Asymptotic Variance of Newton-Cotes Quadratures based on Randomized Sampling Points

In this paper we consider the problem of numerical integration when sampling nodes are random, and we suggest to use Newton-Cotes quadrature rules to exploit smoothness properties of the integrand. In previous papers it was shown that a Riemann sum approach can cause a severe variance inflation when the sampling points are not equidistant. However, under some integrability conditions on the typical point-distance, we show that Newton-Cotes quadratures based on a stationary point process in R yield unbiased estimators for the integral and that the aforementioned variance inflation can be avoided if a Newton-Cotes quadrature of sufficiently high order is applied. In a stereological application, this corresponds to the estimation of volume of a compact object from area measurements on parallel sections.