On estimation in state space models using the bootstrap particle filter

Jimmy Olsson

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2005

ISSN 1404-028X

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Abstract:

This thesis is based on two papers concerning the subjects of estimation, filtering and prediction in state space models using stochastic particle filters. Particle filters offer a means to obtain approximations of filter and predictive conditional distributions in a state space model that includes non-linear/non-Gaussian components.

The aim of Paper A is to study the bias of Monte Carlo integration estimates obtained by the bootstrap particle filter. A bound on this bias, which is geometrically growing in time and inversely proportional to the number N of particles of the system, is derived. Under suitable mixing assumptions on the latent Markov model, a bound of the bias which is uniform with respect to the time parameter and inversely proportional to N is obtained. In the last part of the paper we investigate the behaviour of the bias as N goes to infinity; it will be seen that the bias, for a fixed time point, is indeed asymptotically inversely proportional to N.

In Paper B we study the asymptotic performance of an approximate maximum 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.

Keywords:

Asymptotic normality, bootstrap particle filter, consistency, hidden Markov model, maximum likelihood, non-linear filtering, sequential Monte Carlo methods, state space models