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.

翻译：我们考虑一个由两个点云生成的空间结构化种群模型：均匀泊松过程$M$（强度$n\to\infty$）作为亲代代际模型，以及考克斯点过程$N$（子代代际模型），其条件强度由$M$与缩放扩散密度$\sigma^{-1}f(\cdot/\sigma)$的卷积给出。基于$M$和$N$的实现，我们研究在所有情形$\sigma=\sigma_n$下$f$的非参数估计以及物理尺度参数$\sigma>0$的估计。我们证明了最优收敛速率并非随尺度单调变化，并据此构造了极小极大估计量——无论$\sigma$已知还是被视为干扰参数（此时可通过插入法估计并实现渐近极小极大性）。统计重建揭示了直接问题与反卷积问题之间的竞争关系。我们的研究特别发现了一种最不利的中间推断尺度的存在，这一现象似乎尚属首次报道。