  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.
