Asymptotic Properties of Recursive Maximum Likelihood Estimation in Non-Linear State-Space Models
This work provides theoretical guarantees for an existing algorithm, which is incremental but important for practitioners in fields like signal processing or finance using non-linear state-space models.
The paper theoretically analyzes a recursive maximum likelihood algorithm for non-linear state-space models, showing it accurately estimates maxima of the average log-likelihood with sufficient particles and provides tight error bounds under mild conditions.
Using stochastic gradient search and the optimal filter derivative, it is possible to perform recursive (i.e., online) maximum likelihood estimation in a non-linear state-space model. As the optimal filter and its derivative are analytically intractable for such a model, they need to be approximated numerically. In [Poyiadjis, Doucet and Singh, Biometrika 2018], a recursive maximum likelihood algorithm based on a particle approximation to the optimal filter derivative has been proposed and studied through numerical simulations. Here, this algorithm and its asymptotic behavior are analyzed theoretically. We show that the algorithm accurately estimates maxima to the underlying (average) log-likelihood when the number of particles is sufficiently large. We also derive (relatively) tight bounds on the estimation error. The obtained results hold under (relatively) mild conditions and cover several classes of non-linear state-space models met in practice.