Randomized quasi-Monte Carlo methods with applications to quantitative risk management

Abstract

We use randomized quasi-Monte Carlo (RQMC) techniques to construct computational tools for working with normal mixture models, which include automatic integration routines for density and distribution function evaluation, as well as fitting algorithms. We also provide open source software with all our methods implemented. In many practical problems, combining RQMC with importance sampling (IS) gives further variance reduction. However, the optimal IS density is typically not known, nor can it be sampled from. We solve this problem in the setting of single index models by finding a near optimal location-scale transform of the original density that approximates the optimal IS density for the univariate index. Sampling from complicated multivariate models, such as generalized inverse Gaussian mixtures, often involves sampling from a multivariate normal by inversion and from another univariate distribution, say W, whose quantile function is not known nor easily approxi- mated. We explore how we can still use RQMC in this setting and propose several methods when sampling of W is only possible via a black box random variate generator. We also study different ways to feed acceptance rejection (AR) algorithms for W with quasi-random numbers. RQMC methods on triangles have recently been developed by K. Basu and A. Owen. We show that one of the proposed sequences has suboptimal projection properties and address this issue by proposing to use their sequence to construct a stratified sampling scheme. Furthermore, we provide an extensible lattice construction for triangles and perform a simulation study.