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We consider the problem of evaluating $I(\varphi):=\int_{[0,1)^s}\varphi(x)dx$ for a function $\varphi \in L^2[0,1)^{s}$. In situations where $I(\varphi)$can be approximated by an estimate of the form$N^{-1}\sum_{n=0}^{N-1}\varphi(x^n)$, with $\{x^n\}_{n=0}^{N-1}$ a point set in$[0,1)^s$, it is now well known that the $O_P(N^{-1/2})$ Monte Carloconvergence rate can be improved by taking for $\{x^n\}_{n=0}^{N-1}$ the first$N=\lambda b^m$ points, $\lambda\in\{1,\dots,b-1\}$, of a scrambled$(t,s)$-sequence in base $b\geq 2$. In this paper we derive a bound for thevariance of scrambled net quadrature rules which is of order $o(N^{-1})$without any restriction on $N$. As a corollary, this bound allows us to providesimple conditions to get, for any pattern of $N$, an integration error of size$o_P(N^{-1/2})$ for functions that depend on the quadrature size $N$. Notably,we establish that sequential quasi-Monte Carlo (M. Gerber and N. Chopin, 2015,\emph{J. R. Statist. Soc. B, to appear.}) reaches the $o_P(N^{-1/2})$convergence rate for any values of $N$. In a numerical study, we show that forscrambled net quadrature rules we can relax the constraint on $N$ without anyloss of efficiency when the integrand $\varphi$ is a discontinuous functionwhile, for sequential quasi-Monte Carlo, taking $N=\lambda b^m$ may onlyprovide moderate gains.

On Integration Methods Based on Scrambled Nets of Arbitrary Size