BCAM Scientific Seminar: Density estimation by Monte Carlo and randomized quasi-Monte Carlo

Abstract:
Estimating the density of a random variable X has been studied exten- sively and is commonly used in settings where a set of n observations of X (a data set) is given a priori and one wishes to estimate the density from that. Popular density estimators include histograms and kernel density estimators (KDE).
In this talk, we are interested instead in a situation where the observations are generated by Monte Carlo simulation from a model. We generate a sample and estimate the density from the sample. Then the question arise of whether variance reduction methods such as stratification, randomized quasi-Monte Carlo (RQMC), conditional Monte Carlo, etc., can be exploited to produce a more accurate density estimator for a given sample size.
We provide both theoretical and empirical results on the convergence rates for histograms and for kernel density estimators, when the observations are generated via RQMC. We also examine the combination of RQMC with a conditional Monte Carlo approach to density estimation, defined by taking the stochastic derivative of a conditional cdf of X. This approach can provide a huge improvement when it applies.
This is based on joint work with Amal Ben Abdellah and Florian Puchhammer, Universit ́e de Montr ́eal and GERAD, and Art B. Owen, Stanford University.

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