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The median trick does not help for fully nested scrambling

Takashi Goda, Kosuke Suzuki

TL;DR

The paper analyzes median-based estimators in randomized quasi-M Monte Carlo with two scrambling schemes for digital nets. It confirms that both fully nested and linear scrambling yield the same variance for scrambled nets, but only linear scrambling allows the median trick to accelerate convergence for smooth integrands; fully nested scrambling does not. The main theoretical result shows that for fully nested scrambling, the median estimator satisfies $\sqrt{r}\,N^{3/2}(M_N^{(r)}-I(f)) \xrightarrow{d} \mathcal{N}\left(0,\frac{\pi\,\sigma^2(f)}{2}\right)$, implying no improvement beyond the $N^{-3/2}$ rate, even as $r\to\infty$. Numerical experiments corroborate the theory, demonstrating normal-like error behavior for fully nested scrambling and faster decay under linear scrambling, particularly for highly smooth integrands.

Abstract

In randomized quasi-Monte Carlo methods for numerical integration, average estimators based on digital nets with fully nested and linear scrambling are known to exhibit the same variance. In this note, we show that this equivalence does not extend to the median estimators. Specifically, while the median estimator with linear scrambling can achieve faster convergence for smooth integrands, the median estimator with fully nested scrambling does not exhibit this advantage.

The median trick does not help for fully nested scrambling

TL;DR

The paper analyzes median-based estimators in randomized quasi-M Monte Carlo with two scrambling schemes for digital nets. It confirms that both fully nested and linear scrambling yield the same variance for scrambled nets, but only linear scrambling allows the median trick to accelerate convergence for smooth integrands; fully nested scrambling does not. The main theoretical result shows that for fully nested scrambling, the median estimator satisfies , implying no improvement beyond the rate, even as . Numerical experiments corroborate the theory, demonstrating normal-like error behavior for fully nested scrambling and faster decay under linear scrambling, particularly for highly smooth integrands.

Abstract

In randomized quasi-Monte Carlo methods for numerical integration, average estimators based on digital nets with fully nested and linear scrambling are known to exhibit the same variance. In this note, we show that this equivalence does not extend to the median estimators. Specifically, while the median estimator with linear scrambling can achieve faster convergence for smooth integrands, the median estimator with fully nested scrambling does not exhibit this advantage.

Paper Structure

This paper contains 9 sections, 1 theorem, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Fully nested scrambling
  4. Linear scrambling
  5. Variance of scrambling
  6. Median estimator of the linear scrambling
  7. Median estimator of the fully nested scrambling
  8. Numerial experiments
  9. Proof of the main theorem

Key Result

Theorem 2.1

Let $f$ have Lipschitz continuous derivative of order $\beta$, and define Let $Q_N$ be the estimator of the fully nested scrambling given in eq:full-estimator and $M_N^{(r)}$ the median estimator using $r$ samples of $Q_N$'s, defined as in eq:median-estimator. Then we have the following convergence in law:

Figures (3)

  • Figure 1: Histograms of rescaled errors for different test functions and values of $N$ with $r=1$.
  • Figure 2: Histograms of rescaled errors for different test functions and values of $N$ with $r=15$.
  • Figure 3: Convergence behavior of the median estimators $M_N^{(r)}$ for both fully nested scrambling and linear scrambling.

Theorems & Definitions (2)

  • Theorem 2.1
  • Remark 2.2