On-line Parameter Estimation in General State-Space Models using a Pseudo-Likelihood Approach

State-space models are a very general class of time series capable of modeling dependent observations in a natural and interpretable way. While optimal state estimation can now be routinely performed using SMC (sequential Monte Carlo) methods, on-line static parameter estimation largely remains an unsolved problem. In Andrieu and Doucet [2003] it was proposed to use a pseudo-likelihood approach. This pseudo-likelihood can be optimised directly using a stochastic gradient algorithm, but we focus on an on-line Expectation-Maximization (EM). We present here novel simple recursions that allow us to estimate confidence intervals on-line and develope new theoretical results concerning the pseudo-likelihood estimate. More precisely we characterise the loss of efficiency compared to that of the maximum likelihood estimate, and also quantify the bias of the estimate in cases where the pseudo-likelihood needs to be approximated. We show in a tractable situation requiring no Monte Carlo simulation that these theoretical results accurately predict performance, pointing to their practical relevance.