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Brownian sheet and uniformity tests on the hypercube

A. Cabaña, E. M. Cabaña

TL;DR

This work addresses multivariate uniformity testing on the hypercube $C=[0,1]^p$ by constructing a $p$-Brownian sheet as a sum of $2^p$ independent Gaussian ramps (the Brownian tents). The authors develop a rigorous $H$-tent/$H$-ramp decomposition of functions vanishing on the boundary, derive the associated Karhunen–Loève expansions, and formulate two consistent tests (m-as and s-as) based on the asymptotic behaviour of the empirical $H$-tents; they also provide finite-sample implementations via Monte Carlo. Empirical results show competitive powers against several uniformity tests, especially for copula-type alternatives, with partial tests offering a computationally efficient option in high dimensions. The framework yields a tractable, flexible approach to multivariate uniformity testing with practical applicability to sampling methods and copula modeling.

Abstract

A construction of $p$-parameter Brownian sheet on the hypercube $C=[0,1]^p$ as a sum of $2^p$ independent Gaussian processes is obtained. The terms are closely related to Brownian pillows, and the probability laws of their $L^2(C)$ squared norms are computed. This allows us to propose consistent tests of uniformity for samples of i.i.d. random vectors on $C$. A comparison of powers of the new tests with those of several uniformity tests found in the statistical literature completes the article. The proposed tests show a good performance in detecting copula alternatives. Keywords: Brownian sheet, multivariate uniformity tests.

Brownian sheet and uniformity tests on the hypercube

TL;DR

This work addresses multivariate uniformity testing on the hypercube by constructing a -Brownian sheet as a sum of independent Gaussian ramps (the Brownian tents). The authors develop a rigorous -tent/-ramp decomposition of functions vanishing on the boundary, derive the associated Karhunen–Loève expansions, and formulate two consistent tests (m-as and s-as) based on the asymptotic behaviour of the empirical -tents; they also provide finite-sample implementations via Monte Carlo. Empirical results show competitive powers against several uniformity tests, especially for copula-type alternatives, with partial tests offering a computationally efficient option in high dimensions. The framework yields a tractable, flexible approach to multivariate uniformity testing with practical applicability to sampling methods and copula modeling.

Abstract

A construction of -parameter Brownian sheet on the hypercube as a sum of independent Gaussian processes is obtained. The terms are closely related to Brownian pillows, and the probability laws of their squared norms are computed. This allows us to propose consistent tests of uniformity for samples of i.i.d. random vectors on . A comparison of powers of the new tests with those of several uniformity tests found in the statistical literature completes the article. The proposed tests show a good performance in detecting copula alternatives. Keywords: Brownian sheet, multivariate uniformity tests.

Paper Structure

This paper contains 18 sections, 5 theorems, 8 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Decomposition of $p$-Brownian sheet as a sum of $2^p$ independent Gaussian terms
  3. Ramp components of a function $g:C\rightarrow \hbox{\bf R}$ that vanishes on $\partial^-C$.
  4. Brownian tents, ramps, and a construction of Brownian sheet
  5. Probability laws of Brownian tents and their squared norms
  6. Uniformity tests based on the $H$-tents of the empirical process
  7. Two consistent tests based on the asymptotic laws of the $H$-tents
  8. Finite samples tests
  9. The minimum test (abbreviated m-test) for samples of size $n$
  10. The sum test (abbreviated s-test) for samples of size $n$
  11. Partial tests
  12. Computing the $H$-tent of $W_{{\cal U}_n}$
  13. A brief empirical description of powers
  14. Empirical comparison of the powers of several uniformity tests
  15. Performance of our partial tests against six-dimensional copulas
  16. ...and 3 more sections

Key Result

Theorem 1

$$

Figures (3)

  • Figure 1: Tents and ramps in $[0,1]^2$
  • Figure 2: Decomposition of Brownian Sheet ($p=2$).
  • Figure 3: A schematic summary of the proposed procedures to test uniformity of a multivariate sample $X$ with significance level $\alpha$

Theorems & Definitions (7)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Corollary 1
  • Remark 1
  • Theorem 3
  • Lemma 1