The Robust Randomized Quasi Monte Carlo method, applications to integrating singular functions
Résumé
We are given a simulation budget of $B$ points to calculate an expectation $\mu=\mathbb{E}\left(F(U)\right)$. {The standard} Monte Carlo method achieves a root mean squared risk of order $1/\sqrt B$, while a Randomized Quasi Monte Carlo (RQMC) method achieves an accuracy $\sigma_B \ll 1/\sqrt B$, as long as $F$ is only assumed square integrable.
The question we address in this work is, given a budget $B$ and a confidence level $1-\delta$, what is the optimal size of error tolerance such that $\mathbb{P}\left(|{\tt Est}-\mu|> {\tt TOL}\right)\leq \delta$ for an estimator ${\tt Est}$ to be determined? We show that a judicious choice of ``robust'' aggregation methods coupled with RQMC methods allows reaching the best {\tt TOL}, with provable optimality as $B\to+\infty$. This study is supported by numerical experiments, ranging from bounded $F(U)$ to heavy-tailed $F(U)$, the latter being well suited to singular $F$.
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