scholarly journals Edgeworth Expansions for the Number of Distinct Components Associated with the Ewens Sampling Formula

2013 ◽  
Vol 43 (1) ◽  
pp. 17-28 ◽  
Author(s):  
Hajime Yamato
Keyword(s):  
Edgeworth Expansions ◽  
Ewens Sampling Formula ◽  
Sampling Formula

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.

A stability property of the Ewens sampling formula

1983 ◽  
Vol 20 (03) ◽  
pp. 449-459
Author(s):  
Stanley Sawyer
Keyword(s):  
Population Size ◽  
Error Bound ◽  
Mutation Rate ◽  
Stability Property ◽  
Ewens Sampling Formula ◽  
Sampling Formula ◽  
Deleterious Alleles

An error bound for convergence to the Ewens sampling formula is given where the population size or mutation rate may vary from generation to generation, or the population is not yet at equilibrium. An application is given to a model of Hartl and Campbell about selectively-equivalent subtypes within a class of deleterious alleles, and a theorem is proven showing that the size of the deleterious class stays within bounds sufficient to apply the first result. Generalizations are discussed.

Random partitions in population genetics

1978 ◽  
Vol 361 (1704) ◽  
pp. 1-20 ◽  
Keyword(s):  
Population Genetics ◽  
Genetic Variation ◽  
Charge State ◽  
Characteristic Property ◽  
Large Population ◽  
Recurrent Mutation ◽  
Random Partitions ◽  
Ewens Sampling Formula ◽  
Sampling Formula ◽  
State Models

This paper is concerned with models for the genetic variation of a sample of gametes from a large population. The need for consistency between different sample sizes limits the mathematical possibilities to what are here called ‘partition structures Distinctive among them is the structure described by the Ewens sampling formula, which is shown to enjoy a characteristic property of non-interference between the different alleles. This characterization explains the robustness of the Ewens formula when neither selection nor recurrent mutation is significant, although different structures arise from selective and ‘charge-state’ models

Explicit Limits of Total Variation Distance in Approximations of Random Logarithmic Assemblies by Related Poisson Processes

1997 ◽  
Vol 6 (1) ◽  
pp. 87-105 ◽  
Author(s):  
DUDLEY STARK
Keyword(s):  
Total Variation ◽  
Local Limit ◽  
Total Variation Distance ◽  
Weighted Sums ◽  
Set Partitions ◽  
Ewens Sampling Formula ◽  
Combinatorial Objects ◽  
Sampling Formula ◽  
Random Mappings ◽  
Variation Distance

Assemblies are labelled combinatorial objects that can be decomposed into components. Examples of assemblies include set partitions, permutations and random mappings. In addition, a distribution from population genetics called the Ewens sampling formula may be treated as an assembly. Each assembly has a size n, and the sum of the sizes of the components sums to n. When the uniform distribution is put on all assemblies of size n, the process of component counts is equal in distribution to a process of independent Poisson variables Zi conditioned on the event that a weighted sum of the independent variables is equal to n. Logarithmic assemblies are assemblies characterized by some θ > 0 for which i[ ]Zi → θ. Permutations and random mappings are logarithmic assemblies; set partitions are not a logarithmic assembly. Suppose b = b(n) is a sequence of positive integers for which b/n → β ε (0, 1]. For logarithmic assemblies, the total variation distance db(n) between the laws of the first b coordinates of the component counting process and of the first b coordinates of the independent processes converges to a constant H(β). An explicit formula for H(β) is given for β ε (0, 1] in terms of a limit process which depends only on the parameter θ. Also, it is shown that db(n) → 0 if and only if b/n → 0, generalizing results of Arratia, Barbour and Tavaré for the Ewens sampling formula. Local limit theorems for weighted sums of the Zi are used to prove these results.

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