Technical Summary: Tighter Confidence Intervals for Quantiles of Heterogeneous Data

Problem Statement

Standard nonparametric quantile inference typically assumes independent and identically distributed (i.i.d.) data. However, in many modern applications, observations are independent but arise from distinct distributions (heterogeneous data). While classical results (e.g., Shorack, 1973; van Zuijlen, 1976) establish that the asymptotic variance of the empirical distribution function is maximized under i.i.d. sampling and can be significantly reduced under heterogeneity, this theoretical advantage has not been practically utilized for inference.

Currently, no asymptotically correct nonparametric method exists to quantify this variance reduction for general heterogeneous settings. Consequently, statisticians are forced to either:

1. Use i.i.d.-based confidence intervals, which are overly conservative (too wide) when heterogeneity is present.
2. Restrict analysis to unrealistic scenarios where heterogeneity is negligible, effectively reverting to the i.i.d. limit.

The specific inferential target in this work is the quantile $\bar{Q}_n(\tau)$ corresponding to the average distribution $\bar{F}_n = n^{-1}\sum_{i=1}^n F_{n,i}$, where $X_i \sim F_{n,i}$ are independent but not identically distributed.

Methodology

Framework and Assumptions

The authors consider a triangular array of independent univariate data $X_i \sim F_{n,i}$ for $1 \le i \le n$. The data is partitioned into $g$ groups, denoted by index sets $I_{n,j}$, such that observations within a group share a common distribution, though distributions may differ arbitrarily across groups.

• Assumption 1: For all $n \ge 2$, there exists a partition of indices such that $F_i = F_{i'}$ for all $i, i'$ in the same group, and each group contains at least two indices.
• Assumption 2: A Lindeberg-type condition holds: $n V_n(\tau) \to \infty$ as $n \to \infty$, where $V_n(\tau)$ is the asymptotic variance.

Variance Reduction and Estimation

The estimation variance of the empirical distribution function $\hat{F}$ at the target quantile is given by:
$$V_n(\tau) = n^{-1} \sum_{i=1}^n F_i(\bar{Q}_n(\tau)) [1 - F_i(\bar{Q}_n(\tau))]$$
Under i.i.d. conditions, $V_n(\tau) \equiv \tau(1-\tau)$. Under heterogeneity, $V_n(\tau) \le \tau(1-\tau)$, with strict inequality generally holding.

The core methodological contribution is the construction of a ratio-consistent estimator $\hat{V}(\tau)$ for this heterogeneous variance. The estimator exploits the group structure:
$$\hat{V}(\tau) = n^{-1} \sum_{j=1}^g \frac{1}{m_j - 1} \left( \sum_{\ell \in I_{n,j}} \mathbb{1}\{X_\ell \le \hat{Q}(\tau)\} \right) \left( \sum_{\ell \in I_{n,j}} \mathbb{1}\{X_\ell > \hat{Q}(\tau)\} \right)$$
where $m_j$ is the size of group $j$ and $\hat{Q}(\tau)$ is the empirical quantile. This estimator is rank-based and invariant under strictly increasing transformations.

Theoretical Results

1. Consistency: Under Assumptions 1 and 2, the estimator $\hat{V}(\tau)$ is ratio-consistent, meaning $\hat{V}(\tau)/V_n(\tau) \xrightarrow{p} 1$.
2. Asymptotic Normality: Combining the consistency of the variance estimator with the Central Limit Theorem for the empirical distribution function, the authors establish:
   $$\left( \frac{n}{\hat{V}(\tau)} \right)^{1/2} (\hat{F}(\bar{Q}_n(\tau)) - \tau) \xrightarrow{d} N(0, 1)$$
3. Confidence Intervals: By inverting the asymptotic result using Galois inequalities (relating distribution and quantile functions), the paper derives an asymptotically correct $100(1-\alpha)\%$ confidence interval for $\bar{Q}_n(\tau)$:
   $$(\hat{Q}(\tau - \hat{c}(\alpha)), \hat{Q}(\tau + \hat{c}(\alpha)))$$
   where $\hat{c}(\alpha) = \hat{V}(\tau)/n^{1/2} \Phi^{-1}(1-\alpha/2)$. Notably, this procedure does not require the existence of a quantile density function.

Simulation Results

The authors conducted a simulation study with $n=350$ and 10,000 Monte Carlo replications to evaluate the method's performance.

• Data Generating Mechanisms: Three types of heterogeneous data were tested (Normal, Exponential, Uniform) with varying degrees of heterogeneity controlled by a parameter $\gamma \in \{1, 2, 3, 4\}$.
• Grouping Designs: Two grouping strategies were compared: "Twin groups" ($m_j=2$, $g=175$) and "Triangular groups" ($m_j=j+1$, $g=25$).
• Findings:
  • Variance Reduction: The estimation error for heterogeneous data was substantially smaller than for i.i.d. samples drawn from the average distribution. The "Twin groups" design, exhibiting higher heterogeneity, showed the greatest reduction in variability.
  • Coverage: The proposed confidence intervals achieved coverage probabilities very close to the nominal 95% level across all scenarios (ranging roughly from 93.0% to 95.2%).
  • Interval Width: The resulting intervals were substantially shorter than those derived from standard i.i.d. assumptions, which were shown to be overly conservative.

Significance and Claims

The paper claims to provide the first general, asymptotically correct nonparametric method for constructing confidence intervals for quantiles in heterogeneous settings. By explicitly modeling the group structure and estimating the reduced variance, the method allows statisticians to:

1. Avoid the excessive conservatism of i.i.d.-based intervals when data is heterogeneous.
2. Obtain valid inference without requiring smoothness conditions (such as the existence of a quantile density) on the underlying distributions.
3. Achieve significantly tighter confidence intervals that maintain correct coverage rates, thereby improving the precision of quantile estimation in modern applications where the i.i.d. assumption is violated.