High-Dimensional Two-Sample Test for Elliptical Symmetry Distribution

Technical Summary: High-Dimensional Two-Sample Test for Elliptical Symmetry Distribution

Problem Statement
The paper addresses the high-dimensional two-sample location problem, specifically testing the equality of two location vectors ($H_0: \theta_1 = \theta_2$) when the dimension $p$ is comparable to or exceeds the sample sizes ($n_1, n_2$). The data are assumed to follow elliptical symmetric distributions with arbitrary dependence structures in the scatter matrix.

Existing literature faces two primary limitations in this setting:

1. Robustness vs. Dependence: While spatial-sign procedures are robust to heavy-tailed data, their standard calibration relies on trace conditions that assume sparse or weakly dependent scatter matrices. Under strong, dense dependence, these tests fail to maintain correct size because the null distribution becomes a weighted chi-square mixture rather than a normal distribution, and their fixed-point diagonal standardizers fail to recover the true diagonal shape of the elliptical distribution.
2. Moment Conditions: Many robust methods require positive moment conditions on the radial variable, which may not hold for heavy-tailed distributions.

Methodology
The authors propose a new spatial-sign test based on coordinatewise pairwise-difference quantile scales (PDQ). The procedure involves the following steps:

1. Diagonal Standardization: Instead of the iterative fixed-point estimator used in prior work (e.g., Feng et al., 2016; Li et al., 2016), the authors estimate the diagonal of the elliptical shape matrix using coordinatewise pairwise-difference quantiles. For a sample $k$ and coordinate $j$, the scale estimator $\hat{d}_{kj}$ is defined as the squared $\alpha$-quantile of the absolute differences $|X_{kij} - X_{k\ell j}|$.

   • This estimator is location-free and does not require positive moment conditions on the radial variable.
   • It recovers the diagonal shape up to a scalar factor specific to the sample, which vanishes after spatial normalization.

2. Spatial Median and Signs: Using the estimated diagonal matrix $\hat{D}_k$, the authors compute the full-sample spatial median $\hat{\theta}_k$ and the standardized spatial signs $\hat{S}_{ki} = U(\hat{D}_k^{-1/2}(X_{ki} - \hat{\theta}_k))$.

3. Test Statistic: The test statistic is a diagonal-deleted quadratic form of the spatial signs:
   $$\hat{R}^{PDQ}_n = -\frac{1}{n_1 n_2} \sum_{i=1}^{n_1} \sum_{j=1}^{n_2} \hat{S}_{1i}^\top \hat{S}_{2j}$$
   To correct for the empirical diagonal terms in the expansion, an empirical diagonal-deletion correction $\hat{b}_n$ is subtracted, yielding the feasible statistic $T^{PDQ}_n = \hat{R}^{PDQ}_n - \hat{b}_n$.

4. Calibration: Recognizing that the null distribution under strong dependence is generally a non-Gaussian weighted chi-square mixture, the authors employ a Rademacher wild bootstrap for calibration. This method consistently estimates the null law without requiring explicit estimation of the unknown spectral weights or eigenvalues of the limiting distribution.

Key Theoretical Contributions

• Explicit-Rate Stochastic Expansion: The authors derive an explicit-rate expansion for the full-sample statistic, establishing the convergence rates of the diagonal estimator, the spatial median, and the plug-in matrices.
• General Null Distribution: Under arbitrary correlation structures, the paper establishes that the oracle diagonal-deleted statistic converges to a general weighted chi-square mixture. The normal approximation is shown to be a special case occurring only when no single eigenvalue dominates the spectrum.
• Diagonal-Deletion Justification: The paper justifies the empirical diagonal-deletion correction, showing that it removes the within-sample diagonal terms necessary for the quadratic expansion without requiring the correction term to estimate the population mean with high precision.
• Bootstrap Consistency: It is proven that the Rademacher wild bootstrap consistently estimates the null law, providing a valid calibration method for both asymptotically normal and non-normal regimes.
• Relaxed Assumptions: The theoretical framework does not require sparsity, bandedness, or weak correlation of the scatter matrices. It accommodates arbitrary dependence structures provided the dimension satisfies $p = o\{n_0^2 / \log^2(p + n_0)\}$ in bounded-spectrum regimes.

Empirical Results
Simulation studies compare the proposed PDQ test against:

• SST: The spatial-sign test of Li et al. (2016) with sign-based diagonal iteration.
• ART: The approximate randomization test of Wang and Xu (2022).
• BF-F: The normal-reference F-type test of Zhu et al. (2024).

The simulations consider multivariate normal, multivariate $t$ (heavy-tailed), and mixture normal distributions under AR(1) and Compound Symmetry (CS) dependence structures with high correlation ($\rho=0.9$).

• Size Control: The proposed PDQ test demonstrates accurate and stable control of the nominal level across all settings, including heavy-tailed and contaminated data. In contrast, SST is systematically liberal (over-rejects) under strong dependence and non-Gaussianity, while BF-F becomes conservative under heavy tails.
• Power: Under dense alternatives, PDQ tracks the performance of competing methods under Gaussian data but significantly outperforms them under heavy-tailed and contaminated distributions. The apparent power gains of SST in some scenarios are attributed to its size distortion (over-rejection under the null) rather than genuine discrimination ability.

Significance and Claims
The paper claims to resolve the gap between arbitrary-covariance mean-based tests and heavy-tail-robust spatial-sign tests. By replacing the fixed-point diagonal standardizer with a pairwise-difference quantile estimator and utilizing a wild bootstrap for calibration, the proposed method:

1. Remains valid under strong, dense dependence where previous spatial-sign theories fail.
2. Requires no positive moment conditions on the radial variable, making it suitable for heavy-tailed distributions.
3. Provides a unified framework that adapts to the underlying dependence structure, yielding a weighted chi-square limit generally and a normal limit only as a special case.

The authors conclude that the proposed procedure offers the most reliable overall performance for high-dimensional two-sample location testing when dependence is strong and distributions are potentially heavy-tailed or contaminated.