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 density estimators, the mean integrated square error converges more slowly than at the canonical rate of $\mathcal{O}(1/n)$. 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 permits one to obtain a smooth estimator of the cumulative distribution function, whose sample derivative is, under certain conditions, an unbiased estimator of the density at any point, and therefore converges at a faster rate than the usual density estimators. We can achieve an even faster rate by combining this with randomized quasi-Monte Carlo to generate the samples.