Maximum-Coordinate Tests for Sparse Mean Shifts

Why Test the Largest Coordinate?

A multivariate mean test can be designed to respond to different shapes of departure from the null. A quadratic statistic accumulates evidence across coordinates; a maximum-coordinate statistic instead asks whether at least one standardized mean is unusually large in magnitude. That focus can be useful when only a small number of coordinates are expected to shift, since a few strong signals need not be diluted by many near-zero coordinates.

The maximum has a calibration challenge: even when every coordinate follows its null distribution, taking a maximum over many coordinates makes an extreme value more likely. The construction below controls this multiple-coordinate error without assuming that the coordinates are independent. We use an exact Gaussian model and known marginal standard deviations so the test and its size calculation are explicit.

Write \(\bar X_j=n^{-1}\sum_{i=1}^n X_{ij}\) for the sample mean in coordinate \(j\). Its standardized value is \(Z_j=\sqrt n\,\bar X_j/\sigma_j\). Under the Gaussian model, \(Z_j\) has a standard normal distribution under \(H_0\), regardless of the dependence between coordinates. The vector of these standardized means can still be dependent.

Calibration Without Coordinate Independence

The test compares all \(p\) coordinates with a threshold chosen to limit the probability of any false rejection. It does not claim that the maximum has a particular exact distribution when coordinates are dependent. Instead, it combines the marginal normal tail probabilities using the union bound.

Proof. Let \(A_j=\{|Z_j|>c_{\alpha,p}\}\) be the rejection event for coordinate \(j\). Under the null, \(Z_j\sim N(0,1)\), so the definition of \(c_{\alpha,p}\) gives \(P(A_j)=2\{1-\Phi(c_{\alpha,p})\}=\alpha/p\). The test rejects on the union of these events. By the union bound,

This argument uses only the marginal probabilities, so it remains valid whether the coordinates are independent, positively correlated, or otherwise dependent. If they are independent, the probability of no coordinate rejection is the product \(\prod_{j=1}^p P(A_j^c)=(1-\alpha/p)^p\); taking its complement gives the stated equality. \(\square\)

The bound can be conservative. Under independence, for example, the exact null probability \(1-(1-\alpha/p)^p\) is generally below \(\alpha\). More refined calibration can use information about the joint distribution, but the Bonferroni threshold is a transparent baseline and does not require a dependence model.

Power Against Sparse Mean Shifts

The relevant signal size for coordinate \(j\) is its mean measured in standard errors: \(\delta_j=\sqrt n\,\mu_j/\sigma_j\). Under an alternative, \(Z_j\sim N(\delta_j,1)\), so a signal is easier to detect when its standardized mean is large in absolute value. The cutoff also depends on \(p\): adding coordinates makes the test guard against more opportunities for a null coordinate to exceed the threshold.

Proof. Since \(Z_j\sim N(\delta_j,1)\), the event \(|Z_j|>c_{\alpha,p}\) is the disjoint union of \(Z_j>c_{\alpha,p}\) and \(Z_j<-c_{\alpha,p}\). Standardizing gives

A coordinate crossing is one way for the maximum to exceed the cutoff, so the overall rejection event contains each coordinate event and its probability is at least \(q_j\). If the standardized means for coordinates in \(S\) are independent, the probability none of those coordinates crosses is \(\prod_{j\in S}(1-q_j)\). Its complement is the probability at least one crosses. That event is contained in the full rejection event, proving the lower bound. Without the stated independence, the single-coordinate lower bounds still hold, but the product expression need not. \(\square\)

This result makes the sparse-signal advantage precise. A few coordinates with large \(|\delta_j|\) can each have substantial crossing probability, and one such crossing is enough for rejection. The test does not require a large aggregate contribution from many coordinates. Conversely, if all \(|\delta_j|\) remain small relative to the cutoff, the maximum may have limited power even when many coordinates shift. The test's sensitivity is governed directly by the largest standardized shifts, not just by the total size of the mean vector.

Interpretation, Assumptions, and a Common Pitfall

The test is most naturally interpreted as a screen for a large coordinate-wise standardized departure. Its choice of scale matters: dividing by \(\sigma_j\) places coordinates with different measurement units or marginal variances on a common scale. The assumption that these standard deviations are known is part of the exact calibration given here. Replacing them with estimates changes the distribution of the statistic and requires a suitable calibration rather than automatically preserving the same cutoff.

A common mistake is to compare each \(|Z_j|\) with the single-coordinate two-sided cutoff \(\Phi^{-1}(1-\alpha/2)\), then claim that the chance of any false rejection is \(\alpha\). That cutoff controls the error probability for one coordinate only. For \(p\) coordinates, the union of the rejection events can have probability greater than \(\alpha\); under independence it is \(1-(1-\alpha)^p\), not \(\alpha\). The correction is to use a multiple-coordinate threshold such as \(c_{\alpha,p}\), for which the union bound proves the desired size control even with dependence.

A second interpretive caution concerns dependence. The Bonferroni size guarantee does not require independent coordinates, but the product power calculation for several signals does. When dependence is unknown, report a valid single-coordinate power lower bound or use a justified joint model; do not multiply marginal non-rejection probabilities as though the coordinates were independent. Likewise, a maximum test is not a universal replacement for other mean tests: it targets unusually large individual standardized shifts, while alternatives spread thinly across many coordinates may call for a different statistic.