Dispersal density estimation across scales

Abstract

We consider a space structured population model generated by two-point clouds: a homogeneous Poisson process M with intensity $\mathit{n}\to \infty$ as a model for a parent generation together with a Cox point process N as offspring generation, with conditional intensity given by the convolution of M with a scaled dispersal density ${\mathit{\sigma }}^{-1}\mathit{f}(·/\mathit{\sigma })$. Based on a realisation of M and N, we study the nonparametric estimation of f and the estimation of the physical scale parameter $\mathit{\sigma }>0$ simultaneously for all regimes $\mathit{\sigma }={\mathit{\sigma }}_{\mathit{n}}$. We establish that the optimal rates of convergence do not depend monotonously on the scale and we construct minimax estimators accordingly whether σ is known or considered as a nuisance, in which case we can estimate it and achieve asymptotic minimaxity by plug-in. The statistical reconstruction exhibits a competition between a direct and a deconvolution problem. Our study reveals in particular the existence of a least favorable intermediate inference scale, a phenomenon that seems to be new.