Abstract
The random search problem has long attracted continuous interest owing to its broad interdisciplinary range of applications, including animal foraging, facilitated target location in biological systems and human motion. In this paper, we address the issue of statistical inference for ordinary Gaussian, Pareto, tempered Pareto and fractional Gaussian random walk models, which are among the most studied random walk models proposed as the best strategy in the random search problem. Based on rigorous analysis of the local asymptotic normality property and the Fisher information, we discuss some issues in unbiased joint estimation of the model parameters, in particular, the maximum-likelihood estimation. We present that there exist both theoretical and practical difficulties in more realistic tempered Pareto and fractional Gaussian random walk models from a statistical standpoint. We discuss our findings in the context of individual animal movement and show how our results may be used to facilitate the analysis of movement data and to improve the understanding of the underlying stochastic process.
Highlights
We address the issue of statistical analysis of animal random walk data using a mathematically rigorous approach borrowed from the general theory of stochastic processes
It remains a largely open question to what extent the inherently continuous animal movement can be adequately described by discrete random walk models as it is intuitively clear
We addressed this issue by considering the problem of parameter estimation for random walk data
Summary
Peculiarities of individual animal movement have been attracting considerable attention over the last three decades (Mandelbrot 1977; Kareiva & Shigesada 1983; Viswanathan et al 1996, 2008; Bartumeus et al 2003; Reynolds et al 2007; Codling et al 2008) as they are thought to hold the key to the understanding of the animal dispersal, and to better understanding of the spatiotemporal phenomena in population dynamics such as biological invasions, pattern formation, and so on (Turchin 1988; Okubo & Levin 2001). Having chosen parameter estimators for the step size distribution, we reveal their convergence rate(s) for a few most commonly used random walk models.
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