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April 2013 Asymptotic theory with hierarchical autocorrelation: Ornstein–Uhlenbeck tree models
Lam Si Tung Ho, Cécile Ané
Ann. Statist. 41(2): 957-981 (April 2013). DOI: 10.1214/13-AOS1105


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.


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Lam Si Tung Ho. Cécile Ané. "Asymptotic theory with hierarchical autocorrelation: Ornstein–Uhlenbeck tree models." Ann. Statist. 41 (2) 957 - 981, April 2013.


Published: April 2013
First available in Project Euclid: 29 May 2013

zbMATH: 1267.62092
MathSciNet: MR3099127
Digital Object Identifier: 10.1214/13-AOS1105

Primary: 62F12, 62M10
Secondary: 62M30, 92B10, 92D15

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 2 • April 2013
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