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Statistical inference for diffusions in genetics
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Sant, Jaromir (2021) Statistical inference for diffusions in genetics. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3756290
Abstract
In this thesis we consider theoretical and practical aspects of conducting inference on data coming from the Wright{Fisher diffusion, which arises as the scaling limit of several discrete models used to describe the way in which allele frequencies change over time. This diffusion evolves on a bounded interval, and thus many standard results in diffusion theory assuming evolution on the entire real line do not apply.
Conditions ensuring the #-uniform ergodicity of positively recurrent diffusions on bounded intervals with entrance or regular boundaries are established, and used to prove uniform in the selection and mutation parameters ergodicity for the Wright{Fisher case. The family of measures induced by the diffusion is further shown to be uniformly locally asymptotically normal, and these results are used to show the uniform (over compact sets in the parameter space) consistency, asymptotic normality, convergence of moments and asymptotic efficiency of the Maximum Likelihood and Bayesian estimators for the selection parameter in a continuous observation regime.
By appealing to a suitable state space augmentation and making use of the exact algorithm for the Wright{Fisher diffusion, we propose an exact Markov Chain Monte Carlo scheme which is able to directly target the joint posterior of the allele age and selection parameter. The method is subsequently tested on simulated data for a variety of prior distributions on both parameters.
Finally, a brief sketch of how #-uniform ergodicity might be extended for the multidimensional Wright{Fisher diffusion is provided. The main techniques granting control over the rate of convergence in the ergodic theorem are developed, with a particular emphasis on why establishing such control is particularly challenging in the Wright{ Fisher case.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Mathematical statistics, Diffusion processes, Population biology -- Mathematical models, Population genetics -- Mathematical models, Markov processes, Diffusion Monte Carlo method | ||||
Official Date: | December 2021 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Jenkins, Paul A. ; Koskela, Jere ; Spanò, Dario | ||||
Format of File: | |||||
Extent: | xi, 175 leaves : illustrations | ||||
Language: | eng |
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