The use of approximating models in Monte Carlo maximum likelihood estimation

Abstract

To obtain the likelihood of a non-Gaussian state-space model, Durbin and Koopman (1997, Biometrika, 84, 669–684) first calculate the likelihood under an approximating linear Gaussian model and then use Monte Carlo methods to estimate the necessary adjustment factor. We show that Durbin and Koopman's method is closely related to a method proposed by Geyer (1994, J. Roy. Statist. Soc. B 56, 261–274) for simulating the likelihood of a random-effects model and to a method proposed by Schall (1991, Biometrika, 78, 719–727) for approximating the maximum likelihood estimate of a generalised linear mixed model. A hybrid method is proposed for approximating the entire likelihood function as opposed to Durbin and Koopman's pointwise approximation. We also suggest an alternative class of approximating models based on conjugate latent process and apply it to approximate the likelihood of a time series model for count data.