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The properties of the generalized Waring distribution defined on the non negative integers are reviewed. Formulas for its moments and its mode are given. A construction as a mixture of negative binomial distributions is also presented. Then we turn to the Petersen model for estimating the population size $N$ in a two-way capture recapture experiment. We construct a Bayesian model for $N$ by combining a Waring prior with the hypergeometric distribution for the number of units caught twice in the experiment. Credible intervals for $N$ are obtained using quantiles of the posterior, a generalized Waring distribution. The standard confidence interval for the population size constructed using the asymptotic variance of Petersen estimator and .5 logit transformed interval are shown to be special cases of the generalized Waring credible interval. The true coverage of this interval is shown to be bigger than or equal to its nominal converage in small populations, regardless of the capture probabilities. In addition, its length is substantially smaller than that of the .5 logit transformed interval. Thus the proposed generalized Waring credible interval appears to be the best way to quantify the uncertainty of the Petersen estimator for populations size.

This paper proposes multivariate copula models for hierarchical data. They account for two types of correlation: one is between variables measured on the same unit and the other is a correlation between units in the same cluster. This model is used to carry out copula regression for hierarchical data that gives cluster specific prediction curves. In the simple case where a cluster contains two units and where two variables are measured on each one, the new model is constructed within a D-vine. Then we focus on situations where two variables are measured on the units of a cluster of arbitrary size. The proposed copula density has an explicit form; it is expressed in terms of three copula families. We study the properties of the model; compare it to the linear mixed model and end with special cases. When the three copula families and the marginal distributions are normal, the model is equivalent to a normal linear mixed model with random, cluster specific, intercepts. The method to select the three copula families and to estimate their parameters are proposed. We perform a Monte Carlo study of the parameter estimators. A data set on the marks of students in several school is used to implement the proposed model and to compare its performance to standard normal mixed linear models.

A multi-state version of an animal movement analysis method based on conditional logistic regression, called Step Selection Function (SSF), is proposed. In ecology SSF is developed from a comparison between the observed location of an animal and randomly sampled locations at each time step. Interpretation of the parameters in the multi-state model and the impact of different sampling schemes for the random locations are discussed. We prove the equivalence between the new model and a random walk model on the plane. This equivalence allows one to use both pure movement and local discrete choice behaviors in identifying the model's hidden states. The new method is used to model the movement behavior of GPS-collared bison in Prince Albert National Park, Canada. The multi-state SSF successfully teases apart areas used to forage and to travel. The analysis thus provides valuable insights into how bison adjust their movement to habitat features, thereby revealing spatial determinants of functional connectivity in heterogeneous landscapes.

In this paper, we propose a general hidden state random walk model to describe the movement of an animal that takes into account movement taxis with respect to features of the environment. A circular-linear process models the direction and distance between two consecutive localizations of the animal. A hidden process structure accounts for the animal's change in movement behavior. The originality of the proposed approach is that several environmental targets can be included in the directional model. An EM algorithm is devised to fit this model and an application to the analysis of the movement of caribou in Canada's boreal forest is presented, Comment: 28 pages

This paper suggests a nonlinear mixed effects model for data points in $\mathit{SO}(3)$, the set of $3\times3$ rotation matrices, collected according to a repeated measure design. Each sample individual contributes a sequence of rotation matrices giving the relative orientations of the right foot with respect to the right lower leg as its ankle moves. The random effects are the five angles characterizing the orientation of the two rotation axes of a subject's right ankle. The fixed parameters are the average value of these angles and their variances within the population. The algorithms to fit nonlinear mixed effects models presented in Pinheiro and Bates (2000) are adapted to the new directional model. The estimation of the random effects are of interest since they give predictions of the rotation axes of an individual ankle. The performance of these algorithms is investigated in a Monte Carlo study. The analysis of two data sets is presented. In the biomechanical literature, there is no consensus on an in vivo method to estimate the two rotation axes of the ankle. The new model is promising. The estimates obtained from a sample of volunteers are shown to be in agreement with the clinically accepted results of Inman (1976), obtained by manipulating cadavers. The repeated measure directional model presented in this paper is developed for a particular application. The approach is, however, general and might be applied to other models provided that the random directional effects are clustered around their mean values., Comment: Published in at http://dx.doi.org/10.1214/10-AOAS342 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)