  We study the class of state-space models and perform maximum likelihood
estimation for the model parameters. We consider a stochastic approximation
expectation-maximization (SAEM) algorithm to maximize the likelihood function
with the novelty of using approximate Bayesian computation (ABC) within SAEM.
The task is to provide each iteration of SAEM with a filtered state of the
system, and this is achieved using an ABC sampler for the hidden state, based
on sequential Monte Carlo (SMC) methodology. It is shown that the resulting
SAEM-ABC algorithm can be calibrated to return accurate inference, and in some
situations it can outperform a version of SAEM incorporating the bootstrap
filter. Two simulation studies are presented, first a nonlinear Gaussian
state-space model then a state-space model having dynamics expressed by a
stochastic differential equation. Comparisons with iterated filtering for
maximum likelihood inference, and Gibbs sampling and particle marginal methods
for Bayesian inference are presented.
