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Register a new userOn the convergence of the maximum likelihood estimator for the transition rate under a 2-state symmetric model
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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.
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The Tree of Life is the graphical structure that represents the evolutionary process from single-cell organisms at the origin of life to the vast biodiversity we see today. Reconstructing this tree from genomic sequences is challenging due to the variety of biological forces that shape the signal in the data, and many of those processes like incomplete lineage sorting and hybridization can produce confounding information. Here, we present the mathematical version of the identifiability proofs of phylogenetic networks under the pseudolikelihood model in SNaQ. We establish that the ability to detect different hybridization events depends on the number of nodes on the hybridization blob, with small blobs (corresponding to closely related species) being the hardest to be detected. Our work focuses on level-1 networks, but raises attention to the importance of identifiability studies on phylogenetic inference methods for broader classes of networks.
We consider a sequence of identically independently distributed random samples from an absolutely continuous probability measure in one dimension with unbounded density. We establish a new rate of convergence of the $infty-$Wasserstein distance between the empirical measure of the samples and the true distribution, which extends the previous convergence result by Trilllos and Slepv{c}ev to the case that the true distribution has an unbounded density.
The mixed fractional Vasicek model, which is an extended model of the traditional Vasicek model, has been widely used in modelling volatility, interest rate and exchange rate. Obviously, if some phenomenon are modeled by the mixed fractional Vasicek model, statistical inference for this process is of great interest. Based on continuous time observations, this paper considers the problem of estimating the drift parameters in the mixed fractional Vasicek model. We will propose the maximum likelihood estimators of the drift parameters in the mixed fractional Vasicek model with the Radon-Nikodym derivative for a mixed fractional Brownian motion. Using the fundamental martingale and the Laplace transform, both the strong consistency and the asymptotic normality of the maximum likelihood estimators have been established for all $Hin(0,1)$, $H eq 1/2$.
In phylogenetic studies, biologists often wish to estimate the ancestral discrete character state at an interior vertex $v$ of an evolutionary tree $T$ from the states that are observed at the leaves of the tree. A simple and fast estimation method --- maximum parsimony --- takes the ancestral state at $v$ to be any state that minimises the number of state changes in $T$ required to explain its evolution on $T$. In this paper, we investigate the reconstruction accuracy of this estimation method further, under a simple symmetric model of state change, and obtain a number of new results, both for 2-state characters, and $r$--state characters ($r>2$). Our results rely on establishing new identities and inequalities, based on a coupling argument that involves a simpler `coin toss approach to ancestral state reconstruction.
We derive Laplace-approximated maximum likelihood estimators (GLAMLEs) of parameters in our Graph Generalized Linear Latent Variable Models. Then, we study the statistical properties of GLAMLEs when the number of nodes $n_V$ and the observed times of a graph denoted by $K$ diverge to infinity. Finally, we display the estimation results in a Monte Carlo simulation considering different numbers of latent variables. Besides, we make a comparison between Laplace and variational approximations for inference of our model.
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