  For the purpose of maximum likelihood estimation of static parameters, we
apply a kernel smoother to the particles in the standard SIR filter for
non-linear state space models with additive Gaussian observation noise. This
reduces the Monte Carlo error in the estimates of both the posterior density of
the states and the marginal density of the observation at each time point. We
correct for variance inflation in the smoother, which together with the use of
Gaussian kernels, results in a Gaussian (Kalman) update when the amount of
smoothing turns to infinity. We propose and study of a criterion for choosing
the optimal bandwidth $h$ in the kernel smoother. Finally, we illustrate our
approach using examples from econometrics. Our filter is shown to be highly
suited for dynamic models with high signal-to-noise ratio, for which the SIR
filter has problems.
