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Quasi-Monte Carlo confidence intervals using quantiles of randomized nets

Zexin Pan

TL;DR

The paper develops quantile-based confidence intervals for high-dimensional QMC integration using randomized digital nets, achieving asymptotically valid coverage without requiring a growing number of replicates. By leveraging the Walsh decomposition and a careful error decomposition into a symmetric part and a vanishing perturbation, it shows that the median-based estimator concentrates and that quantile intervals capture the target integral with the nominal probability. The analysis covers the complete random design and extends to general randomizations, including random linear scrambling, with explicit conditions on the integrand and the randomization scheme. Empirically, the method yields narrower intervals and reliable coverage compared to conventional $t$-intervals, particularly in skewed, high-dimensional settings, offering a practical path to robust uncertainty quantification in QMC.

Abstract

Recent advances in quasi-Monte Carlo integration have demonstrated that the median trick significantly enhances the convergence rate of linearly scrambled digital net estimators. In this work, we leverage the quantiles of such estimators to construct confidence intervals with asymptotically valid coverage for high-dimensional integrals. By analyzing the distribution of the integration error for a class of infinitely differentiable integrands, we prove that as the sample size grows, the error decomposes into an asymptotically symmetric component and a vanishing perturbation, which guarantees that a quantile-based interval for the median estimator asymptotically captures the target integral with the nominal coverage probability.

Quasi-Monte Carlo confidence intervals using quantiles of randomized nets

TL;DR

The paper develops quantile-based confidence intervals for high-dimensional QMC integration using randomized digital nets, achieving asymptotically valid coverage without requiring a growing number of replicates. By leveraging the Walsh decomposition and a careful error decomposition into a symmetric part and a vanishing perturbation, it shows that the median-based estimator concentrates and that quantile intervals capture the target integral with the nominal probability. The analysis covers the complete random design and extends to general randomizations, including random linear scrambling, with explicit conditions on the integrand and the randomization scheme. Empirically, the method yields narrower intervals and reliable coverage compared to conventional -intervals, particularly in skewed, high-dimensional settings, offering a practical path to robust uncertainty quantification in QMC.

Abstract

Recent advances in quasi-Monte Carlo integration have demonstrated that the median trick significantly enhances the convergence rate of linearly scrambled digital net estimators. In this work, we leverage the quantiles of such estimators to construct confidence intervals with asymptotically valid coverage for high-dimensional integrals. By analyzing the distribution of the integration error for a class of infinitely differentiable integrands, we prove that as the sample size grows, the error decomposes into an asymptotically symmetric component and a vanishing perturbation, which guarantees that a quantile-based interval for the median estimator asymptotically captures the target integral with the nominal coverage probability.
Paper Structure (16 sections, 27 theorems, 237 equations, 7 figures)

This paper contains 16 sections, 27 theorems, 237 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background and notation
  3. Digital nets and randomization
  4. Fourier-Walsh decomposition
  5. Notations involving $\boldsymbol{k}$ and $\boldsymbol{\kappa}$
  6. Bounds on Walsh coefficients
  7. Proof of main results
  8. Proof of Step 1
  9. Proof of Step 2
  10. Proof of Step 3
  11. Main results
  12. Generalization to other randomization
  13. Numerical experiments
  14. One-dimensional example
  15. Eight-dimensional example
  16. ...and 1 more sections

Key Result

Lemma 1

For $f\in C([0,1]^s)$, the error of $\hat{\mu}_\infty$ defined by equation eqn:muEdef satisfies where

Figures (7)

  • Figure 1: Deviation of $\Pr(\hat{\mu}_E > \mu)$ from $1/2$ for $f(x)=x^{33}\exp(x)$.
  • Figure 2: $90$th percentile interval lengths of quantile intervals and $t$-intervals for $f(x)=x^{33}\exp(x)$.
  • Figure 3: Coverage levels of quantile intervals and $t$-intervals for $f(x)=x^{33}\exp(x)$.
  • Figure 4: Deviation of $\Pr(\hat{\mu}_E > \mu)$ from $1/2$ for $f(\boldsymbol{x})=\prod_{j=1}^8 x_j\exp(x_j)$.
  • Figure 5: $90$th percentile interval lengths of quantile intervals and $t$-intervals for $f(\boldsymbol{x})=\prod_{j=1}^8 x_j\exp(x_j)$.
  • ...and 2 more figures

Theorems & Definitions (61)

  • Lemma 1
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • Theorem 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 51 more