Variance of the Isotropic Uniform Systematic Sampling

Abstract

The integral of a smooth function with bounded support over a set with finite perimeter in Euclidean space ℝ^d is estimated using a periodic grid in an isotropic uniform random position. Extension term in the estimator variance is proportional to the integral of the squared modulus of the function over the object boundary and to the grid scaling factor raised to the power of d+1. Our result generalizes the Kendall-Hlawka-Matheron formula for the variance of the isotropic uniform systematic estimator of volume.