Derivative-Free Estimation of the Score Vector and Observed Information Matrix with Application to State-Space Models
Résumé
Ionides, King et al. (see e.g. Inference for nonlinear dynamical systems, PNAS 103) have recently introduced an original approach to perform maximum likelihood parameter estimation in state-space models which only requires being able to simulate the latent Markov model according its prior distribution. Their methodology relies on an approximation of the score vector for general statistical models based upon an artificial posterior distribution and bypasses the calculation of any derivative. Building upon this insightful work, we provide here a simple "derivative-free" estimator of the observed information matrix based upon this very artificial posterior distribution. However for state-space models where sequential Monte Carlo computation is required, these estimators have too high a variance and need to be modified. In this specific context, we derive new derivative-free estimators of the score vector and observed information matrix which are computed using sequential Monte Carlo approximations of smoothed additive functionals associated with a modified version of the original state-space model.