Monte Carlo and Quasi–Monte Carlo Density Estimation via Conditioning

Abstract

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 nonparametric density estimators, the mean integrated square error converges more slowly than at the canonical rate of [Formula: see text]. 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 uniform random numbers. Summary of Contribution: Stochastic simulation is commonly used to estimate the mathematical expectation of some output random variable X together with a confidence interval for this expectation. But the simulations usually provide information to do much more, such as estimating the entire distribution (or density) of X. Histograms are routinely provided by standard simulation software, but they are very primitive density estimators. Kernel density estimators perform better, but they are trickier to use, have bias, and their mean square error converges more slowly than the canonical rate of O(1/n) with n independent samples. In this paper, we explain how to construct unbiased density estimators that converge at the canonical rate and even much faster when combined with randomized quasi–Monte Carlo. The key idea is to use conditional Monte Carlo to hide appropriate information and obtain a computable (random) conditional density, which acts (under certain conditions) as an unbiased density estimator. Moreover, this sample density is typically smoother than the classic density estimators as a function of the underlying uniform random numbers, so it can get along much better with randomized quasi–Monte Carlo methods. This offers an opportunity to further improve the O(1/n) rate. We observe rates near O(1/n2) on some examples, and we give conditions under which this type of rate provably holds. The proposed approach is simple, easy to implement, and extremely effective, so it provides a significant addition to the stochastic simulation toolbox.