The distribution of the frequencies of age-ordered alleles in a diffusion model

1990 ◽  
Vol 22 (3) ◽  
pp. 519-532 ◽  
Author(s):  
S. N. Ethier
Keyword(s):  
Diffusion Model ◽  
Dirichlet Distribution ◽  
Random Variables ◽  
Sampling Formula

We prove that the frequencies of the oldest, second-oldest, third-oldest, … alleles in the stationary infinitely-many-neutral-alleles diffusion model are distributed as X1, (1 − X1)X2, (1 − X1)(1 − X2)X3, …, where X1, X2,X3, … are independent beta (1, θ) random variables, θ being twice the mutation intensity; that is, the frequencies of age-ordered alleles have the so-called Griffiths–Engen–McCloskey, or GEM, distribution. In fact, two proofs are given, the first involving reversibility and the size-biased Poisson–Dirichlet distribution, and the second relying on a result of Donnelly and Tavaré relating their age-ordered sampling formula to the GEM distribution.

The distribution of the frequencies of age-ordered alleles in a diffusion model

1990 ◽  
Vol 22 (03) ◽  
pp. 519-532 ◽  
Author(s):  
S. N. Ethier
Keyword(s):  
Diffusion Model ◽  
Dirichlet Distribution ◽  
Random Variables ◽  
Sampling Formula

We prove that the frequencies of the oldest, second-oldest, third-oldest, … alleles in the stationary infinitely-many-neutral-alleles diffusion model are distributed as X 1, (1 − X 1)X 2, (1 − X 1)(1 − X 2)X 3, …, where X 1, X 2, X 3, … are independent beta (1, θ) random variables, θ being twice the mutation intensity; that is, the frequencies of age-ordered alleles have the so-called Griffiths–Engen–McCloskey, or GEM, distribution. In fact, two proofs are given, the first involving reversibility and the size-biased Poisson–Dirichlet distribution, and the second relying on a result of Donnelly and Tavaré relating their age-ordered sampling formula to the GEM distribution.

Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics

1986 ◽  
Vol 23 (4) ◽  
pp. 1008-1012 ◽  
Author(s):  
Fred M. Hoppe
Keyword(s):  
Characteristic Property ◽  
Dirichlet Distribution ◽  
Biased Sampling ◽  
Sampling Formula

A characteristic property of the Ewen sampling formula is shown to follow from the invariance under size-biased sampling of the Poisson–Dirichlet distribution.

The infinitely-many-neutral-alleles diffusion model by ages

1990 ◽  
Vol 22 (01) ◽  
pp. 1-24 ◽  
Author(s):  
S. N. Ethier
Keyword(s):  
Diffusion Model ◽  
Probability Distributions ◽  
Ewens Sampling Formula ◽  
Sample Paths ◽  
Sampling Formula ◽  
Frequent Allele

We discuss two formulations of the infinitely-many-neutral-alleles diffusion model that can be used to study the ages of alleles. The first one, which was introduced elsewhere, assumes values in the set of probability distributions on the set of alleles, and the ages of the alleles can be inferred from its sample paths. We illustrate this approach by proving a result of Watterson and Guess regarding the probability that the most frequent allele is oldest. The second diffusion model, which is new, assumes values in the set of probability distributions on the set of pairs (x, a), where x is an allele and a is its age. We illustrate this second approach by proving an extension of the Ewens sampling formula to age-ordered samples due to Donnelly and Tavaré.

scholarly journals The sampling formula and Laplace transform associated with the two-parameter Poisson-Dirichlet distribution

2011 ◽  
Vol 43 (04) ◽  
pp. 1066-1085
Author(s):  
Fang Xu
Keyword(s):  
Laplace Transform ◽  
Random Sampling ◽  
Dirichlet Distribution ◽  
Probability Generating Function ◽  
Fixed Time ◽  
Infinite Dimensional ◽  
Sampling Formula ◽  
Dimensional Distribution ◽  
The Relationship ◽  
Two Parameter

In this paper we investigate the relationship between the sampling formula and Laplace transform associated with the two-parameter Poisson-Dirichlet distribution. We conclude that they are equivalent to determining the corresponding infinite-dimensional distribution. With these tools, a central limit theorem is established associated with the infinitely-many-neutral-alleles model at any fixed time. We also obtain the probability generating function of random sampling from a generalized two-parameter diffusion process. At the end of the paper a selection case is considered.

scholarly journals Gaussian Limits Associated with the Poisson-Dirichlet Distribution and the Ewens Sampling Formula

2002 ◽  
Vol 12 (1) ◽  
pp. 101-124 ◽  
Author(s):  
Paul Joyce ◽  
Stephen M. Krone ◽  
Thomas G. Kurtz
Keyword(s):  
Dirichlet Distribution ◽  
Ewens Sampling Formula ◽  
Sampling Formula

Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics

1986 ◽  
Vol 23 (04) ◽  
pp. 1008-1012
Author(s):  
Fred M. Hoppe
Keyword(s):  
Characteristic Property ◽  
Dirichlet Distribution ◽  
Biased Sampling ◽  
Sampling Formula

A characteristic property of the Ewen sampling formula is shown to follow from the invariance under size-biased sampling of the Poisson–Dirichlet distribution.

Modeling Repeated Count Data: Some Extensions of the Rasch Poisson Counts Model

1995 ◽  
Vol 20 (3) ◽  
pp. 241-258 ◽  
Author(s):  
Marijtje A. J. van Duijn ◽  
Margo G. H. Jansen
Keyword(s):  
Count Data ◽  
Dirichlet Distribution ◽  
Random Variables ◽  
Data Sets ◽  
Unknown Parameters ◽  
Intensity Parameters ◽  
Test Parameters ◽  
Within Subjects ◽  
The Subject ◽  
Poisson Counts

We consider data that can be summarized as an N × K table of counts—for example, test data obtained by administering K tests to N subjects. The cell entries yij are assumed to be conditionally independent Poisson-distributed random variables, given the NK Poisson intensity parameters μij. The Rasch Poisson Counts Model (RPCM) postulates that the intensity parameters are products of test difficulty and subject ability parameters. We expand the RPCM by assuming that the subject parameters are random variables having a common gamma distribution with fixed unknown parameters and that the vectors of test difficulty parameters per subject follow a common Dirichlet distribution with fixed unknown parameters. Further, we show how additional structures can be imposed on the test parameters, modeling a within-subjects design. Methods for testing the fit and estimating the parameters of these models are presented and illustrated with the analysis of two empirical data sets.

Size-biased filtering of Poisson–Dirichlet samples with an application to partition structures in genetics

1986 ◽  
Vol 23 (04) ◽  
pp. 1008-1012 ◽  
Author(s):  
Fred M. Hoppe
Keyword(s):  
Characteristic Property ◽  
Dirichlet Distribution ◽  
Biased Sampling ◽  
Sampling Formula

A characteristic property of the Ewen sampling formula is shown to follow from the invariance under size-biased sampling of the Poisson–Dirichlet distribution.

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