The Annals of Statistics

Asymptotic normality of scrambled geometric net quadrature

Kinjal Basu and Rajarshi Mukherjee

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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 $\mathbb{R}^{d}$.

Article information

Ann. Statist., Volume 45, Number 4 (2017), 1759-1788.

Received: February 2016
Revised: July 2016
First available in Project Euclid: 28 June 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory
Secondary: 62D05: Sampling theory, sample surveys 65D30: Numerical integration

Asymptotic normality numerical integration quasi-Monte Carlo scrambled geometric net Stein’s method


Basu, Kinjal; Mukherjee, Rajarshi. Asymptotic normality of scrambled geometric net quadrature. Ann. Statist. 45 (2017), no. 4, 1759--1788. doi:10.1214/16-AOS1508.

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Supplemental materials

  • Supplement to “Asymptotic normality of scrambled geometric net quadrature”. The supplementary material contain the proofs of supporting lemmas.