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J Math Biol. 2019 Nov 21. doi: 10.1007/s00285-019-01453-1. [Epub ahead of print]

On the convergence of the maximum likelihood estimator for the transition rate under a 2-state symmetric model.

Author information

1
Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, Canada. Lam.Ho@dal.ca.
2
Department of Mathematical Sciences, University of Delaware, Newark, USA.
3
Program in Computational Biology, Fred Hutchinson Cancer Research Center, Seattle, USA.
4
Departments of Biomathematics, Biostatistics and Human Genetics, University of California, Los Angeles, USA.

Abstract

Maximum likelihood estimators are used extensively to estimate unknown parameters of stochastic trait evolution models on phylogenetic trees. Although the MLE has been proven to converge to the true value in the independent-sample case, we cannot appeal to this result because trait values of different species are correlated due to shared evolutionary history. In this paper, we consider a 2-state symmetric model for a single binary trait and investigate the theoretical properties of the MLE for the transition rate in the large-tree limit. Here, the large-tree limit is a theoretical scenario where the number of taxa increases to infinity and we can observe the trait values for all species. Specifically, we prove that the MLE converges to the true value under some regularity conditions. These conditions ensure that the tree shape is not too irregular, and holds for many practical scenarios such as trees with bounded edges, trees generated from the Yule (pure birth) process, and trees generated from the coalescent point process. Our result also provides an upper bound for the distance between the MLE and the true value.

KEYWORDS:

2-state symmetric model; Coalescent point process; Maximum likelihood estimator; Phylogenetics; Trait evolution; Yule process

PMID:
31754778
DOI:
10.1007/s00285-019-01453-1

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