problems from a GibbsPoisson 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 Pk of distinct sampled species. We investigate the estimation of γ problem from the sample size k and the observed value of Pk . 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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T. Huillet ( ) Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise, CNRS UMR-8089, Site de Saint Martin, 2 avenue Adolphe-Chauvin, 95302 Cergy-Pontoise, France e-mail: [email protected] S. Martínez Depto. Ingenieria Matematica and Centro Modelamiento Matematico, Universidad de Chile, UMI 2071, Uchile-Cnrs, Casilla 170-3 Correo 3, Santiago, Chile e-mail: [email protected]
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T. Huillet, S. Martínez
Keywords Occupancy distributions · Sampling from Gibbs-Poisson distribution · Species abundance and frequencies · Biodiversity · Combinatorial probability · Subordinators
1 Introduction and Outline of Main Results Estimating the number n of unseen species from a k-sample displaying only p ≤ k distinct sampled species has been a challenging problem since t
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