On Integration Methods Based on Scrambled Nets of Arbitrary Size

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