Open Access
August 2017 Asymptotic normality of scrambled geometric net quadrature
Kinjal Basu, Rajarshi Mukherjee
Ann. Statist. 45(4): 1759-1788 (August 2017). DOI: 10.1214/16-AOS1508


In a very recent work, Basu and Owen [Found. Comput. Math. 17 (2017) 467–496] propose the use of scrambled geometric nets in numerical integration when the domain is a product of $s$ arbitrary spaces of dimension $d$ having a certain partitioning constraint. It was shown that for a class of smooth functions, the integral estimate has variance $O(n^{-1-2/d}(\log n)^{s-1})$ for scrambled geometric nets compared to $O(n^{-1})$ for ordinary Monte Carlo. The main idea of this paper is to expand on the work by Loh [Ann. Statist. 31 (2003) 1282–1324] to show that the scrambled geometric net estimate has an asymptotic normal distribution for certain smooth functions defined on products of suitable subsets of $\mathbb{R}^{d}$.


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Kinjal Basu. Rajarshi Mukherjee. "Asymptotic normality of scrambled geometric net quadrature." Ann. Statist. 45 (4) 1759 - 1788, August 2017.


Received: 1 February 2016; Revised: 1 July 2016; Published: August 2017
First available in Project Euclid: 28 June 2017

zbMATH: 1383.62042
MathSciNet: MR3670195
Digital Object Identifier: 10.1214/16-AOS1508

Primary: 62E20
Secondary: 62D05 , 65D30

Keywords: asymptotic normality , numerical integration , quasi-Monte Carlo , scrambled geometric net , Stein’s method

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 4 • August 2017
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