# Phylogenetic confidence intervals for the optimal trait value

## TL;DR

This paper introduces a new method to calculate confidence intervals for optimal trait values in evolutionary models involving related species.

## Contribution

The novelty lies in deriving confidence interval formulae for the optimal trait value using limit theorems from a stochastic evolutionary model.

## Key findings

- Three limit theorems for the sample mean are established based on adaptation rate domains.
- For fast adaptation, the normalized sample mean is approximately normally distributed for large n.
- New confidence interval formulae for the optimal trait value are developed using these theorems.

## Abstract

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.