Mean-Field Nonparametric Estimation of Interacting Particle Systems

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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.