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Tighter confidence intervals for quantiles of heterogeneous data

John H. J. Einmahl, Yi He

TL;DR

This work tackles nonparametric quantile inference under heterogeneous, independent data by treating the data as a triangular array with within-group equality of distributions. It derives the asymptotic variance $V_n( au)=n^{-1}\sum_{i=1}^{n}F_iigl(ar{Q}_n( au)igr)igl[1-F_iigl(ar{Q}_n( au)igr)igr]$ and constructs a ratio-consistent estimator $\,igl\hat{V}( au)\bigr$ based on within-group cross-counts, enabling a CLT for $n^{1/2}(\,\,\,)$ without requiring the quantile density. This yields asymptotically valid confidence intervals for $ar{Q}_n( au)$ of the form $(\hat{Q}( au-\hat{c}()),\hat{Q}( au+\hat{c}()))$ with $\hat{c}()=\hat{V}( au) n^{-1/2}\Phi^{-1}(1-\alpha/2)$ via the Galois connection. Simulations across multiple heterogeneous mechanisms show substantially tighter intervals than i.i.d.-based ones while attaining near-nominal coverage, highlighting the practical gains in quantile inference under heterogeneity. The approach is fully rank-based, invariant to monotone transformations, and does not rely on the existence of the quantile density.

Abstract

It is well known that the asymptotic variance of sample quantiles can be reduced under heterogeneity relative to the i.i.d. setting. However, asymptotically correct confidence intervals for quantiles are not yet available. We propose a novel, consistent estimator of the reduced asymptotic variance arising when quantiles are computed from groups of observations, leading to asymptotically correct confidence intervals. Simulation studies show that our confidence intervals are substantially shorter than those in the i.i.d. case and attain nearly correct coverage across a wide range of heterogeneous settings.

Tighter confidence intervals for quantiles of heterogeneous data

TL;DR

This work tackles nonparametric quantile inference under heterogeneous, independent data by treating the data as a triangular array with within-group equality of distributions. It derives the asymptotic variance and constructs a ratio-consistent estimator based on within-group cross-counts, enabling a CLT for without requiring the quantile density. This yields asymptotically valid confidence intervals for of the form with via the Galois connection. Simulations across multiple heterogeneous mechanisms show substantially tighter intervals than i.i.d.-based ones while attaining near-nominal coverage, highlighting the practical gains in quantile inference under heterogeneity. The approach is fully rank-based, invariant to monotone transformations, and does not rely on the existence of the quantile density.

Abstract

It is well known that the asymptotic variance of sample quantiles can be reduced under heterogeneity relative to the i.i.d. setting. However, asymptotically correct confidence intervals for quantiles are not yet available. We propose a novel, consistent estimator of the reduced asymptotic variance arising when quantiles are computed from groups of observations, leading to asymptotically correct confidence intervals. Simulation studies show that our confidence intervals are substantially shorter than those in the i.i.d. case and attain nearly correct coverage across a wide range of heterogeneous settings.
Paper Structure (4 sections, 4 theorems, 41 equations, 1 figure, 1 table)

This paper contains 4 sections, 4 theorems, 41 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Simulations
  4. Proofs

Key Result

Proposition 1

Under Assumptions ass:pair and ass:lindeberg, for every $\tau\in(0,1)$, where $\xrightarrow{p}$ denotes convergence in probability.

Figures (1)

  • Figure 1: Comparison of quantile–level estimation errors for heterogeneous data (blue) and i.i.d. samples from the average distribution (red) under twin groups (top) and triangular groups (bottom).

Theorems & Definitions (8)

  • Proposition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof : Proof of Proposition \ref{['lem:consistency']}
  • proof : Proof of Theorem \ref{['thm:clt']}