[Paper Review] Convergence of Multivariate Quantile Surfaces

Adil Ahidar-Coutrix, Philippe Berthet|arXiv (Cornell University)|Jul 9, 2016

Statistical Methods and Inference14 references3 citations

TL;DR

This paper introduces a nonparametric framework for multivariate quantile surfaces anchored at any observer point O ∈ ℝᵈ, generalizing univariate quantiles to arbitrary distributions without requiring symmetry, density, or a unique center. It establishes a dimension-free Bahadur-Kiefer representation with rate n⁻¹/⁴(log n)¹/²(log log n)¹⁴, enabling uniform weak convergence and non-asymptotic Gaussian approximation for joint confidence regions of Tukey half-spaces.

ABSTRACT

We define the quantile set of order $α\in \left[ 1/2,1 ight) $ associated to a law $P$ on $\mathbb{R}^{d}$ to be the collection of its directional quantiles seen from an observer $O\in \mathbb{R}^{d}$. Under minimal assumptions these star-shaped sets are closed surfaces, continuous in $(O,α)$ and the collection of empirical quantile surfaces is uniformly consistent.\ Under mild assumptions -- no density or symmetry is required for $P$ -- our uniform central limit theorem reveals the correlations between quantile points and a non asymptotic Gaussian approximation provides joint confident enlarged quantile surfaces. Our main result is a dimension free rate $n^{-1/4} (\log n)^{1/2}(\log\log n) ^{1/4} $ of Bahadur-Kiefer embedding by the empirical process indexed by half-spaces. These limit theorems sharply generalize the univariate quantile convergences and fully characterize the joint behavior of Tukey half-spaces.

Motivation & Objective

• To generalize univariate quantile convergence to multivariate settings without requiring symmetry, density, or a unique center.
• To develop a nonparametric framework for quantile surfaces using directional projections from an arbitrary observer point O ∈ ℝᵈ.
• To establish uniform consistency and weak convergence of empirical quantile surfaces under minimal assumptions on the underlying distribution P.
• To derive a dimension-free Bahadur-Kiefer representation for the empirical process indexed by half-spaces, enabling joint inference on multivariate quantiles.
• To provide non-asymptotic Gaussian approximation for joint confidence regions of quantile surfaces, applicable to robust multivariate inference.

Proposed method

• Define directional quantiles from an observer O ∈ ℝᵈ as the set of points where the probability of a half-space through O reaches order α ∈ [1/2, 1).
• Construct quantile surfaces as star-shaped sets formed by directional quantiles, ensuring continuity in (O, α) under minimal assumptions.
• Use Talagrand’s inequality and moment inequalities for empirical processes indexed by half-spaces to control uniform deviations.
• Apply the Berthet-Mason strong approximation theorem to embed the empirical process into a Gaussian process with a dimension-free rate.
• Establish a uniform Bahadur-Kiefer representation for the empirical process over the class of half-spaces, achieving rate n⁻¹/⁴(log n)¹/²(log log n)¹⁴.
• Derive non-asymptotic Gaussian approximation for joint confidence regions of quantile surfaces using the strong approximation result.

Experimental results

Research questions

• RQ1 Can multivariate quantile surfaces be consistently estimated without assuming symmetry, density, or a unique center?
• RQ2 What is the optimal rate of convergence for empirical quantile surfaces in high-dimensional settings?
• RQ3 How can joint confidence regions for multivariate quantiles be constructed with non-asymptotic validity?
• RQ4 Can the empirical process indexed by half-spaces be represented via a Gaussian process with a dimension-free rate?
• RQ5 How do quantile surfaces generalize univariate quantile convergence in the multivariate setting?

Key findings

• Empirical quantile surfaces are uniformly consistent under minimal assumptions, without requiring density or symmetry of the underlying distribution P.
• The paper establishes a uniform weak convergence result for empirical quantile surfaces, extending classical univariate weak convergence to multivariate settings.
• A dimension-free Bahadur-Kiefer representation is achieved with rate n⁻¹/⁴(log n)¹/²(log log n)¹⁴ for the empirical process indexed by half-spaces.
• Non-asymptotic Gaussian approximation is provided for joint confidence regions of quantile surfaces, enabling robust inference on multivariate data clouds.
• The quantile surfaces are continuous in both observer position O and quantile level α, ensuring stable geometric interpretation across the parameter space.
• The framework generalizes Tukey half-spaces and enables joint inference on depth-based contours without requiring a central median point.