## The Annals of Statistics

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

### Asymptotic normality of scrambled geometric net quadrature

Kinjal Basu and Rajarshi Mukherjee

#### Abstract

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

**Source**

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

**Dates**

Received: February 2016

Revised: July 2016

First available in Project Euclid: 28 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1498636873

**Digital Object Identifier**

doi:10.1214/16-AOS1508

**Mathematical Reviews number (MathSciNet)**

MR3670195

**Zentralblatt MATH identifier**

1383.62042

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 62D05: Sampling theory, sample surveys 65D30: Numerical integration

**Keywords**

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

#### Citation

Basu, Kinjal; Mukherjee, Rajarshi. Asymptotic normality of scrambled geometric net quadrature. Ann. Statist. 45 (2017), no. 4, 1759--1788. doi:10.1214/16-AOS1508. https://projecteuclid.org/euclid.aos/1498636873

#### Supplemental materials

- Supplement to “Asymptotic normality of scrambled geometric net quadrature”. The supplementary material contain the proofs of supporting lemmas.Digital Object Identifier: doi:10.1214/16-AOS1508SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.