Estimation of age and rate of increase of rare variants.

The problem considered is that of estimating the age or rate of increase of a variant on the basis of the present number of replicates observed in a population. In place of previous diffusion equation analyses of age probability distributions, the likelihood for the age is studied on the basis of a discrete branching process model. It is shown that variations inherent in the process of gene evolution in natural populations make it impossible to provide a reliable point estimate of the age of a specified variant, although the likelihood analysis provides a confidence interval which may place useful bounds on the period in which a variant originated. The observed distribution of numbers of several variants may also provide useful information. The problems of estimation are discussed with reference to rare variants arising in American Indian populations.