Extreme-Value Calibration of Maximum Tests

Why Maximum Tests Need Extreme-Value Calibration

A coordinatewise maximum test rejects when at least one among many standardized coordinates is unusually large. Even if every coordinate follows its null distribution, the largest value tends to increase as the number of coordinates grows. A fixed single-coordinate cutoff therefore does not generally control the probability of at least one false rejection. Extreme-value calibration accounts for that growth by centering and scaling the maximum.

The formulas below concern independent Gaussian coordinates. This setting gives both a clean asymptotic limit and exact finite-dimensional cutoffs. In a mean-testing application, the standardized coordinates might be \(Z_j=\sqrt{n}\,\bar X_j/\sigma_j\), where the null mean is zero and the variances \(\sigma_j^2\) are known. If the observations are Gaussian and these coordinates are independent under the null, then each \(Z_j\) is standard normal. Estimating variances or allowing dependence requires additional treatment; independence is an assumption here, not a consequence of taking a maximum.

The Gaussian Extreme-Value Limit

For the one-sided maximum, define the normalizing constants

The scale \(1/a_m\) shrinks as the dimension grows. The centering \(b_m\) tracks the typical size of the largest coordinate, while the scale retains the non-degenerate fluctuations around that size. The limiting distribution is the standard Gumbel distribution, whose cumulative distribution function is \(G(x)=\exp\{-e^{-x}\}\).

Proof. Put \(u_m=b_m+x/a_m\). Independence gives

where \(\bar\Phi(u)=1-\Phi(u)\) is the standard normal upper-tail probability. Since \(u_m\to\infty\), the normal tail satisfies Mills' ratio \(\bar\Phi(u)\sim \phi(u)/u\), with \(\phi(u)=(2\pi)^{-1/2}e^{-u^2/2}\). Write \(u_m=a_m+d_m/a_m\), where \(d_m=x-\tfrac12\{\log\log m+\log(4\pi)\}\). Then

Here \(d_m=O(\log\log m)\), so \(d_m^2/a_m^2\to0\), and \(u_m/a_m\to1\). Consequently,

For the last limit, substituting \(d_m\) and \(a_m=\sqrt{2\log m}\) cancels the factor involving \(\log m\) and \(2\pi\), leaving \(e^{-x}\). In particular, \(\bar\Phi(u_m)\to0\) and \(m\bar\Phi(u_m)^2\to0\). The expansion \(\log(1-y)=-y+O(y^2)\) as \(y\to0\) therefore yields

This is the asserted limit. \(\square\)

For a two-sided test, the maximum is over absolute values, so a large coordinate can occur in either tail. The centering must reflect both tails. The scale remains \(a_m\), but the constant in the centering changes:

Indeed, the probability of exceeding a high positive threshold in either direction is approximately twice the one-sided tail. That factor of two changes the constant from \(\log(4\pi)\) to \(\log\pi\).

Derivation. Set \(v_m=b_m^{\mathrm{abs}}+x/a_m=a_m+q_m/a_m\), where \(q_m=x-\tfrac12\{\log\log m+\log\pi\}\). Symmetry and Mills' ratio give \(P(|Z_1|>v_m)=2\bar\Phi(v_m)\sim 2\phi(v_m)/v_m\). As in the preceding proof, \(v_m/a_m\to1\), \(q_m^2/a_m^2\to0\), and \(v_m^2/2=\log m+q_m+o(1)\). Thus

Independence now gives \(P(T_m\le v_m)=\{1-P(|Z_1|>v_m)\}^m\). Applying the same logarithm limit as in the one-sided proof yields \(\exp\{-e^{-x}\}\), as claimed. \(\square\)

Turning the Limit into a Critical Value

For a target level \(\alpha\), let \(x_{1-\alpha}\) be the \(1-\alpha\) quantile of the standard Gumbel distribution. Solving \(G(x)=1-\alpha\) gives

The extreme-value approximation then gives the one-sided and two-sided critical values

Reject when \(M_m>c_{m,\alpha}^{+}\) for a one-sided maximum, or when \(T_m>c_{m,\alpha}^{\mathrm{abs}}\) for a two-sided maximum. The convergence results justify these rules asymptotically: because \(G\) is continuous and \(G(x_{1-\alpha})=1-\alpha\), the null rejection probability tends to \(\alpha\) as \(m\to\infty\). This is a limiting calibration, not an assertion that the finite-\(m\) rejection probability equals \(\alpha\).

These expressions follow directly from independence. For the one-sided case, \(P(M_m\le c)=\Phi(c)^m\); setting this equal to \(1-\alpha\) gives the first formula. For the absolute maximum, \(P(T_m\le c)=\{P(-c\le Z_1\le c)\}^m=\{2\Phi(c)-1\}^m\); equating this to \(1-\alpha\) gives the second.

Interpretation and Common Pitfalls

Extreme-value calibration matters because the maximum summarizes a growing collection of opportunities for a large null fluctuation. The normalization makes that growth explicit: the leading size of the maximum is about \(\sqrt{2\log m}\), while the Gumbel quantile supplies a finer adjustment for the desired tail probability. In applications, this lets a sparse-signal test use a cutoff appropriate to the number of coordinates rather than a single-coordinate quantile.

• Using a single-coordinate cutoff for a maximum. A cutoff that gives a 5% exceedance probability for one coordinate will generally be exceeded with probability greater than 5% when applied to many coordinates. Calibrate the maximum itself.
• Mixing one-sided and two-sided constants. The \(\log(4\pi)\) centering is for \(\max_j Z_j\); the \(\log\pi\) centering is for \(\max_j|Z_j|\). Select the statistic first, then use its matching constants.
• Treating a limit as an exact finite-sample result. The Gumbel cutoff is asymptotic. When independent Gaussian coordinates are justified, the exact formulas provide finite-\(m\) calibration.
• Assuming independence without support. The product probabilities and exact cutoffs above rely on independent coordinates. Correlation changes the distribution of the maximum; the independent-coordinate formulas cannot simply be presumed valid.

The central distinction is between a maximum's null distribution and the distribution of any one coordinate. A threshold calibrated for the former accounts for the number of coordinates and the tail convention. In a real analysis, those assumptions and the meaning of one- versus two-sided rejection should be made explicit.