arXiv subjects

Yuguang F. Ipsen

Publications and source records attributed to Yuguang F. Ipsen.

4 recordsLinked to original sources

arXiv2019

Models for Genetic Diversity Generated by Negative Binomial Point Processes

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.

stat.AP↗

arXiv2018

Convergence to stable limits for ratios of trimmed Levy processes and their jumps

We derive characteristic function identities for conditional distributions of an r-trimmed Levy process given its r largest jumps up to a designated time t. Assuming the underlying Levy process is in the domain of attraction of a stable process as t goes to 0, these identities are applied to show joint convergence of the trimmed process divided by its large jumps to corresponding quantities constructed from a stable limiting process. This generalises related results in the 1-dimensional subordinator case developed in Kevei & Mason (2014) and produces new discrete distributions on the infinite simplex in the limit.

math.PR↗

arXiv2017

Functional Laws for Trimmed Levy Processes

Two different ways of trimming the sample path of a stochastic process in D[0, 1]: global ("trim as you go") trimming and record time ("lookback") trimming are analysed to find conditions for the corresponding operators to be continuous with respect to the (strong) J1-topology. A key condition is that there should be no ties among the largest ordered jumps of the limit process. As an application of the theory, via the continuous mapping theorem we prove limit theorems for trimmed Levy processes, using the functional convergence of the underlying process to a stable process. The results are applied to a reinsurance ruin time problem.

math.PR↗

arXiv2017

Generalised Poisson-Dirichlet Distributions and the Negative Binomial Point Process

When $S=(S_t)_{t\ge 0}$ is an $α$-stable subordinator, the sequence of ordered jumps of $S$, up till time $1$, omitting the $r$ largest of them, and taken as proportions of their sum $^{(r)}S_t$, defines a 2-parameter distribution on the infinite dimensional simplex, $\nabla_{\infty}$, which we call the $\mathrm{PD}_α^{(r)}$ distribution. When $r=0$ it reduces to the $\mathrm{PD}_α$ distribution introduced by Kingman in 1975. We observe a serendipitous connection between $\mathrm{PD}_α^{(r)}$ and the negative binomial point process of Gregoire (1984), which we exploit to analyse in detail a size-biased version of $\mathrm{PD}_α^{(r)}$. As a consequence we derive a stick-breaking representation for the process and a useful form for its distribution. This program produces a large new class of distributions available for a variety of modelling purposes.

math.PR↗