图隆平均场粒子系统的非参数估计
Non-parametric estimates for graphon mean-field particle systems
AI总结：
针对图隆平均场粒子系统，研究基于连续时间轨迹构建密度、漂移系数及图隆函数的非参数估计，证明收敛性并通过极小极大分析验证估计量的逐点最优性。
摘要：
我们研究Bayraktar、Chakraborty与Wu的工作中提出的graphon（图隆）平均场系统，它是一类异质相互作用扩散粒子系统的大群体极限，其相互作用为平均场类型，权重由底层graphon函数刻画。通过观测粒子系统的连续时间轨迹，我们构建了该平均场系统的粒子密度、漂移系数的插件估计量，并由此得到graphon相互作用权重的估计量。其中密度与漂移的估计量是经验数据核插值的直接结果，底层graphon函数的估计量则通过反卷积方法得到。我们证明，当粒子数量增加时，graphon估计量逐点收敛于真实graphon函数，进而在割度量下收敛。此外，我们在一类特定的粒子系统中开展了极小极大分析，验证了密度与漂移估计量的逐点最优性。
英文摘要：
We consider the graphon mean-field system introduced in the work of Bayraktar, Chakraborty, and Wu. It is the large-population limit of a heterogeneously interacting diffusive particle system, where the interaction is of mean-field type with weights characterized by an underlying graphon function. Through observation of continuous-time trajectories within the particle system, we construct plug-in estimators of the particle density, the drift coefficient, and thus the graphon interaction weights of the mean-field system. Our estimators for the density and drift are direct results of kernel interpolation on the empirical data, and a deconvolution method leads to an estimator of the underlying graphon function. We show that, as the number of particles increases, the graphon estimator converges to the true graphon function pointwisely, and as a consequence, in the cut metric. Besides, we conduct a minimax analysis within a particular class of particle systems to justify the pointwise optimality of the density and drift estimators.
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