Locally lattice sampling designs for isotropic random fields

• 成果类型: Article
• 署名作者: Stein, ML
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1995
• 页码: 1991-2012
• 关键词: estimating integrals; STOCHASTIC-PROCESSES

摘要

For predicting integral(G)v(x)Z(x) dx, where v is a fixed known function and Z is a stationary random field, a good sampling design should have a greater density of observations where v is relatively large in absolute value. Designs using this idea when G = [0, 1] have been studied for some time. For G a region in two dimensions, very little is known about the statistical properties of cubature rules based on designs with varying density. This work proposes a class of designs that are locally parallelogram lattices but whose densities can vary. The asymptotic variance of the cubature error for these designs is obtained for a class of isotropic random fields and an asymptotically optimal sequence of cubature rules within this class is found. I conjecture that this sequence of cubature rules is asymptotically optimal with respect to all cubature rules.