Asymptotic Properties of Recursive Particle Maximum Likelihood Estimation

Using stochastic gradient search and the optimal filter derivative, it is possible to perform recursive maximum likelihood estimation in a non-linear state-space model. As its derivative are analytically intractable for such model, they need be approximated numerically. In Poyiadjis et al. (G. Poyiadjis, A. Doucet, S. Singh, Biometrika, vol. 98, no. 1, pp. 65–80, 2011), algorithm based on particle approximation has been proposed studied through numerical simulations. This asymptotic behavior here analyzed theoretically. Under regularity conditions, we show that accurately estimates maxima of underlying log-likelihood rate when number particles sufficiently large. We also provide qualitative upper bounds error terms particles.