A Consistent Estimator of the Evolutionary Rate

Abstract

We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate {lambda}, conditioned on the observed number of particles to be equal n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance {sigma}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 [Formula] of the evolutionary rate{rho} 2 ={sigma}2/{lambda}. The estimator [Formula] is proportional to the sample variance [Formula] computed from n trait values. We find an approximate formula for the standard error of [Formula] based on a neat asymptotic relation for the variance of [Formula].