High Dimensional Bootstrap and Asymptotic Expansion for the $k$-th Largest Coordinate

1. Problem Statement

The paper addresses a gap in high-dimensional statistical inference regarding the $k$-th largest coordinate of a normalized sum of independent random vectors.

• Context: While high-dimensional Gaussian approximation and bootstrap theory for the maximum ($k=1$) are well-established (e.g., Chernozhukov et al., Koike), the theory for general order statistics ($k > 1$) remains sparse.
• Challenge: Existing second-order asymptotic expansions for maxima rely on the event $\{T_n \le t\}$ being a rectangle (or hyperrectangle). However, for the $k$-th order statistic, the event $\{T_{n,[k]} \le t\}$ is equivalent to the exceedance count $N_n(t) \le k-1$. This event is not a rectangle; its local structure is governed by the count of coordinates exceeding a threshold rather than a single boundary.
• Goal: To develop a second-order accurate bootstrap theory (specifically for coverage probabilities and critical values) for the $k$-th largest coordinate in high dimensions, where the dimension $d$ can grow with the sample size $n$.

2. Methodology

The author proposes a novel approach that bridges the gap between order statistics and existing Edgeworth expansion machinery for maxima.

A. Combinatorial Representation (Inclusion-Exclusion)

The core innovation is the use of factorial moments and weighted inclusion-exclusion.

• The event $\{T_{n,[k]} \le t\}$ is represented via the exceedance count $N_n(t) = \sum_{j=1}^d \mathbb{1}\{S_{n,j} > t\}$.
• Using a finite inclusion-exclusion identity (Lemma A.1), the probability $P(N_n(t) \ge k)$ is expanded into a sum of probabilities of intersections of orthants (rare events where specific coordinates exceed $t$).
• This reduces the problem to analyzing a collection of rare-orthant probabilities of dimension $s$ (where $s$ ranges from $k$ to a truncation point $k_0 \approx A \log n$).

B. Local Edgeworth Expansions

The paper transfers the second-order expansion machinery developed by Koike (2026) for maxima to these low-dimensional projections:

• Stein Kernels: The method relies on the existence of Stein kernels for the random vectors to handle non-smoothness and high dimensionality without requiring classical Cramér conditions.
• Projected Densities: For every subset of coordinates $I \subset [d]$, the paper derives Edgeworth expansions for the projected sums $S_{n,I}$.
• Weighted Aggregation: The expansions for individual orthants are aggregated using the inclusion-exclusion weights to form a global expansion for the $k$-th order statistic.

C. Bootstrap Variants

The theory is applied to two bootstrap schemes:

1. Single Wild Bootstrap: Uses multipliers $w_i$. The paper derives the coverage error expansion, showing that matching the third moment ($\gamma = E[w^3] = 1$) eliminates the leading linear error term.
2. Prepivoted Double Wild Bootstrap: A two-step procedure where the first bootstrap generates a $p$-value, which is then used as the input for a second bootstrap. This achieves second-order accuracy even without third-moment matching.

D. Dependence Structures

The paper addresses two dependence regimes:

1. Maximal Correlation: The standard assumption where $\rho_d \log d \to 0$ (weak dependence).
2. Stationary Gaussian Exponential Mixing: A structural alternative where coordinates form a stationary sequence with exponentially decaying strong-mixing coefficients. This allows for local clusters of high correlation. The cost is an explicit remainder term $r_d$ representing the effect of local exceedance clustering.

