Models for Genetic Diversity Generated by Negative Binomial Point Processes
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We develop a model based on a generalised Poisson-Dirichlet distribution for the analysis of genetic diversity, and illustrate its use on microsatellite data for the genus Dasyurus (the quoll, a marsupial carnivore listed as near-threatened in Australia). Our class of distributions, termed PD_α^(r), is constructed from a negative binomial point process, generalizing the usual one-parameter PD_α model, which is constructed from a Poisson point process. Both models have at their heart a Stable(α) process, but in PD_α^(r), an extra parameter r>0 adds flexibility, analogous to the way the negative binomial distribution allows for "overdispersion" in the analysis of count data. A key result obtained is a generalised version of Ewens' sampling formula for PD_α^(r). We outline the theoretical basis for the model, and, for the quolls data, estimate the parameters α and r by least squares, showing how the extra parameter r aids in the interpretability of the data by comparison with the standard PD_α model. The methods potentially have implications for the management and conservation of threatened populations.
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