Density estimation by randomized quasi-Monte Carlo

Abstract:

Randomized quasi-Monte Carlo (RQMC) is commonly used to estimate the mathematical expectation of a random variable expressed as an integral over the $s$-dimensional unit cube $(0, 1)^s$. Under certain conditions the RQMC estimator converges faster than the crude Monte Carlo (MC) estimator.

In this talk we examine how RQMC can improve the convergence rate of the mean integrated square error when estimating the density of a random variable $X$, defined as a function over $(0,1)^s$, in comparison to MC.

We provide both theoretical and empirical results on the convergence rates of density estimators defined by histograms kernel density estimators, when the observations are generated via RQMC.

We also discuss the combination of RQMC with a \emph{conditional Monte Carlo} approach to density estimation, in which each observation of $X$ is replaced by a conditional density given less information, and the estimator is the average of those conditional densities.

This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer