发表机构
Duisburg-Essen University（杜伊斯堡-埃森大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对多元McKean--Vlasov扩散过程，提出基于稀疏ReQU神经网络的筛最大似然估计量，证明其在平稳密度与漂移系数估计上的收敛速率，且该速率达到极小极大最优性。
AI 中文摘要
本研究致力于从同一时刻的独立观测值中，对多元McKean--Vlasov扩散过程中依赖密度的漂移系数以及平稳密度进行非参数估计。在对（已知）势函数的若干假设下，我们将该问题简化为一维问题，并基于受结构约束和Hölder约束的稀疏ReQU神经网络构造筛最大似然估计量。利用适配端点的分级近似方法，我们在真实与估计的平稳密度之间的Kullback-Leibler散度上取得了$(b_n\text{log}n/n)^{2(\beta+1)/(2\beta+3)}$的速率，其中$b_n$至多为对数因子。类似地，研究表明所构造的漂移系数估计量在$L^2$度量下以$(b_n\text{log}n/n)^{\beta/(2\beta+3)}$的速率收敛于真实漂移系数。匹配的Assouad下界证明了该速率界在对数因子范围内达到极小极大最优性。
英文摘要
The present research is devoted to the nonparametric estimation of a density-dependent drift coefficient in a multivariate McKean--Vlasov diffusion from independent observations at a common time, as well as the stationary density. Under certain assumptions on the (known) potential, we reduce the problem to the one-dimensional one and construct a sieve maximum-likelihood estimator based on sparse ReQU neural networks subject to structural and Hölder constraints. Using the endpoint-adapted graded approximation, we achieve the rate of $\left(b_n\log n/n\right)^{2(β+1)/(2β+3)}$ for the Kullback-Leibler divergence between the true and estimated stationary densities, with $b_n$ being at most a logarithmic factor. Similarly, it is shown that the constructed estimator for the drift coefficient converges to the true one at the rate of $\left(b_n\log n/n\right)^{β/(2β+3)}$ in the $L^2$-metric. A matching Assouad lower bound proves minimax optimality of this bound up to logarithmic factors.
Comments34 pages, 2 figures
