On Bounds and Asymptotics of Sequential Monte Carlo Methods for Filtering, Smoothing, and Maximum Likelihood Estimation in State Space Models

Jimmy Olsson

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology
2007

ISBN 91-628-7048-3
LUTFMS--1029--2006

Abstract:

This thesis is based on four papers (A-D) treating filtering, smoothing, and maximum likelihood (ML) estimation in general state space models using stochastic particle filters (also referred to as sequential Monte Carlo (SMC) methods).

The aim of Paper A is to study the bias of Monte Carlo integration estimates produced by the so-called bootstrap particle filter. A bound on this bias which is inversely proportional to the number N of particles of the system is established. In addition, we refine the analysis by deriving the asymptotic bias as N tends to infinity and, under suitable mixing assumptions on the latent Markov model, a time uniform bound.

In Paper B we consider ML estimation based on EM (Expectation-Maximization) methods. In this context, the key ingredient is the computation of smoothed sum functionals of the hidden states for given values of the model parameters. It has been observed by several authors that using standard SMC methods for this smoothing assignment may be unreliable for larger observations sizes. Thus we study a simple variant, based on forgetting ideas of the state space model dynamics, of the basic sequential smoothing approach which is transparent in terms of computation time and reduces the variability of the sum functional approximation. Under suitable regularity assumptions, it is shown that this modification indeed allows a tighter control of the Lp error and the bias of the approximation.

To perform ML estimation in state space models, the log-likelihood function must be approximated. In Paper C we study such approximations based on particle filters, and in particular conditions for consistency and asymptotic normality of the corresponding approximate ML estimators. Numerical results illustrate the theory.

Paper D is devoted to the study asymptotic properties of weighted particle samples produced by the so-called two-stage sampling (TSS) particle filter, which is a generalization of the auxiliary particle filter proposed by Pitt and Shephard (1999). Besides establishing a central limit theorem (CLT) for smoothed particle estimates, we also derive bounds on the Lp error and bias of the same for a finite particle sample size. Setting out from the recursive formula for the asymptotic variance of the CLT, we discuss some possible improvements of the TSS algorithm.

Key words:
Asymptotic normality, consistency, EM algorithm, maximum likelihood, particle filter, sequential Monte Carlo, smoothing, state space models, two-stage sampling