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 cumulative density function (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.
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Acknowledgements
This work has been supported by a NSERC Discovery Grant and an IVADO Grant to P. L’Ecuyer. F. Puchhammer was also supported by Spanish and Basque governments fundings through BCAM (ERDF, ESF, SEV-2017-0718, PID2019-108111RB-I00, PID2019-104927GB-C22, BERC 2018e2021, EXP. 2019/00432, ELKARTEK KK-2020/00049), and the computing infrastructure of i2BASQUE academic network and IZO-SGI SGIker (UPV).
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L’Ecuyer, P., Puchhammer, F. (2022). Density Estimation by Monte Carlo and Quasi-Monte Carlo. In: Keller, A. (eds) Monte Carlo and Quasi-Monte Carlo Methods. MCQMC 2020. Springer Proceedings in Mathematics & Statistics, vol 387. Springer, Cham. https://doi.org/10.1007/978-3-030-98319-2_1
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