On integration methods based on scrambled nets of arbitrary size

We consider the problem of evaluating I(φ):=∫[0,1)sφ(x)dx for a function φ∈L2[0,1)sφ∈L2[0,1)s. In situations where I(φ)I(φ) can be approximated by an estimate of the form N−1∑n=0N−1φ(xn), with {xn}n=0N−1 a point set in [0,1)s[0,1)s, it is now well known that the OP(N−1/2)OP(N−1/2) Monte Carlo convergence rate can be improved by taking for {xn}n=0N−1 the first N=λbmN=λbm points, λ∈{1,…,b−1}λ∈{1,…,b−1}, of a scrambled (t,s)(t,s)-sequence in base b≥2b≥2. In this paper we derive a bound for the variance of scrambled net quadrature rules which is of order O(N−1)O(N−1) without any restriction on NN. As a corollary, this bound allows us to provide simple conditions to get, for any pattern of NN, an integration error of size OP(N−1/2)OP(N−1/2) for functions that depend on the quadrature size NN. Notably, we establish that sequential quasi-Monte Carlo (Gerber & Chopin, 2015) reaches the OP(N−1/2)OP(N−1/2) convergence rate for any values of NN. In a numerical study, we show that for scrambled net quadrature rules we can relax the constraint on NN without any loss of efficiency when the integrand φφ is a discontinuous function while, for sequential quasi-Monte Carlo, taking N=λbmN=λbm may only provide moderate gains.