Time-calibrated species phylogenies are critical for addressing a wide range of questions in evolutionary biology, such as those that elucidate historical biogeography or uncover patterns of coevolution and diversification. Because molecular sequence data are not informative on absolute time, external data--most commonly, fossil age estimates--are required to calibrate estimates of species dive...

In [12], it was shown that a generalized maximum likelihood estimation problem on a (canonical) α-power-law model (M-family) can be solved by solving a system of linear equations. This was due to an orthogonality relationship between the M-family and a linear family with respect to the relative α-entropy (or the Iα-divergence). Relative α-entropy is a generalization of the usual relative entrop...

Van Dorp and Kotz (2003) introduced a general class of two-sided power distribution. We provide several examples of this distribution by considering a non negative, non decreasing differentiable function g on 0 1 satisfying g 1 − g 0 = 1. For the considered distributions the maximum likelihood estimation and moment estimation of parameters are obtained. Moments, moment generating functions, and...

A. The von Mises Expansion Before diving into the auxiliary results of Section 5, let us first derive some properties of the von Mises expansion. It is a simple calculation to verify that the Gateaux derivative is simply the functional derivative of in the event that T (F ) = R (f). Lemma 8. Let T (F ) = R (f)dμ where f = dF/dμ is the Radon-Nikodym derivative, is differentiable and let G be som...

Bayesian methods provide a powerful way to estimate species divergence times by combining information from molecular sequences with information from the fossil record. With the explosive increase of genomic data, divergence time estimation increasingly uses data of multiple loci (genes or site partitions). Widely used computer programs to estimate divergence times use independent and identicall...