scholarly journals Closed-Form Asymptotic Sampling Distributions under the Coalescent with Recombination for an Arbitrary Number of Loci

2012 ◽  
Vol 44 (02) ◽  
pp. 391-407 ◽  
Author(s):  
Anand Bhaskar ◽  
Yun S. Song
Keyword(s):  
Asymptotic Expansion ◽  
Closed Form ◽  
Recombination Rate ◽  
Functional Form ◽  
Asymptotic Series ◽  
Sampling Distribution ◽  
Challenging Problem ◽  
Sampling Distributions ◽  
Asymptotic Sampling ◽  
New Framework

Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on an asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. These expressions are universal in the sense that their functional form in terms of the marginal one-locus distributions applies to all finite- and infinite-alleles models of mutation.

scholarly journals Closed-Form Asymptotic Sampling Distributions under the Coalescent with Recombination for an Arbitrary Number of Loci

2012 ◽  
Vol 44 (2) ◽  
pp. 391-407 ◽  
Author(s):  
Anand Bhaskar ◽  
Yun S. Song
Keyword(s):  
Asymptotic Expansion ◽  
Closed Form ◽  
Recombination Rate ◽  
Functional Form ◽  
Asymptotic Series ◽  
Sampling Distribution ◽  
Challenging Problem ◽  
Sampling Distributions ◽  
Asymptotic Sampling ◽  
New Framework

Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on an asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. These expressions are universal in the sense that their functional form in terms of the marginal one-locus distributions applies to all finite- and infinite-alleles models of mutation.

scholarly journals Closed-Form Two-Locus Sampling Distributions: Accuracy and Universality

Genetics ◽  
2009 ◽  
Vol 183 (3) ◽  
pp. 1087-1103 ◽  
Author(s):  
Paul A. Jenkins ◽  
Yun S. Song
Keyword(s):  
Closed Form ◽  
Recombination Rate ◽  
Extensive Study ◽  
Composite Likelihood ◽  
Likelihood Method ◽  
Sampling Formula ◽  
Sampling Distributions ◽  
Mutation Model ◽  
Asymptotic Sampling

Sampling distributions play an important role in population genetics analyses, but closed-form sampling formulas are generally intractable to obtain. In the presence of recombination, there is no known closed-form sampling formula that holds for an arbitrary recombination rate. However, we recently showed that it is possible to obtain useful closed-form sampling formulas when the population-scaled recombination rate ρ is large. Specifically, in the case of the two-locus infinite-alleles model, we considered an asymptotic expansion of the sampling formula in inverse powers of ρ and obtained closed-form expressions for the first few terms in the expansion. In this article, we generalize this result to an arbitrary finite-alleles mutation model and show that, up to the first few terms in the expansion that we are able to compute analytically, the functional form of the asymptotic sampling formula is common to all mutation models. We carry out an extensive study of the accuracy of the asymptotic formula for the two-locus parent-independent mutation model and discuss in detail a concrete application in the context of the composite-likelihood method. Furthermore, using our asymptotic sampling formula, we establish a simple sufficient condition for a given two-locus sample configuration to have a finite maximum-likelihood estimate (MLE) of ρ. This condition is the first analytic result on the classification of the MLE of ρ and is instantaneous to check in practice, provided that one-locus probabilities are known.

scholarly journals Asymptotic Inference for Optimal Rerandomization Designs

2021 ◽  
Vol 1 (1) ◽  
pp. 49-58
Author(s):  
Mårten Schultzberg ◽  
Per Johansson
Keyword(s):  
Experimental Design ◽  
Normal Distribution ◽  
Mahalanobis Distance ◽  
Design Strategy ◽  
Sampling Distribution ◽  
Optimal Designs ◽  
Asymptotic Inference ◽  
Optimal Allocations ◽  
Asymptotic Sampling

AbstractRecently a computational-based experimental design strategy called rerandomization has been proposed as an alternative or complement to traditional blocked designs. The idea of rerandomization is to remove, from consideration, those allocations with large imbalances in observed covariates according to a balance criterion, and then randomize within the set of acceptable allocations. Based on the Mahalanobis distance criterion for balancing the covariates, we show that asymptotic inference to the population, from which the units in the sample are randomly drawn, is possible using only the set of best, or ‘optimal’, allocations. Finally, we show that for the optimal and near optimal designs, the quite complex asymptotic sampling distribution derived by Li et al. (2018), is well approximated by a normal distribution.

A property of the asymptotic series for a class of Titchmarsh–Weyl m-functions

1986 ◽  
Vol 102 (3-4) ◽  
pp. 253-257 ◽  
Author(s):  
B. J. Harris
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Linear Differential Equation ◽  
Asymptotic Series ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
The Real ◽  
Linear Differential ◽  
Image Position

SynopsisIn an earlier paper [6] we showed that if q ϵ CN[0, ε) for some ε > 0, then the Titchmarsh–Weyl m(λ) function associated with the second order linear differential equationhas the asymptotic expansionas |A| →∞ in a sector of the form 0 < δ < arg λ < π – δ.We show that if the real valued function q admits the expansionin a neighbourhood of 0, then

Hyperasymptotics

1990 ◽  
Vol 430 (1880) ◽  
pp. 653-668 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Transition Point ◽  
Asymptotic Series ◽  
Stokes Phenomenon ◽  
Numerical Computations ◽  
Borel Summation ◽  
Nested Sequence ◽  
Exponentially Small

We develop a technique for systematically reducing the exponentially small (‘superasymptotic’) remainder of an asymptotic expansion truncated near its least term, for solutions of ordinary differential equations of Schrödinger type where one transition point dominates. This is achieved by repeatedly applying Borel summation to a resurgence formula discovered by Dingle, relating the late to the early terms of the original expansion. The improvements form a nested sequence of asymptotic series truncated at their least terms. Each such ‘hyperseries’ involves the terms of the original asymptotic series for the particular function being approximated, together with terminating integrals that are universal in form, and is half the length of its predecessor. The hyperasymptotic sequence is therefore finite, and leads to an ultimate approximation whose error is less than the square of the original superasymptotic remainder. The Stokes phenomenon is automatically and exactly incorporated into the scheme. Numerical computations confirm the efficacy of the technique.

