Sensitivity Of Covariance Tests To Nonnormality

Covariance Structure Is Not the Same as Distributional Shape

A likelihood-ratio test for a covariance restriction is often derived under a multivariate normal model. That derivation supplies a convenient reference distribution, but two assumptions are involved: the covariance matrix has the specified structure, and the observations follow a Gaussian distribution. These assumptions are distinct. A population can satisfy a covariance restriction while having heavy tails, skewness, or dependence that is not captured by its covariance matrix.

This distinction matters because the Gaussian likelihood-ratio statistic is calculated from sample covariance matrices. Its variation across samples depends on fourth moments, not only on the population covariance. If those moments differ from their Gaussian values, a chi-square reference distribution can be poorly calibrated even when the covariance null hypothesis is true.

For independent and identically distributed observations \(X_1,\ldots,X_n\) with mean \(\mu\), define the sample covariance matrix using divisor \(n\) by \(S=n^{-1}\sum_{i=1}^n(X_i-\bar X)(X_i-\bar X)'\), where \(\bar X=n^{-1}\sum_iX_i\). Many Gaussian covariance likelihood-ratio statistics are functions of \(S\), including determinant ratios. Under normality, their large-sample chi-square calibration follows from the normal likelihood theory. Outside normality, the statistic can still be computed, but that calibration is not automatic.

Why Fourth Moments Enter

The entries of \(S\) are averages of products of pairs of observations. The variability of a product such as \((X_j-\mu_j)(X_k-\mu_k)\) depends on expectations involving up to four factors. Thus, finite fourth moments and their values determine the usual large-sample variability of sample covariance.

To see why, first use the population mean in the covariance average. The summands \(\operatorname{vec}[(X_i-\mu)(X_i-\mu)'-\Sigma]\) are independent, have mean zero, and have finite covariance \(\Omega\). The multivariate central limit theorem therefore gives the stated limit for their normalized sum. Replacing \(\mu\) by \(\bar X\) changes the covariance matrix by \((\bar X-\mu)(\bar X-\mu)'\). Since finite second moments imply \(\bar X-\mu=O_p(n^{-1/2})\), this matrix is \(O_p(n^{-1})\); after multiplication by \(\sqrt n\), it is \(o_p(1)\). This proves the result.

For a Gaussian vector, the fourth moments have a particular form, and hence \(\Omega\) has a particular value. For a nonnormal vector with the same \(\Sigma\), \(\Omega\) need not be the same. A Gaussian likelihood-ratio test uses its Gaussian-based curvature and calibration; if the actual sampling variability differs, its null statistic need not have the nominal chi-square distribution.

A Derived Example: Sphericity Under Elliptical Tails

A particularly transparent calculation is the Gaussian likelihood-ratio test of sphericity, whose null hypothesis is \(\Sigma=\sigma^2I_p\) for some \(\sigma^2>0\). Its statistic, up to the usual large-sample convention, is \(T_n=n\{p\log(\operatorname{tr}S/p)-\log\det S\}\). Under a Gaussian null, the asymptotic degrees of freedom are \(q=p(p+1)/2-1\), the number of free covariance parameters removed by the restriction.

Suppose instead that the observations are independent and identically distributed, have mean zero and covariance \(\sigma^2I_p\), have finite fourth moments, and are spherical with fourth moments \(E[X_iX_jX_kX_l]=(1+\kappa)\sigma^4(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})\). Here \(\delta_{ij}\) is 1 when \(i=j\) and 0 otherwise; this equation defines the elliptical fourth-moment parameter \(\kappa\) for this calculation. The Gaussian case has \(\kappa=0\).

For the derivation, write \(S=\sigma^2I_p+\Delta\), where \(\Delta=O_p(n^{-1/2})\) under the stated moment conditions. Taylor expansion around \(\sigma^2I_p\) gives \(\log\det S=p\log\sigma^2+\operatorname{tr}(\Delta)/\sigma^2-\operatorname{tr}(\Delta^2)/(2\sigma^4)+O_p(\|\Delta\|^3)\). Expanding the other term gives \(p\log(\operatorname{tr}S/p)=p\log\sigma^2+\operatorname{tr}(\Delta)/\sigma^2-\{\operatorname{tr}(\Delta)\}^2/(2p\sigma^4)+O_p(\|\Delta\|^3)\). The linear terms cancel. Since \(n\|\Delta\|^3=o_p(1)\), it follows that \(T_n=\frac{n}{2\sigma^4}\{\operatorname{tr}(\Delta^2)-(\operatorname{tr}\Delta)^2/p\}+o_p(1)\).

Let \(W_n=\sqrt n\,\Delta\). The expression in braces is the squared Frobenius length of the component of \(W_n\) in the space of traceless symmetric matrices. That space has dimension \(q\). For any traceless symmetric matrix \(A\), the fourth-moment formula gives \(\operatorname{Var}\{\operatorname{tr}(A W_n)\}\longrightarrow 2(1+\kappa)\sigma^4\operatorname{tr}(A^2)\). Indeed, the term involving \((\operatorname{tr}A)^2\) vanishes because \(A\) is traceless. Choose an orthonormal basis of this space under the Frobenius inner product. The multivariate central limit theorem makes the limiting coordinates independent centered normal variables, each with variance \(2(1+\kappa)\sigma^4\). Substituting their squared lengths into the expansion gives \((1+\kappa)\) times the sum of \(q\) independent squared standard normal variables, which proves the result. \(\square\)

Misspecification Does Not Mean the Covariance Null Is False

The sphericity calculation used an elliptical distribution, where one scalar parameter changed the limiting scale. Outside that setting, nonnormality may change different directions of covariance variation by different amounts. The resulting limit can be a weighted sum of chi-square variables rather than a single scaled chi-square. If fourth moments are infinite, even the usual square-root-\(n\) covariance limit may fail, so neither the Gaussian calibration nor the finite-fourth-moment argument above is justified.

Interpretation and Practical Caution

A Gaussian covariance test applied to nonnormal observations has two possible vulnerabilities. First, its null calibration may be wrong because the sampling variability of \(S\) depends on fourth moments. Second, when the data contain very heavy tails, the sample covariance itself may be unstable. Neither issue changes the definition of the population covariance hypothesis, but both affect what can be concluded from a reported test statistic or p-value.

The key distinction is between the target of the hypothesis and the assumptions used to calibrate the test. For example, a diagonal-covariance null says that population covariances are zero; it does not assert independence, normality, or light tails. Conversely, finding nonnormal observations is not evidence, by itself, that the covariance matrix violates the specified structure.

When normality is doubtful, interpretation should state that the Gaussian calibration may be unreliable, especially when sample sizes are modest or tails are heavy. A procedure that accounts for nonnormal covariance variability may be appropriate, but its validity also depends on its own assumptions, including adequate moment conditions and the sampling design. A Gaussian parametric simulation does not resolve a concern that the Gaussian model is wrong, because it reproduces the very distributional assumption in question.

Common Mistake

• Wrong reasoning: “The data are not multivariate normal, so the covariance null hypothesis must be false.”
• Correction: Normality concerns the full distribution; a covariance null concerns only \(\Sigma\). The diagonal-covariance example above is nonnormal yet has \(\Sigma=I_2\). Nonnormality can undermine the Gaussian reference distribution without making the covariance restriction false.
• Another wrong inference: “The test did not reject, so nonnormality is irrelevant.”
• Correction: A nonrejection does not verify Gaussian calibration. The test’s size and power may both be affected by the actual fourth moments, and finite samples can make that effect consequential.