TY - JOUR
T1 - Limiting Distributions of Generalised Poisson–Dirichlet Distributions Based on Negative Binomial Processes
AU - Ipsen, Yuguang
AU - Maller, Ross
AU - Shemehsavar, Soudabeh
N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/12/1
Y1 - 2020/12/1
N2 - The PDα(r) distribution, a two-parameter distribution for random vectors on the infinite simplex, generalises the PD α distribution introduced by Kingman, to which it reduces when r= 0. The parameter α∈ (0 , 1) arises from its construction based on ratios of ordered jumps of an α-stable subordinator, and the parameter r> 0 signifies its connection with an underlying negative binomial process. Herein, it is shown that other distributions on the simplex, including the Poisson–Dirichlet distribution PD (θ) , occur as limiting cases of PDα(r), as r→ ∞. As a result, a variety of connections with species and gene sampling models, and many other areas of probability and statistics, are made.
AB - The PDα(r) distribution, a two-parameter distribution for random vectors on the infinite simplex, generalises the PD α distribution introduced by Kingman, to which it reduces when r= 0. The parameter α∈ (0 , 1) arises from its construction based on ratios of ordered jumps of an α-stable subordinator, and the parameter r> 0 signifies its connection with an underlying negative binomial process. Herein, it is shown that other distributions on the simplex, including the Poisson–Dirichlet distribution PD (θ) , occur as limiting cases of PDα(r), as r→ ∞. As a result, a variety of connections with species and gene sampling models, and many other areas of probability and statistics, are made.
KW - Ewens sampling formula
KW - Generalised Poisson–Dirichlet laws
KW - Negative binomial point process
KW - Size-biased sampling
KW - Species and gene abundance distributions
KW - Trimmed α-stable subordinator
UR - http://www.scopus.com/inward/record.url?scp=85069463760&partnerID=8YFLogxK
U2 - 10.1007/s10959-019-00928-7
DO - 10.1007/s10959-019-00928-7
M3 - Article
SN - 0894-9840
VL - 33
SP - 1974
EP - 2000
JO - Journal of Theoretical Probability
JF - Journal of Theoretical Probability
IS - 4
ER -