A Consistent Estimator of the Evolutionary Rate

TL;DR

This paper introduces an unbiased estimator for the evolutionary rate in a branching particle system modeled by a Yule process with Brownian motion, providing a practical tool for analyzing trait evolution in related species.

Contribution

It proposes a new unbiased estimator for the evolutionary rate based on sample variance, with an approximate standard error formula, advancing methods for trait evolution analysis.

Findings

The estimator Rn2 is proportional to the sample variance of trait values.

An approximate formula for the standard error of Rn2 is derived.

The method applies to related species modeled by a Yule process with Brownian motion.

Abstract

We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate L, 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 s2. 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 Rn2 of the evolutionary rate r2=s2/L. The estimator Rn2 is proportional to the sample variance Sn2 computed from n trait values. We find an approximate formula for the standard error of Rn2 based on a neat asymptotic relation for the variance of Sn2.