(In)Consistency of Maximum Likelihood Estimators in Ornstein-Uhlenbeck Autocorrelation Tree Models

• Activity Number: 245
• Type: Contributed
• Date/Time: Monday, August 1, 2011 : 2:00 PM to 3:50 PM
• Sponsor: ENAR
• Author(s): Cecile Ane*+ and Lam Ho
• Companies: University of Wisconsin and University of Wisconsin
• Keywords: tree autocorrelation ; dependence ; microergodicity ; Ornstein-Uhlenbeck ; phylogenetics ; evolution

Abstract

We consider linear models with strong hierarchical autocorrelation, and prove the consistency and asymptotic normality -or lack thereof- of maximum likelihood estimators. Observations are modeled at the tips of a tree and assumed to arise from a Brownian motion or an Ornstein-Uhlenbeck process along the tree. The autocorrelation between two tips increases with the length of their shared path from the root. These models are most often applied in evolutionary biology, where different tips represent different species. We show that the maximum likelihood estimators of some parameters are not consistent in standard asymptotic frameworks. In fact, these parameters are not microergodic: no estimator can ever be consistent for such parameters. We will show the analogy and differences with Ornstein-Uhlenbeck models in spatial statistics. For microergodic parameters, we will present consistency and asymptotic normality of their maximum likelihood estimators under a regular asymptotic framework. We will discuss the consequences of these results for sampling design, in application to evolutionary comparative studies.