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 1404028X

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
nonlinear/nonGaussian 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, nonlinear filtering, sequential Monte Carlo methods,
state space models

