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A generalised Dickman distribution and the number of species in a negative binomial process model
Published online by Cambridge University Press: 01 July 2021
Abstract
We derive the large-sample distribution of the number of species in a version of Kingman’s Poisson–Dirichlet model constructed from an
$\alpha$
-stable subordinator but with an underlying negative binomial process instead of a Poisson process. Thus it depends on parameters
$\alpha\in (0,1)$
from the subordinator and
$r>0$
from the negative binomial process. The large-sample distribution of the number of species is derived as sample size
$n\to\infty$
. An important component in the derivation is the introduction of a two-parameter version of the Dickman distribution, generalising the existing one-parameter version. Our analysis adds to the range of Poisson–Dirichlet-related distributions available for modeling purposes.
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- © The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust
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