Dispersal density estimation across scales

We consider a space structured population model generated by two point clouds: a homogeneous Poisson process $M$ with intensity $n\to\infty$ 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 $\sigma^{-1}f(\cdot/\sigma)$. Based on a realisation of $M$ and $N$, we study the nonparametric estimation of $f$ and the estimation of the physical scale parameter $\sigma>0$ simultaneously for all regimes $\sigma=\sigma_n$. We establish that the optimal rates of convergence do not depend monotonously on the scale and we construct minimax estimators accordingly whether $\sigma$ 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 favourable intermediate inference scale, a phenomenon that seems to be new.