Nonlinear non-Gaussian state-space models are ubiquitous in statistics, econometrics, information engineering and signal processing. Particle methods, also known as Sequential Monte Carlo (SMC) methods, provide reliable numerical approximations to the associated state inference problems. However, in most applications, the state-space model of interest also depends on unknown static parameters that need to be estimated from the data. In this context, standard particle methods fail and it is necessary to rely on more sophisticated algorithms. The aim of this paper is to present a comprehensive review of particle methods that have been proposed to perform static parameter estimation in state-space models. We discuss the advantages and limitations of these methods and illustrate their performance on simple models.Comment: Published at http://dx.doi.org/10.1214/14-STS511 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Particle methods are popular computational tools for Bayesian inference in nonlinear non-Gaussian state space models. For this class of models, we present two particle algorithms to compute the score vector and observed information matrix recursively. The first algorithm is implemented with computational complexity O(N ) and the second with complexity O(N 2 ), where N is the number of particles. Although cheaper, the performance of the O(N ) method degrades quickly, as it relies on the approximation of a sequence of probability distributions whose dimension increases linearly with time. In particular, even under strong mixing assumptions, the variance of the estimates computed with the O(N ) method increases at least quadratically in time. The more expensive O(N 2 ) method relies on a nonstandard particle implementation and does not suffer from this rapid degradation. It is shown how both methods can be used to perform batch and recursive parameter estimation.

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Nonlinear non-Gaussian state-space models arise in numerous applications in control and signal processing. Sequential Monte Carlo (SMC) methods, also known as Particle Filters, provide very good numerical approximations to the associated optimal state estimation problems. However, in many scenarios, the state-space model of interest also depends on unknown static parameters that need to be estimated from the data. In this context, standard SMC methods fail and it is necessary to rely on more sophisticated algorithms. The aim of this paper is to present a comprehensive overview of SMC methods that have been proposed to perform static parameter estimation in general state-space models. We discuss the advantages and limitations of these methods.