Density Estimation by Monte Carlo and Quasi-Monte Carlo

TL;DR

This paper reviews advanced Monte Carlo and Quasi-Monte Carlo techniques for density estimation from simulation data, highlighting variance reduction methods that improve accuracy over traditional approaches.

Contribution

It introduces and compares methods combining RQMC with kernel density and derivative estimators, demonstrating their theoretical and practical advantages.

Findings

RQMC improves density estimation accuracy

Variance reduction techniques enhance convergence rates

Numerical results show reduced mean square error

Abstract

Estimating the density of a continuous random variable X has been studied extensively in statistics, in the setting where n independent observations of X are given a priori and one wishes to estimate the density from that. Popular methods include histograms and kernel density estimators. In this review paper, we are interested instead in the situation where the observations are generated by Monte Carlo simulation from a model. Then, one can take advantage of variance reduction methods such as stratification, conditional Monte Carlo, and randomized quasi-Monte Carlo (RQMC), and obtain a more accurate density estimator than with standard Monte Carlo for a given computing budget. We discuss several ways of doing this, proposed in recent papers, with a focus on methods that exploit RQMC. A first idea is to directly combine RQMC with a standard kernel density estimator. Another one is to adapt a simulation-based derivative estimation method such as smoothed perturbation analysis or the likelihood ratio method to obtain a continuous estimator of the cdf, whose derivative is an unbiased estimator of the density. This can then be combined with RQMC. We summarize recent theoretical results with these approaches and give numerical illustrations of how they improve the convergence of the mean square integrated error.