Asymptotic properties of the bootstrap particle filter maximum likelihood estimator for state space models

Jimmy Olsson and Tobias Rydén

Abstract:
We study the asymptotic performance of an approximatemaximum likelihood estimator for state space models obtained via the bootstrap particle filter. The state space of the latent Markov chain and the parameter space are assumed to be compact. The approximate estimate is computed by, firstly, running possibly dependent particle filters on a fixed grid in the parameter space, yielding a pointwise approximation of the likelihood function. Secondly, the estimate is obtained by maximizing this approximation over the grid. In this setting we formulate criteria for how to increase the number of particles, and how to vary the grid size in order to produce an estimate that is consistent and asymptotically normal.

Key words:: Asymptotic normality, bootstrap particle filter, consistency, hidden Markov model, maximum likelihood, sequential Monte Carlo methods, state space models.