We consider the problem of evaluating I(Ο):=β«[0,1)sβΟ(x)dx for a function ΟβL2[0,1)s. In situations where I(Ο)
can be approximated by an estimate of the form
Nβ1βn=0Nβ1βΟ(xn), with {xn}n=0Nβ1β a point set in
[0,1)s, it is now well known that the OPβ(Nβ1/2) Monte Carlo
convergence rate can be improved by taking for {xn}n=0Nβ1β the first
N=Ξ»bm points, Ξ»β{1,β¦,bβ1}, of a scrambled
(t,s)-sequence in base bβ₯2. In this paper we derive a bound for the
variance of scrambled net quadrature rules which is of order o(Nβ1)
without any restriction on N. As a corollary, this bound allows us to provide
simple conditions to get, for any pattern of N, an integration error of size
oPβ(Nβ1/2) 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 oPβ(Nβ1/2)
convergence rate for any values of N. In a numerical study, we show that for
scrambled net quadrature rules we can relax the constraint on N without any
loss of efficiency when the integrand Ο is a discontinuous function
while, for sequential quasi-Monte Carlo, taking N=Ξ»bm may only
provide moderate gains.Comment: 27 pages, 2 figures (final version, to appear in The Journal of
Complexity