Convergence of likelihood ratios and estimators for selection in nonneutral Wright–Fisher diffusions

Abstract

AbstractA number of discrete time, finite population size models in genetics describing the dynamics of allele frequencies are known to converge (subject to suitable scaling) to a diffusion process termed the Wright–Fisher diffusion. This diffusion evolves on a bounded interval, such that many standard results in diffusion theory, assuming evolution on the real line, no longer apply. In this article we derive conditions to establish ‐uniform ergodicity for diffusions on bounded intervals, and use them to prove that the Wright–Fisher diffusion is uniformly in the selection and mutation parameters ergodic, and that the measures induced by the solution to the stochastic differential equation are uniformly locally asymptotically normal. We subsequently use these results to show that the maximum likelihood and Bayesian estimators for the selection parameter are uniformly over compact sets consistent, asymptotically normal, display moment convergence, and are asymptotically efficient for a suitable class of loss functions.