  Theoretical results for importance sampling rely on the existence of certain
moments of the importance weights, which are the ratios between the proposal
and target densities. In particular, a finite variance ensures square root
convergence and asymptotic normality of the importance sampling estimate, and
can be important for the reliability of the method in practice. We derive
conditions for the existence of any required moments of the weights for
Gaussian proposals and show that these conditions are almost necessary and
sufficient for a wide range of models with latent Gaussian components.
Important examples are time series and panel data models with measurement
densities which belong to the exponential family. We introduce practical and
simple methods for checking and imposing the conditions for the existence of
the desired moments. We develop a two component mixture proposal that allows us
to flexibly adapt a given proposal density into a robust importance density.
These methods are illustrated on a wide range of models including generalized
linear mixed models, non-Gaussian nonlinear state space models and panel data
models with autoregressive random effects.
