Critical Values for Correlated Gaussian Maxima

Why the Gaussian Maximum Sets the Critical Value

When several estimates are reported together, simultaneous coverage concerns all of their errors at once. A critical value for one normal component does not generally provide the desired coverage for a vector of estimates: the chance that every component lies within its interval depends on their joint distribution. In the asymptotic setting, that dependence is summarized by the correlation matrix of the standardized errors.

The relevant random quantity is the largest absolute component of a correlated Gaussian vector. Its quantile calibrates a common multiplier for the componentwise standard errors, producing intervals whose joint coverage is approximately the target level. The calibration uses the entire correlation structure, not merely the number of components.

The event defining the quantile is a rectangle probability: \(P(M\leq c)=P(-c\leq Z_j\leq c,\ j=1,\ldots,m)\). Thus, even though every marginal distribution is standard normal, the probability of this rectangle changes with \(R\). For general \(R\), the quantile is usually found by numerical multivariate normal probability evaluation followed by a one-dimensional root search.

From a Gaussian Quantile to Simultaneous Intervals

Suppose \(\hat\theta=(\hat\theta_1,\ldots,\hat\theta_m)^\mathsf{T}\) estimates \(\theta=(\theta_1,\ldots,\theta_m)^\mathsf{T}\). Let \(s_j\) estimate the asymptotic standard deviation of \(\hat\theta_j\), and suppose the standardized error vector satisfies \(( (\hat\theta_j-\theta_j)/s_j )_{j=1}^m\xrightarrow{d}N_m(0,R)\), with \(s_j\) consistent for the corresponding standard error scale and \(R\) a fixed correlation matrix. This is the joint-normal setting used in asymptotic simultaneous intervals.

Proof. Write \(T_n=\max_j|(\hat\theta_j-\theta_j)/s_j|\). The map \(x\mapsto\max_j|x_j|\) is continuous, since it is the maximum of finitely many continuous functions. The continuous mapping theorem therefore gives \(T_n\xrightarrow{d}M\). The event that all intervals cover their respective parameters is exactly \(\{|\hat\theta_j-\theta_j|\leq c s_j\text{ for every }j\}=\{T_n\leq c\}\). Convergence in distribution implies \(P(T_n\leq c)\to P(M\leq c)\) at every continuity point \(c\). By the definition of the critical value and continuity, \(P(M\leq c)=1-\alpha\). Hence the limiting joint coverage is \(1-\alpha\). \(\square\)

This is an asymptotic calibration, not an assertion of exact finite-sample coverage. Its validity depends on the joint normal approximation and on using standard errors and a correlation matrix that represent the standardized errors in that approximation. The earlier material on asymptotic simultaneous intervals gives the interval form; the Gaussian maximum quantile determines its multiplier.

Two Useful Exact Cases

When the coordinates are independent, the rectangle probability factors into \(m\) identical marginal probabilities. When they are perfectly correlated in the same direction, all coordinates are copies of one standard normal variable, and the maximum absolute value is just the absolute value of that variable. These cases provide useful checks on a numerical calculation.

Derivation. For each standard normal component, symmetry gives \(P(|Z_j|\leq c)=\Phi(c)-\Phi(-c)=2\Phi(c)-1\). Independence makes the probability of the intersection the product of these probabilities, yielding the first formula. Solving \([2\Phi(c)-1]^m=1-\alpha\) gives the stated quantile. In the perfectly correlated case, \(M=|Z_1|\); solving \(2\Phi(c)-1=1-\alpha\) gives the second formula. \(\square\)

Computing a Correlated Gaussian Maximum

For a general correlation matrix, evaluate the probability of the centered rectangle \([-c,c]^m\) under \(N_m(0,R)\). A practical procedure is to choose a candidate \(c\), calculate that probability, and adjust \(c\) until it equals \(1-\alpha\). The probability calculation may use a multivariate normal CDF routine, deterministic numerical integration, or a carefully controlled simulation method. Root-finding then locates the quantile. Accuracy should be checked by evaluating the rectangle probability again at the reported critical value.

A two-dimensional case makes the dependence effect especially visible. If \(Z_1,Z_2\) are standard normal with correlation \(\rho\), write \(\Phi_2(a,b;\rho)=P(Z_1\leq a,Z_2\leq b)\). Symmetry and inclusion-exclusion give

This is the rectangle probability to solve for the desired coverage. It is not the product of marginal probabilities unless \(\rho=0\). For this bivariate symmetric rectangle, an alternative useful identity is

The identity follows by differentiating the bivariate normal rectangle probability with respect to its correlation parameter \(r\), which gives the integrand, and integrating from \(r=0\), where independence gives the product term. It is useful for checking a two-component numerical calculation.

How Correlation Matters—and a Common Pitfall

The correlation structure determines how often large componentwise errors occur together. With independent components, extreme values can arise in different coordinates, so the maximum tends to require a larger cutoff as the number of coordinates grows. With perfectly correlated copies, all coordinates move together and adding copies does not change the maximum. Intermediate dependence lies between these familiar examples in particular models, but there is no universal rule that a certain sign or average size of correlations determines the critical value for every correlation matrix. The full matrix matters.

A common mistake is to use a single-component normal critical value for every coordinate and then call the resulting intervals jointly 95% intervals. For two independent coordinates, using 1.960 gives joint coverage \([2\Phi(1.960)-1]^2\approx0.95^2=0.9025\), not 0.95. Another mistake is to use the independent-components formula when the coordinates are correlated. That formula is correct only when the standardized Gaussian components are independent; it is not justified merely because every marginal is standard normal.