On Integration Methods Based on Scrambled Nets of Arbitrary Size

We consider the problem of evaluating $I(\varphi)=\int_{\ui^s}\varphi(\bx) \dx$ for a function $\varphi \in L_{2}[0,1)^{s}$. In situations where $I(\varphi)$ can be approximated by an estimate of the form $N^{-1}\sum_{n=0}^{N-1}\varphi(\bx^n)$, with $\{\bx^n\}_{n=0}^{N-1}$ a point set in $\ui^s$, \citet{Owen1995, Owen1997a,Owen1997b,Owen1998} shows that the $\bigO_P(N^{-1/2})$ Monte Carlo convergence rate can be improved by taking for $\{\bx^n\}_{n=0}^{N-1}$ the first $N=\lambda b^m$ points, $1\leq\lambda<b$, of a scrambled $(t,s)$-sequence in base $b\geq 2$. For more complex integration problems, \citet{Gerber2014} have recently developed a sequential quasi-Monte Carlo (SQMC) algorithm which has an error of size $\smallo_P(N^{-1/2})$ for bounded and continuous functions $\varphi$, when it uses the first $N=\lambda b^m$ points of scrambled $(t,s)$-sequences as inputs. In this paper we extend these results by relaxing the constraint on $N$. Our main contribution is to provide a bound for the variance of scrambled net quadratures which is of order $\smallo(N^{-1})$, without any restriction on $N$. This bound allows us to provide simple conditions to get an integration error of size $\smallo_P(N^{-1/2})$ for functions that depend on the quadrature size $N$ and, as a corollary, to establish that SQMC reaches the $\smallo_P(N^{-1/2})$ convergence rate for any patterns of $N$. Finally, we show in a numerical study that for scrambled net quadrature rules we can relax the constraint on $N$ without any loss of efficiency when the integrand $\varphi$ is a discontinuous function while, for the univariate version of SQMC, taking $N=\lambda b^m$ may only provide moderate gains.