Likelihood Ratio Density Estimation for Simulation Models

We consider the problem of estimating the density of a random variable X which is the output of a simulation model. We show how an unbiased density estimator can be constructed via the classical likelihood ratio derivative estimation method proposed over 35 years ago by Glynn, Rubinstein, and others. We then extend this density estimation method to cover situations where it does not apply directly. What we obtain is closely related to the generalized likelihood ratio method proposed recently by Peng and his co-authors, although the assumptions differ. We compare the methods and assumptions on some examples.