Gene genealogy and properties of test statistics of neutrality under population growth

We consider the Wright-Fisher model with exponential population growth and investigate effects of population growth on the shape of genealogy and the distributions of several test statistics of neutrality. In the limiting case as the population grows rapidly, the rapid-growth-limit genealogy is characterized. We obtained approximate expressions for expectations and variances of test statistics in the rapid-growth-limit genealogy and star genealogy. The distributions in the star genealogy are narrower than those in the cases of the simulated and rapid-growth-limit genealogies. The expectations and variances of the test statistics are monotone decreasing functions of the time length of the expansion, and the higher power of R₂ against population growth is suggested to be due to their smaller variances rather than to change of the expectations. We also investigated by simulation how quickly the distributions of test statistics approach those of the rapid-growth-limit genealogy.