https://hal-univ-tours.archives-ouvertes.fr/hal-01354100Zoroa, NoemiNoemiZoroaDepartamento de Estadística e Investigación Operativa, Facultad de Matematicas - Universidad de MurciaLesigne, EmmanuelEmmanuelLesigneLMPT - Laboratoire de Mathématiques et Physique Théorique - UT - Université de Tours - CNRS - Centre National de la Recherche ScientifiqueFernandez-Saez, JoséJoséFernandez-SaezDepartamento de Estadística e Investigación Operativa, Facultad de Matematicas - Universidad de MurciaZoroa, PPZoroaDepartamento de Estadística e Investigación Operativa, Facultad de Matematicas - Universidad de MurciaCasas, JéromeJéromeCasasIRBI - Institut de recherche sur la biologie de l'insecte UMR7261 - UT - Université de Tours - CNRS - Centre National de la Recherche ScientifiqueThe coupon collector urn model with unequal probabilities in ecology and evolutionHAL CCSD2017Coupon collector's problemparasitoidstochastic dominancestrong dominance[MATH.MATH-PR] Mathematics [math]/Probability [math.PR][SDV.EE] Life Sciences [q-bio]/Ecology, environmentLesigne, Emmanuel2016-08-18 19:41:482022-05-24 16:32:012016-08-19 10:50:05enJournal articleshttps://hal-univ-tours.archives-ouvertes.fr/hal-01354100/document10.1098/rsif.2016.0643application/x-msword1The sequential sampling of populations with unequal probabilities and with replacement in a closed population is a recurrent problem in ecology and evolution. Examples range from biodiversity sampling, epidemiology to the estimation of signal repertoire in animal communication. Many of these ques- tions can be reformulated as urn problems, often as special cases of the coupon collector problem, most simply expressed as the number of coupons that must be collected to have a complete set. We aimed to apply the coupon collector model in a comprehensive manner to one example—hosts (balls) being searched (draws) and parasitized (ball colour change) by parasitic wasps— to evaluate the influence of differences in sampling probabilities between items on collection speed. Based on the model of a complete multinomial process over time, we define the distribution, distribution function, expectation and variance of the number of hosts parasitized after a given time, as well as the inverse problem, estimating the sampling effort. We develop the relationship between the risk distribution on the set of hosts and the speed of parasitization and propose a more elegant proof of the weak stochastic dominance among speeds of parasitization, using the concept of Schur convexity and the ‘Robin Hood transfer’ numerical operation. Numerical examples are provided and a conjecture about strong dominance—an ordering characteristic of random variables—is proposed. The speed at which new items are discovered is a function of the entire shape of the sampling probability distribution. The sole comparison of values of variances is not sufficient to compare speeds associated with different distributions, as generally assumed in ecological studies.