Extensible grids: uniform sampling on a space filling curve

• 成果类型: Article
• 署名作者: He, Zhijian; Owen, Art B.
• 署名单位: Tsinghua University; Stanford University
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/rssb.12132
• 发表日期: 2016
• 页码: 917-931
• 关键词: quasi-monte-carlo

摘要

We study the properties of points in [0,1]d generated by applying Hilbert's space filling curve to uniformly distributed points in [0,1]. For deterministic sampling we obtain a discrepancy of O(n-1/d) for d2. For random stratified sampling, and scrambled van der Corput points, we derive a mean-squared error of O(n-1-2/d) for integration of Lipschitz continuous integrands, when d3. These rates are the same as those obtained by sampling on d-dimensional grids and they show a deterioration with increasing d. The rate for Lipschitz functions is, however, the best possible at that level of smoothness and is better than plain independent and identically distributed sampling. Unlike grids, space filling curve sampling provides points at any desired sample size, and the van der Corput version is extensible in n. We also introduce a class of piecewise Lipschitz functions whose discontinuities are in rectifiable sets described via Minkowski content. Although these functions may have infinite variation in the sense of Hardy and Krause, they can be integrated with a mean-squared error of O(n-1-1/d). It was previously known only that the rate was o(n-1). Other space filling curves, such as those due to Sierpinski and Peano, also attain these rates, whereas upper bounds for the Lebesgue curve are somewhat worse, as if the dimension were log2(3) times as high.