The Annals of Statistics

Asymptotic theory with hierarchical autocorrelation: Ornstein–Uhlenbeck tree models

Lam Si Tung Ho and Cécile Ané

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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.

Article information

Ann. Statist., Volume 41, Number 2 (2013), 957-981.

First available in Project Euclid: 29 May 2013

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Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62M30: Spatial processes 92D15: Problems related to evolution 92B10: Taxonomy, cladistics, statistics

Tree autocorrelation dependence microergodic Ornstein–Uhlenbeck evolution phylogenetics


Ho, Lam Si Tung; Ané, Cécile. Asymptotic theory with hierarchical autocorrelation: Ornstein–Uhlenbeck tree models. Ann. Statist. 41 (2013), no. 2, 957--981. doi:10.1214/13-AOS1105.

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