Quasi-Monte Carlo for unbounded integrands with importance sampling

TL;DR

Superisingly, it is found that using importance sampling with t distribution as the proposal can improve the root mean squared error of RQMC from $O(n^{-1+2M+\epsilon})$ to $O ( n^{-3/2+\EPsilon)$ for any $M\in(0,1/2)$.

Abstract

We consider the problem of estimating an expectation $ \mathbb{E}\left[ h(W)\right]$ by quasi-Monte Carlo (QMC) methods, where $ h $ is an unbounded smooth function on $ \mathbb{R}^d $ and $ W$ is a standard normal distributed random variable. To study rates of convergence for QMC on unbounded integrands, we use a smoothed projection operator to project the output of $W$ to a bounded region, which differs from the strategy of avoiding the singularities along the boundary of the unit cube $ [0,1]^d $ in 10.1137/S0036144504441573. The error is then bounded by the quadrature error of the transformed integrand and the projection error. If the function $h(\boldsymbol{x})$ and its mixed partial derivatives do not grow too fast as the Euclidean norm $|\boldsymbol{x}|$ goes to infinity, we obtain an error rate of $O(n^{-1+ε})$ for QMC and randomized QMC (RQMC) with a sample size $n$ and an arbitrarily small $ε>0$. However, the rate turns out to be $O(n^{-1+2M+ε})$ if the functions grow exponentially with a rate of $O(\exp\{M|\boldsymbol{x}|^2\})$ for a constant $M\in(0,1/2)$. Superisingly, we find that using importance sampling with t distribution as the proposal can improve the root mean squared error of RQMC from $O(n^{-1+2M+ε})$ to $O( n^{-3/2+ε})$ for any $M\in(0,1/2)$.

Figures (3)

• Figure 1: One-dimensional smoothed projection operator with $\varepsilon = 1$
• Figure 1: RMSEs for the test function $h$ with $d = 5$ and $\nu = 3$. The RMSEs are computed based on $100$ repetitions. The slopes of the gray dashed lines are $-1$ and $-3/2$.
• Figure 2: RMSEs for the test function $h$ with $d = 30$ and $\nu = 3$. The RMSEs are computed based on $100$ repetitions. The slopes of the gray dashed lines are $-1$ and $-3/2$.

Theorems & Definitions (53)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Proposition 2.7
• Definition 3.1
• Definition 3.2
• Definition 3.3