We consider a stochastic system of $N$ interacting particles with constant diffusion coefficient and drift linear in space, time-depending on two unknown deterministic functions. Our concern here is the nonparametric estimation of these functions from a continuous observation of the process on $[0,T]$ for fixed $T$ and large $N$. We define two collections of projection estimators belonging to finite-dimensional subspaces of ${𝕃}^2([0,T])$. We study the ${𝕃}^2$-risks of these estimators, where the risk is defined either by the expectation of an empirical norm or by the expectation of a deterministic norm. Afterwards, we propose a data-driven choice of the dimensions and study the risk of the adaptive estimators. The results are illustrated by numerical experiments on simulated data.

中文翻译：

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相互作用粒子系统的非参数自适应估计

我们考虑一个随机系统$氮$具有恒定扩散系数和空间线性漂移的相互作用粒子，时间取决于两个未知的确定性函数。我们在这里关注的是通过对过程的连续观察来对这些函数进行非参数估计$[0,\mathrm{时间}]$对于固定的$\mathrm{时间}$和大$氮$。我们定义了属于有限维子空间的两个投影估计量集合${𝕃}^2（[0,\mathrm{时间}]）$。我们研究的是${𝕃}^2$-这些估计量的风险，其中风险由经验规范的期望或确定性规范的期望定义。之后，我们提出了数据驱动的维度选择，并研究了自适应估计器的风险。结果通过模拟数据的数值实验来说明。