An importance sampling method based on a one-step look-ahead density from a Markov chain

We propose a new importance sampling method that constructs an importance sampling density which approximates the zero-variance sampling density nonparametrically as follows. In a first stage, it generates a sample (possibly approximately) from the zero-variance density using, for example, Markov chain Monte Carlo methodology. In a second stage, the method constructs a kernel density estimator of the zero-variance density based on the sample in the first stage. The most important aspect of the method is that, unlike other kernel estimation methods, the kernel of the estimator is defined as the one-step transition density of a Markov chain whose stationary distribution is the zero-variance one. We give examples where this one-step transition density is available analytically and provide numerical illustrations in which the method performs very well.