This paper delves into a nonparametric estimation approach for the interaction function within diffusion-type particle system models. We introduce two estimation methods based upon an empirical risk minimization. Our study encompasses an analysis of the stochastic and approximation errors associated with both procedures, along with an examination of certain minimax lower bounds. In particular, we show that there is a natural metric under which the corresponding minimax estimation error of the interaction function converges to zero with parametric rate. This result is rather suprising given complexity of the underlying estimation problem and rather large classes of interaction functions for which the above parametric rate holds.

翻译：本文深入探讨了扩散型粒子系统模型中交互函数的非参数估计方法。我们提出了两种基于经验风险最小化的估计算法。研究涵盖了对两种算法相关随机误差与近似误差的分析，并考察了若干极小化最优下界。特别地，我们发现在某一自然度量下，交互函数对应的极小化最优估计误差以参数速率收敛至零。考虑到基础估计问题的复杂性以及适用上述参数速率的交互函数类别之广泛，这一结果出人意料。