Cauchy Aggregation of Ridge-Regularized Hotelling Tests for High-Dimensional Change-Point Detection

Technical Summary: Cauchy Aggregation of Ridge-Regularized Hotelling Tests for High-Dimensional Change-Point Detection

1. Problem Formulation
The paper addresses the problem of detecting changes in the mean vector of a high-dimensional time series, where the dimension $p$ is comparable to the sample size $n$. The data model is $X_j = \mu_j + \Sigma_p^{1/2}Z_j$, where the covariance matrix $\Sigma_p$ is time-invariant, and the goal is to test the null hypothesis $H_0: \mu_1 = \dots = \mu_n$ against alternatives involving one or more abrupt mean shifts.

In high-dimensional settings, the sample covariance matrix $S_n$ is often singular or ill-conditioned, rendering classical Hotelling-type tests based on $S_n^{-1}$ inapplicable. Ridge-regularized Hotelling (RHT) tests, which replace $S_n^{-1}$ with $(S_n + \lambda I_p)^{-1}$, offer a solution. However, the power of RHT tests depends critically on the choice of the ridge parameter $\lambda$. The optimal $\lambda$ is determined by the unknown covariance structure and the unknown direction/sparsity of the mean shift. Selecting a single fixed $\lambda$ risks substantial power loss if the choice is misspecified relative to the true underlying signal.

2. Methodology
The authors propose an aggregation approach that avoids selecting a single optimal ridge parameter. Instead, the method operates as follows:

• Deterministic Grid: A finite, deterministic grid of ridge parameters $\Lambda_n = \{\lambda_{1,n}, \dots, \lambda_{L,n}\}$ is fixed in advance. These values are scaled by the ratio $\gamma_n = p/(n-1)$ to ensure they remain bounded away from zero and infinity as $n, p \to \infty$.
• Fixed-Ridge Statistics: For each $\lambda_\ell \in \Lambda_n$, the authors compute the marginal RHT scan statistic $T_{\lambda_\ell}$ and its corresponding $p$-value $P_{\lambda_\ell}$. Under standard random-matrix assumptions, the marginal distribution of $T_{\lambda_\ell}$ converges to a pivotal Gaussian process supremum, making the marginal $p$-values asymptotically valid and free of $\Sigma_p$.
• Cauchy Combination: The marginal $p$-values are aggregated using the Cauchy combination rule (Liu and Xie [12]). The test statistic is defined as $C_n = \sum_{\ell=1}^L w_\ell \tan\{\pi(1/2 - P_{\lambda_\ell})\}$, where $w_\ell$ are fixed weights summing to 1. The analytic $p$-value is computed as $P_{CCT} = 1/2 - (1/\pi)\arctan(C_n)$.

3. Key Theoretical Contributions
The paper establishes the theoretical validity of this aggregation strategy under standard random-matrix conditions (Assumption 1):

• Finite-Grid Joint Null Limit: The primary theoretical contribution is the derivation of the joint weak convergence of the vector of ridge processes $\{D_{\lambda_\ell}(s)\}_{\ell=1}^L$ to a centered Gaussian vector process $\{G_\ell(s)\}_{\ell=1}^L$. While the marginal processes are pivotal, the cross-ridge covariance structure depends on the limiting spectral distribution of $\Sigma_p$.
• Size Validity: Two calibration schemes are analyzed:
  1. Joint-Limit Calibration: Using the critical value from the joint limit distribution of the p-values guarantees asymptotically exact fixed-level size.
  2. Analytic Cauchy Calibration: The standard analytic Cauchy $p$-value provides a simple, dependence-robust approximation. The paper proves that this analytic $p$-value is valid in the "small-tail" sense (i.e., $\lim_{\alpha \downarrow 0} \lim_{n \to \infty} P(P_{CCT} \le \alpha)/\alpha = 1$), though it may not yield exact size at conventional levels (e.g., 0.05) without joint calibration.
• Adaptive Consistency: The authors prove that if there exists at least one grid point $\lambda_{\ell_0}$ where the signal drift diverges to infinity, the aggregated test achieves power tending to one, provided the other terms do not asymptotically cancel the diverging term. This establishes the method's ability to adapt to unknown signal structures without explicitly estimating the optimal $\lambda$.

4. Simulation Results
Monte Carlo experiments were conducted across various covariance structures (Identity, Toeplitz, Polynomial decay, Exponential decay, Compound Symmetry) and signal types (dense independent, covariance-aligned, and sparse shifts).

• Size Control: The simulations show that moderate ridge values on the deterministic grid yield empirical sizes close to the nominal level (5%). Very small ridge values can be liberal when $p/n$ is large, motivating the exclusion of extremely small parameters from the grid.
• Power Performance: The Cauchy-aggregated test consistently achieves power close to the "oracle" best fixed ridge choice (the best $\lambda$ selected from the grid after the fact).
• Robustness: The method tracks the performance of the best fixed ridge across diverse covariance and signal configurations. Table 3 quantifies the cumulative power gap relative to the oracle, showing that the Cauchy-combined test ($\Delta_{CCT}$) consistently has the smallest loss compared to fixed choices of $\lambda/\gamma_n = 0.1$ or $0.2$. The advantage is particularly pronounced when the optimal ridge value varies significantly with the spectral structure (e.g., under Poly Decay or covariance-aligned signals).

5. Significance and Claims
The paper claims that its aggregation approach offers a practical solution to the tuning parameter dilemma in high-dimensional change-point detection. By avoiding the estimation of a single power-optimal ridge value, the method:

• Maintains stable size behavior through the use of a deterministic grid.
• Achieves power comparable to the best possible fixed-ridge choice across a wide range of unknown covariance and signal configurations.
• Provides a theoretically grounded justification for using the Cauchy rule in this context, specifically clarifying the distinction between exact joint-limit calibration and the analytic small-tail approximation.

The authors conclude that aggregating a stable deterministic ridge grid recovers most of the oracle fixed-ridge power without requiring the selection of a single, potentially suboptimal, ridge parameter. They note that the method relies on the assumption that the grid is bounded away from zero to avoid finite-sample instability.