Monte Carlo and Quasi-Monte Carlo Density Estimation via Conditioning

Estimating the unknown density from which a given independent sample originates is more difficult than estimating the mean in the sense that, for the best popular non-parametric density estimators, the mean integrated square error converges more slowly than at the canonical rate of O(1/n). When the sample is generated from a simulation model and we have control over how this is done, we can do better. We examine an approach in which conditional Monte Carlo yields, under certain conditions, a random conditional density that is an unbiased estimator of the true density at any point. By averaging independent replications, we obtain a density estimator that converges at a faster rate than the usual ones. Moreover, combining this new type of estimator with randomized quasi-Monte Carlo to generate the samples typically brings a larger improvement on the error and convergence rate than for the usual estimators because the new estimator is smoother as a function of the underlying uniformrandom numbers.