A consistent estimator of the evolutionary rate.

We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate λ, conditioned on the observed number of particles to be equal to n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance σ(2). In this paper we treat n particles as a sample of related species. The spatial Brownian motion of a particle describes the development of a trait value of interest (e.g. log-body-size). We propose an unbiased estimator Rn(2) of the evolutionary rate ρ(2)=σ(2)/λ. The estimator Rn(2) is proportional to the sample variance Sn(2) computed from n trait values. We find an approximate formula for the standard error of Rn(2) based on a neat asymptotic relation for the variance of Sn(2).