Dispersal density estimation across scales

We consider a space structured population model generated by two point clouds: a homogeneous Poisson process $M=\sum_{j}\delta_{X_{j}}$ with intensity of order $n\to\infty$ as a model for a parent generation together with a Cox point process $N=\sum_{j}\delta_{Y_{j}}$ as offspring generation, with conditional intensity of order $M\ast(\sigma^{-1}f(\cdot/\sigma))$, where $\ast$ denotes convolution, $f$ is the so-called dispersal density, the unknown parameter of interest, and $\sigma>0$ is a physical scale parameter. Based on a realisation of $M$ and $N$, we study the nonparametric estimation of $f$, for several regimes $\sigma=\sigma_{n}$. We establish that the optimal rates of convergence do not depend monotonously on the scale $\sigma$ and construct minimax estimators accordingly. Depending on $\sigma$, the reconstruction problem exhibits a competition between a direct and a deconvolution problem. Our study reveals in particular the existence of a least favourable intermediate inference scale.