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Statistics and Computing

, Volume 29, Issue 1, pp 53–65 | Cite as

Asymptotic normality of extensible grid sampling

  • Zhijian HeEmail author
  • Lingjiong Zhu
Article
First Online:

Abstract

Recently, He and Owen (J R Stat Soc Ser B 78(4):917–931, 2016) proposed the use of Hilbert’s space filling curve (HSFC) in numerical integration as a way of reducing the dimension from \(d>1\) to \(d=1\). This paper studies the asymptotic normality of the HSFC-based estimate when using one-dimensional stratification inputs. In particular, we are interested in using scrambled van der Corput sequence in any base \(b\ge 2\) with sample sizes of the form \(n=b^m\), for which the sampling scheme is extensible in the sense of multiplying the sample size by a factor of b. We show that the estimate has an asymptotic normal distribution for functions in \(C^1([0,1]^d)\), excluding the trivial case of constant functions. The asymptotic normality also holds for discontinuous functions under mild conditions. Previously, it was only known that scrambled (0, m, d)-net quadratures enjoy the asymptotic normality for smooth enough functions, whose mixed partial gradients satisfy a Hölder condition. As a by-product, we find lower bounds for the variance of the HSFC-based estimate. Particularly, for non-trivial functions in \(C^1([0,1]^d)\), the lower bound is of order \(n^{-1-2/d}\), which matches the rate of the upper bound established in He and Owen (2016).

Keywords

Asymptotic normality Hilbert’s space filling curve Van der Corput sequence Randomized quasi-Monte Carlo Extensible grid sampling 
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Notes

Acknowledgements

The authors are enormously grateful to the Editor, the Associate Editor, and three anonymous referees whose suggestions and comments have greatly improved the quality of the paper. The authors also thank Professor Art B. Owen for the helpful comments.

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© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of MathematicsSouth China University of TechnologyGuangzhouChina
  2. 2.Department of MathematicsFlorida State UniversityTallahasseeUSA

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