Asymptotic Transformations of q-Series

1998 ◽  
Vol 50 (2) ◽  
pp. 412-425 ◽  
Author(s):  
Richard J. McIntosh
Keyword(s):  
Asymptotic Expansion ◽  
Closed Form ◽  
Asymptotic Expansions ◽  
General Class ◽  
Theta Functions ◽  
Mock Theta Functions

AbstractFor the q–series we construct a companion q–series such that the asymptotic expansions of their logarithms as q → 1– differ only in the dominant few terms. The asymptotic expansion of their quotient then has a simple closed form; this gives rise to a new q–hypergeometric identity. We give an asymptotic expansion of a general class of q–series containing some of Ramanujan's mock theta functions and Selberg's identities.

scholarly journals Computational Inference Beyond Kingman's Coalescent

2015 ◽  
Vol 52 (02) ◽  
pp. 519-537 ◽  
Author(s):  
Jere Koskela ◽  
Paul Jenkins ◽  
Dario Spanò
Keyword(s):  
Importance Sampling ◽  
Genetic Data ◽  
Likelihood Inference ◽  
Data Sets ◽  
Conditional Sampling ◽  
Challenging Problem ◽  
Computational Inference ◽  
Sampling Distributions ◽  
Full Likelihood ◽  
Kingman’S Coalescent

Full likelihood inference under Kingman's coalescent is a computationally challenging problem to which importance sampling (IS) and the product of approximate conditionals (PAC) methods have been applied successfully. Both methods can be expressed in terms of families of intractable conditional sampling distributions (CSDs), and rely on principled approximations for accurate inference. Recently, more general Λ- and Ξ-coalescents have been observed to provide better modelling fits to some genetic data sets. We derive families of approximate CSDs for finite sites Λ- and Ξ-coalescents, and use them to obtain ‘approximately optimal’ IS and PAC algorithms for Λ-coalescents, yielding substantial gains in efficiency over existing methods.

scholarly journals Sampling for IPM Decision Making: Where Should We Invest Time and Resources?

1999 ◽  
Vol 89 (11) ◽  
pp. 1104-1111 ◽  
Author(s):  
Jan P. Nyrop ◽  
Michael R. Binns ◽  
Wopke van der Werf
Keyword(s):  
Decision Making ◽  
Sample Size ◽  
Operating Characteristic ◽  
Crop Protection ◽  
Critical Density ◽  
Sampling Distribution ◽  
Fine Tuning ◽  
Sampling Distributions ◽  
Sample Plan

Guides for making crop protection decisions based on assessments of pest abundance or incidence are cornerstones of many integrated pest management systems. Much research has been devoted to developing sample plans for use in these guides. The development of sampling plans has usually focused on collecting information on the sampling distribution of the pest, describing this sampling distribution with a mathematical model, formulating a sample plan, and sometimes, but not always, evaluating the performance of the proposed sample plan. For crop protection decision making, classification of density or incidence is usually more appropriate than estimation. When classification is done, the average outcome of classification (the operating characteristic) is frequently robust to large changes in the sampling distribution, including estimates of the variance of pest counts, and to sample size. In contrast, the critical density, or critical incidence, about which classifications are made, has a large influence on the operating characteristic. We suggest that rather than investing resources in elaborate descriptions of sampling distributions, or in fine-tuning sample size to achieve desired levels of precision, greater emphasis should be placed on characterizing pest densities that signal the need for management action and on designing decision guides that will be adopted by practitioners.

Corrigendum and addendum to ‘asymptotic series for double zeta, double gamma and Hecke L-functions’

2002 ◽  
Vol 132 (2) ◽  
pp. 377-384 ◽  
Author(s):  
KOHJI MATSUMOTO
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Series ◽  
Gamma Functions ◽  
Double Zeta ◽  
Error Terms

Refined expressions are given for the error terms in the asymptotic expansion formulas for double zeta and double gamma functions, proved in the author's former paper [2]. Some inaccurate claims in [2] are corrected.

scholarly journals Empirical cumulant function based parameter estimation in stable laws

2019 ◽  
Vol 22 (2) ◽  
pp. 311-338 ◽  
Author(s):  
Annika Krutto
Keyword(s):  
Closed Form ◽  
Selection Rule ◽  
Empirical Characteristic Function ◽  
Challenging Problem ◽  
Positive Real ◽  
Infinitely Divisible ◽  
Asymptotically Normal ◽  
Cumulant Function ◽  
Independent Identically Distributed ◽  
Infinite Moments

Stable distributions are a subclass of infinitely divisible distributions that form the only family of possible limiting distributions for sums of independent identically distributed random variables. A challenging problem is estimating their parameters because many have densities with no explicit form and infinite moments. To address this problem, a class of closed-form estimators, called cumulant estimators, has been introduced. Cumulant estimators are derived from the logarithm of empirical characteristic function at two arbitrary distinct positive real arguments. This paper extends cumulant estimators in two directions: (i) it is proved that they are asymptotically normal and (ii) a sample based rule for selecting the two arguments is proposed. Extensive simulations show that under the provided selection rule, the closed-form cumulant estimators generally outperform the well-known algorithmic methods.

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