The goal of these lectures is to review some mathematical aspects of random tree models used in evolutionary biology to model species trees.

We start with stochastic models of tree shapes (finite trees without edge lengths), culminating in the $\beta$-family of Aldous’ branching models.

We next introduce real trees (trees as metric spaces) and show how to study them through their contour, provided they are properly measured and ordered.

We then focus on the reduced tree, or coalescent tree, which is the tree spanned by species alive at the same fixed time. We show how reduced trees, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. Beautiful examples of random combs include the Kingman coalescent and coalescent point processes.

We end up displaying some recent biological applications of coalescent point processes to the inference of species diversification, to conservation biology and to epidemiology.

We consider a particle system studied by E. Brunet and B. Derrida (Phys. Rev. E 70 (2004) 016106), which evolves according to a branching mechanism with selection of the fittest keeping the population size fixed and equal to $N$. The particles remain grouped and move like a travelling front driven by a random noise with a deterministic speed. Because of its mean-field structure, the model can be further analysed as $N\to\infty $. We focus on the case where the noise lies in the max-domain of attraction of the Weibull extreme value distribution and show that under mild conditions the correction to the speed has universal features depending on the tail probabilities.

In many practical situations, we face data which are not only random but vague as well. To deal with these two types of uncertainties, it is necessary to incorporate fuzzy concept into statistical technique. In this paper, we investigate the maximum likelihood estimation and Bayesian estimation for Lindley distribution when the available observations are reported in the form of fuzzy data. We employ the EM algorithm to determine the maximum likelihood estimate (MLE) of the parameter and construct approximate confidence interval by using the asymptotic normality of the MLE. In the Bayesian setting, we use an approximation based on the Laplace approximation as well as a Markov Chain Monte Carlo technique to compute the Bayes estimate of the parameter. In addition, the highest posterior density credible interval of the unknown parameter is obtained. Extensive simulations are performed to compare the performances of the different proposed methods.

We define and study a new generalization of the complementary Weibull geometric distribution introduced by Tojeiro et al. (J. Stat. Comput. Simul. 84 (2014) 1345–1362). The new lifetime model is referred to as the Kumaraswamy complementary Weibull geometric distribution and includes twenty three special models. Its hazard rate function can be constant, increasing, decreasing, bathtub and unimodal shaped. Some of its mathematical properties, including explicit expressions for the ordinary and incomplete moments, generating and quantile functions, Rényi entropy, mean residual life and mean inactivity time are derived. The method of maximum likelihood is used for estimating the model parameters. We provide some simulation results to assess the performance of the proposed model. Two applications to real data sets show the flexibility of the new model compared with some nested and non-nested models.

Power series generalized nonlinear models [Comput. Statist. Data Anal. 53 (2009) 1155–1166] can be used when the Poisson assumption of equidispersion is not valid. In these models, we consider a more general family of discrete distributions for the response variable and a nonlinear structure for the regression parameters, although the dispersion parameter and other shape parameters are assumed known. We derive a general matrix formula for the second-order bias of the maximum likelihood estimate of the regression parameter vector in these models. We use the results by [J. Roy. Statist. Soc. B 30 (1968) 248–275] and bootstrap technique [Ann. Statist. 7 (1979) 1–26] to obtain the bias-corrected maximum likelihood estimate. Simulation studies are performed using different estimates. We also present an empirical application.

In this paper, we derive analytical expressions for the curvature influence statistic proposed by Cook [J. Roy. Statist. Soc. Ser. B 48 (1986) 133–169] in elliptical regression models under a data perturbation scheme. A relationship between the curvature statistics and the residuals is established and the effects of the shape parameter are assessed. The results reveal the role of the shape parameter in applying the curvature influence diagnostics technique.

We consider the recently introduced Transformation-based Markov Chain Monte Carlo (TMCMC) (Stat. Methodol. 16 (2014) 100–116), a methodology that is designed to update all the parameters simultaneously using some simple deterministic transformation of a one-dimensional random variable drawn from some arbitrary distribution on a relevant support. The additive transformation based TMCMC is similar in spirit to random walk Metropolis, except the fact that unlike the latter, additive TMCMC uses a single draw from a one-dimensional proposal distribution to update the high-dimensional parameter. In this paper, we first provide a brief tutorial on TMCMC, exploring its connections and contrasts with various available MCMC methods.

Then we study the diffusion limits of additive TMCMC under various set-ups ranging from the product structure of the target density to the case where the target is absolutely continuous with respect to a Gaussian measure; we also consider the additive TMCMC within Gibbs approach for all the above set-ups. These investigations lead to appropriate scaling of the one-dimensional proposal density. We also show that the optimal acceptance rate of additive TMCMC is 0.439 under all the aforementioned set-ups, in contrast with the well-established 0.234 acceptance rate associated with optimal random walk Metropolis algorithms under the same set-ups. We also elucidate the ramifications of our results and clear advantages of additive TMCMC over random walk Metropolis with ample simulation studies and Bayesian analysis of a real, spatial dataset with which $160$ unknowns are associated.

This paper proposes the Bayesian inference for flexible measurement error models, in which their systematic components include explanatory variable vectors with and without measurement errors, as well as nonlinear effects that are approximated by using B-splines. The model investigated is the structural version, as the error-prone variables follow scale mixtures of normal distributions such as Student-$t$, slash, contaminated normal, Laplace and symmetric hyperbolic distributions. To draw samples of the posterior distribution of the model parameters, an MCMC algorithm is proposed. The performance of this algorithm is assessed through simulations. In addition, the function fmem() of the R package BayesGESM is presented, which provides an easy way to apply the methodology presented in this paper. The proposed methodology is applied to a real data set, which shows that ignoring measurement errors (i.e., analyze the data by using the traditional methodology) can lead to wrong conclusions.

Test functions play an important role in Markov chain theory. Stability of a Markov chain can be demonstrated by constructing a test function of the chain that satisfies a stochastic drift criterion. The test function defines a class of functions of the process for which limit laws hold, yields bounds on the convergence of the Markov chain transition probabilities to the stationary distribution, and provides information concerning the mixing properties of the chain. Under certain conditions, these results can be improved by using a new test function derived from a known test function of a Markov chain.

We propose an extension of Hidden Markov Model (HMM) to support second-order Markov dependence in the observable random process. We propose a Bayesian method to estimate the parameters of the model and the non-observable sequence of states. We compare and select the best model, including the dependence order and number of states, using model selection criteria like Bayes factor and deviance information criterion (DIC). We apply the procedure to several simulated datasets and verify the good performance of the estimation procedure. Tests with a real dataset show an improved fitting when compared with usual first order HMMs demonstrating the usefulness of the proposed model.