Asymptotic theory with hierarchical autocorrelation: Ornstein-Uhlenbeck tree models

TL;DR

This paper develops an asymptotic theory for hierarchical autocorrelation models based on Ornstein-Uhlenbeck processes along trees, revealing limitations in estimating certain parameters and the varying convergence rates of estimators.

Contribution

It provides the first asymptotic analysis of OU tree models, identifying which parameters are microergodic and demonstrating different convergence rates for estimators.

Findings

Mean parameter is not microergodic, no consistent estimator exists.

Covariance parameters can be microergodic under certain conditions.

MLE of autocorrelation converges slower than other parameters.

Abstract

Hierarchical autocorrelation in the error term of linear models arises when sampling units are related to each other according to a tree. The residual covariance is parametrized using the tree-distance between sampling units. When observations are modeled using an Ornstein-Uhlenbeck (OU) process along the tree, the autocorrelation between two tips decreases exponentially with their tree distance. These models are most often applied in evolutionary biology, when tips represent biological species and the OU process parameters represent the strength and direction of natural selection. For these models, we show that the mean is not microergodic: no estimator can ever be consistent for this parameter and provide a lower bound for the variance of its MLE. For covariance parameters, we give a general sufficient condition ensuring microergodicity. This condition suggests that some parameters may not be estimated at the same rate as others. We show that, indeed, maximum likelihood estimators of the autocorrelation parameter converge at a slower rate than that of generally microergodic parameters. We showed this theoretically in a symmetric tree asymptotic framework and through simulations on a large real tree comprising 4507 mammal species.