This paper concerns the nonparametric estimation problem of the distribution-state dependent drift vector field in an interacting $N$-particle system. Observing single-trajectory data for each particle, we derive the mean-field rate of convergence for the maximum likelihood estimator (MLE), which depends on both Gaussian complexity and Rademacher complexity of the function class. In particular, when the function class contains $\alpha$-smooth H{\"o}lder functions, our rate of convergence is minimax optimal on the order of $N^{-\frac{\alpha}{d+2\alpha}}$. Combining with a Fourier analytical deconvolution argument, we derive the consistency of MLE for the external force and interaction kernel in the McKean-Vlasov equation.
翻译:本文涉及一个互动的 $N美元 粒子系统中分布状态依赖的漂移矢量字段的非参数估计问题。 观察每个粒子的单轨数据,我们得出最大可能性估计器(MLE)的平均汇合率(MLE),这取决于功能等级的高萨复杂度和拉德马赫复杂度。 特别是当函数等级包含$\alpha$- smooth H o}lder 函数时, 我们的汇合率在 $N ⁇ -\frac halpha ⁇ d+2\alpha ⁇ 的顺序上是最小的。 结合四级分析演进论, 我们得出MLE对麦肯- 弗拉索夫方程式中的外部力量和互动内核的一致性。