Dependence Corrections for Maximum Tests

Why Dependence Changes a Maximum Test

A maximum-coordinate test rejects when at least one standardized coordinate is unusually large. Its calibration is therefore governed not just by the marginal tail probability of each coordinate, but by how often large values occur together. If coordinates are independent, the chance of at least one exceedance accumulates across all coordinates. If coordinates move together, many exceedances may represent essentially one extreme event.

We focus on a Gaussian null model. Let \(Z=(Z_1,\ldots,Z_p)^\top\) have mean zero, unit marginal variances, and correlation matrix \(R=(\rho_{jk})\). The two-sided maximum statistic is \(M_p=\max_{1\le j\le p}|Z_j|\). A level-\(\alpha\) threshold \(u\) should satisfy \(P(M_p>u)\le\alpha\), with equality for an exact continuous calibration. The issue is how to choose \(u\) when the coordinates are dependent.

Three Calibration Principles

Under independence, \(P(M_p\le u)=[1-2\{1-\Phi(u)\}]^p\), where \(\Phi\) is the standard normal distribution function. In the high-dimensional regime, this leads to the extreme-value threshold for \(2p\) one-sided normal tails. Define

For independent standard normal coordinates, \(P(M_p\le u_p(x))\to \exp\{-e^{-x}\}\). Thus an asymptotic level-\(\alpha\) threshold uses \(x_\alpha=-\log[-\log(1-\alpha)]\). Dependence correction asks when this same calibration remains valid and, when it does not, what alternative threshold is justified.

One threshold is valid without any dependence assumptions. The union bound gives \(P(M_p>u)\le 2p\{1-\Phi(u)\}\). Consequently, choosing \(u=\Phi^{-1}(1-\alpha/(2p))\) controls the test size at \(\alpha\) for every joint distribution with standard normal marginals. It can be conservative, because it does not account for overlap between coordinate exceedances.

A Weak-Dependence Result

A more informative result gives a sufficient condition for the independence-based extreme-value calibration to survive dependence. The condition below controls the aggregate effect of all pairwise correlations at the relevant tail threshold; it is useful even when the coordinates are not exactly independent.

Here is a proof. Let \(Q_p(u)=P(|Z_j|\le u\text{ for all }j)\), and let \(Q_p^{(0)}(u)=[P(|N(0,1)|\le u)]^p\) be the corresponding probability for independent standard normal coordinates. A Gaussian rectangle comparison bound gives

for a universal constant \(C\), when \(|\rho_{jk}|<1\). To see the source of this bound, interpolate between the identity correlation matrix and \(R\). Differentiating the Gaussian probability of the rectangle \([-u,u]^p\) with respect to an interpolated pairwise correlation produces four boundary terms. Each is bounded by the corresponding bivariate normal density at a corner, since the conditional probability for the remaining coordinates is at most one. At correlations \(t\rho_{jk}\), \(0\le t\le1\), those densities are bounded by a constant times \((1-\rho_{jk}^2)^{-1/2}\exp\{-u^2/(1+|\rho_{jk}|)\}\). Integrating the derivative over \(t\) contributes the factor \(|\rho_{jk}|\); summing over pairs yields the displayed inequality. Singular covariance matrices follow by adding an independent Gaussian perturbation, applying the bound, and taking the perturbation variance to zero.

The assumed sum therefore implies that the dependent and independent rectangle probabilities differ by a quantity tending to zero at \(u=u_p(x)\). For the independent probability, write \(q_p=2\{1-\Phi(u_p(x))\}\). The normal tail asymptotics give \(2p\{1-\Phi(u_p(x))\}\to e^{-x}\), so \(pq_p\to e^{-x}\) and \(q_p\to0\). Hence \([1-q_p]^p\to\exp\{-e^{-x}\}\), which proves the asserted limit.

For the stated simpler sufficient condition, put \(r_p=\max_{j\ne k}|\rho_{jk}|\). When \(r_p\log p\to0\), eventually \(r_p<1/2\), and the sum in the theorem is bounded by a constant times \(p^2r_p\exp\{-u_p(x)^2/(1+r_p)\}\). At this threshold, \(\exp\{-u_p(x)^2\}\) is of order \((\log p)/p^2\). Also, \(\exp\{u_p(x)^2r_p/(1+r_p)\}\) stays bounded because \(r_p\log p\to0\). The bound is therefore of order \(r_p\log p\), which tends to zero. This completes the proof. \(\square\)

Structured Dependence: Calibrate the Effective Extremes

Weak pairwise correlations are not the only setting in which a valid threshold can be obtained. When the dependence has a simple structure, one can often calibrate the maximum directly. The number of recorded coordinates need not equal the number of distinct opportunities for an extreme event.

Common Pitfall and Practical Interpretation

A common mistake is to use the independent-coordinate threshold merely because every coordinate has a standard normal marginal distribution. Marginals determine individual tail probabilities, but they do not determine the probability of a union of exceedance events. With strongly dependent coordinates, an independent calibration may be too high and yield a test that is much more conservative than intended. With a different dependence pattern, simply reducing the effective coordinate count by intuition can instead produce an invalid, liberal test.

A sound calibration starts by identifying what is known about the joint null distribution. If only the Gaussian marginals are trusted, the Bonferroni threshold provides finite-sample size control. If correlations are weak in the aggregate, the comparison result justifies the independent extreme-value limit. If dependence has a specified structure, conditioning or reducing to independent blocks can yield a more accurate threshold. These conclusions assume the coordinates have been correctly standardized; using marginal variances that are not one changes the tail probabilities and invalidates the stated thresholds.