No abstract

Predator-prey interactions are vital to the stability of many ecosystems. Yet, few studies have considered how they are mediated due to substantial challenges in quantifying behavior over appropriate temporal and spatial scales. Here, we employ high-resolution sonar imaging to track the motion and interactions among predatory fish and their schooling prey in a natural environment. In particular, we address the relationship between predator attack behavior and the capacity for prey to respond both directly and through collective propagation of changes in velocity by group members. To do so, we investigated a large number of attacks and estimated per capita risk during attack and its relation to the size, shape, and internal structure of prey groups. Predators were found to frequently form coordinated hunting groups, with up to five individuals attacking in line formation. Attacks were associated with increased fragmentation and irregularities in the spatial structure of prey groups, features that inhibit collective information transfer among prey. Prey group fragmentation, likely facilitated by predator line formation, increased (estimated) per capita risk of prey, provided prey schools were maintained below a threshold size of approximately 2 m(2). Our results highlight the importance of collective behavior to the strategies employed by both predators and prey under conditions of considerable informational constraints.

a b s t r a c tWe consider a log-linear model for time series of counts. This type of model provides a framework where both negative and positive association can be taken into account. In addition time dependent covariates are accommodated in a straightforward way. We study its probabilistic properties and maximum likelihood estimation. It is shown that a perturbed version of the process is geometrically ergodic, and, under some conditions, it approaches the non-perturbed version. In addition, it is proved that the maximum likelihood estimator of the vector of unknown parameters is asymptotically normal with a covariance matrix that can be consistently estimated. The results are based on minimal assumptions and can be extended to the case of log-linear regression with continuous exogenous variables. The theory is applied to aggregated financial transaction time series. In particular, we discover positive association between the number of transactions and the volatility process of a certain stock.

This paper considers geometric ergodicity and likelihood based inference for linear and nonlinear Poisson autoregressions. In the linear case the conditional mean is linked linearly to its past values as well as the observed values of the Poisson process. This also applies to the conditional variance, implying an interpretation as an integer valued GARCH process. In a nonlinear conditional Poisson model, the conditional mean is a nonlinear function of its past values and a nonlinear function of past observations. As a particular example an exponential 1 autoregressive Poisson model for time series is considered. Under geometric ergodicity the maximum likelihood estimators of the parameters are shown to be asymptotically Gaussian in the linear model. In addition we provide a consistent estimator of the asymptotic covariance, which is used in the simulations and the analysis of some transaction data. Our approach to verifying geometric ergodicity proceeds via Markov theory and irreducibility. Finding transparent conditions for proving ergodicity turns out to be a delicate problem in the original model formulation. This problem is circumvented by allowing a perturbation of the model. We show that as the perturbations can be chosen to be arbitrarily small, the differences between the perturbed and non-perturbed versions vanish as far as the asymptotic distribution of the parameter estimates is concerned.

We derive an asymptotic theory of nonparametric estimation for a time series regression model $Z_t=f(X_t)+W_t$, where \ensuremath\{X_t\} and \ensuremath\{Z_t\} are observed nonstationary processes and $\{W_t\}$ is an unobserved stationary process. In econometrics, this can be interpreted as a nonlinear cointegration type relationship, but we believe that our results are of wider interest. The class of nonstationary processes allowed for $\{X_t\}$ is a subclass of the class of null recurrent Markov chains. This subclass contains random walk, unit root processes and nonlinear processes. We derive the asymptotics of a nonparametric estimate of f(x) under the assumption that $\{W_t\}$ is a Markov chain satisfying some mixing conditions. The finite-sample properties of $\hat{f}(x)$ are studied by means of simulation experiments.Comment: Published at http://dx.doi.org/10.1214/009053606000001181 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org