Quasi-Random Sampling for Multivariate Distributions via Generative Neural Networks

Abstract

Generative moment matching networks (GMMNs) are introduced for generating approximate quasi-random samples from multivariate models with any underlying copula to compute estimates with variance reduction. So far, quasi-random sampling for multivariate distributions required a careful design, exploiting specific properties (such as conditional distributions) of the implied parametric copula or the underlying quasi-Monte Carlo (QMC) point set, and was only tractable for a small number of models. Using GMMNs allows one to construct approximate quasi-random samples for a much larger variety of multivariate distributions without such restrictions, including empirical ones from real data with dependence structures not well captured by parametric copulas. Once trained on pseudo-random samples from a parametric model or on real data, these neural networks only require a multivariate standard uniform randomized QMC point set as input and are thus fast in estimating expectations of interest under dependence with variance reduction. Numerical examples are considered to demonstrate the approach, including applications inspired by risk management practice.

KEYWORDS:

• Copulas
• Expected shortfall
• Generative moment matching networks
• Maximum mean discrepancy
• Quasi-random numbers
• Sums of dependent random variables

Additional information

Funding

Marius Hofert acknowledges support from NSERC (grant RGPIN-5010-2015). Avinash Prasad acknowledges support from NSERC (PGS D Scholarship). Mu Zhu acknowledges support from NSERC (RGPIN-2016-03876).