We study the problem of parameter estimation for large exchangeable interacting particle systems when a sample of discrete observations from a single particle is known. We propose a novel method based on martingale estimating functions constructed by employing the eigenvalues and eigenfunctions of the generator of the mean field limit, where the law of the process is replaced by the (unique) invariant measure of the mean field dynamics. We then prove that our estimator is asymptotically unbiased and asymptotically normal when the number of observations and the number of particles tend to infinity, and we provide a rate of convergence towards the exact value of the parameters. Finally, we present several numerical experiments which show the accuracy of our estimator and corroborate our theoretical findings, even in the case the mean field dynamics exhibit more than one steady states.

翻译：我们研究了当从单个粒子的离散观测样本已知时，大规模可交换相互作用粒子系统的参数估计问题。我们提出了一种基于鞅估计函数的新方法，该方法通过利用平均场极限生成元的特征值和特征函数构建，其中过程的分布被平均场动力学的（唯一）不变测度替代。接着，我们证明当观测数量与粒子数量趋于无穷大时，所提出的估计量渐近无偏且渐近正态，并给出了参数精确值的收敛速率。最后，我们通过多个数值实验展示了该估计量的精度，验证了理论结果，即使在平均场动力学呈现多个稳态的情况下也同样适用。