Small-time approximation of the transition density for diffusions with singularities. Application to the Wright-Fisher model

TL;DR

This paper develops new approximation methods for the transition density of Wright-Fisher diffusions, especially near boundary states, improving modeling of allele frequency dynamics over time.

Contribution

It introduces the Asymptotic Expansion and Gaussian approximation functions, extending density estimation techniques for diffusion models with singularities.

Findings

AE provides accurate density estimates including extreme allele values

GaussA's validity range is characterized

Proposed methods outperform Beta and Gaussian distributions in simulations

Abstract

The Wright-Fisher (W-F) diffusion model serves as a foundational framework for interpreting population evolution through allele frequency dynamics over time. Despite the known transition probability between consecutive generations, an exact analytical expression for the transition density at arbitrary time intervals remains elusive. Commonly utilized distributions such as Gaussian or Beta inadequately address the fixation issue at extreme allele frequencies (0 or 1), particularly for short periods. In this study, we introduce two alternative parametric functions, namely the Asymptotic Expansion (AE) and the Gaussian approximation (GaussA), derived through probabilistic methodologies, aiming to better approximate this density. The AE function provides a suitable density for allele frequency distributions, encompassing extreme values within the interval [0,1]. Additionally, we outline the range of validity for the GaussA approximation. While our primary focus is on W-F diffusion, we demonstrate how our findings extend to other diffusion models featuring singularities. Through simulations of allele frequencies under a W-F process and employing a recently developed adaptive density estimation method, we conduct a comparative analysis to assess the fit of the proposed densities against the Beta and Gaussian distributions.