3. Key Contributions

A. Theoretical Advancement

• Second-Order Expansion for Order Statistics: The paper proves that the $k$-th largest coordinate admits a Koike-type second-order bootstrap expansion. This extends the validity of Edgeworth and Cornish-Fisher expansions from maxima to general order statistics.
• Coverage Error Rates:
  • For the single wild bootstrap, if the third moment is matched ($\gamma=1$), the coverage error is $O(\varepsilon_n^2)$, where $\varepsilon_n \approx \sqrt{\frac{\log^3(dn)}{n}}$. This is a significant improvement over the standard first-order rate $O(\varepsilon_n)$.
  • The double wild bootstrap achieves the same $O(\varepsilon_n^2)$ rate without requiring third-moment matching.
• Explicit Remainder Terms: In the stationary mixing setting, the paper provides an explicit remainder $r_d$ that isolates the impact of local correlation clusters. It shows that if $d$ grows sufficiently fast relative to $n$ (specifically $d \gtrsim n \log n$), this remainder is also of order $O(n^{-1})$.

B. Technical Innovations

• Factorial Moment Reduction: Successfully reducing the complex geometry of order statistics to a sum of low-dimensional orthant probabilities.
• Handling Non-Rectangular Events: Providing a rigorous framework to handle the non-rectangular nature of $\{T_{n,[k]} \le t\}$ using combinatorial identities and local smoothing inequalities.
• Robustness to Dependence: Demonstrating that second-order accuracy holds even under stationary exponential mixing, provided the dimension is large enough to dilute the local clustering effects.

4. Key Results

• Theorem 2.1 (Single Wild Bootstrap): Establishes the coverage expansion:
  $$P(T_{n,[k]} \ge \hat{c}_{1-\alpha, k}) = \alpha - (1-\gamma)Q_{n,k}(c^G_{1-\alpha, k}) - E\{R_{n,k}(\alpha)\} + O(\varepsilon_n^2)$$
  where $Q_{n,k}$ is a deterministic linear term and $R_{n,k}$ is a quadratic Cornish-Fisher correction.
• Corollary 2.1: If $\gamma=1$ (third-moment matching), the linear distortion vanishes, yielding second-order accuracy $O(\varepsilon_n^2)$.
• Theorem 2.2 (Double Wild Bootstrap): The prepivoted double bootstrap achieves the same $O(\varepsilon_n^2)$ coverage error, removing the need for moment matching.
• Theorem 2.3 (Stationary Mixing): Replaces the maximal correlation condition with exponential mixing. The error bounds include an explicit remainder $r_d$, which is $O(n^{-1})$ when $d$ is sufficiently large relative to $n$.
• Simulation Results:
  • Empirical studies ($d=400, n \in \{200, 400\}$) compare Empirical Bootstrap (EB), Gaussian Wild (GB), Mammen (MB), Rademacher (RB), Beta (BB), and Double (DB) bootstraps.
  • Findings: EB is uniformly conservative. GB is sensitive to asymmetry (liberal). Mammen (MB) and Beta (BB) wild bootstraps (which match the third moment) are the most robust and stable across asymmetric and dependent designs. The Double Bootstrap (DB) is often closest to the nominal level in asymmetric cases but exhibits a slight liberal bias.

5. Significance

• Bridging Theory and Practice: This work moves high-dimensional inference beyond the maximum statistic, enabling more nuanced analysis of tail behavior and extreme values (e.g., the top 5 or 10 coordinates) which are crucial in fields like finance (portfolio risk), genomics (gene expression), and signal processing.
• Improved Accuracy: By achieving second-order accuracy, the proposed methods significantly reduce coverage errors in finite samples compared to standard first-order Gaussian approximations, especially in high-dimensional settings where $d \gg n$.
• Methodological Framework: The combination of factorial moments, inclusion-exclusion, and local Edgeworth expansions provides a powerful template for analyzing other non-rectangular functionals of high-dimensional sums.
• Practical Guidance: The simulation results offer concrete guidance on which bootstrap multipliers to use (favoring Mammen or Beta distributions) to ensure robust inference under various dependence and distributional assumptions.

In summary, Long Feng's paper successfully generalizes the state-of-the-art second-order bootstrap theory from maxima to the $k$-th order statistic, providing rigorous theoretical guarantees and practical algorithms for high-dimensional inference.