Robust Tests of a Specified Covariance Matrix

Why Test a Specified Covariance Matrix?

A covariance matrix describes both the variability of each component and the dependence between components. In applications, a scientific model may specify all of these quantities at once: for example, a reference covariance structure for repeated measurements or a covariance matrix used to define a multivariate monitoring target. A joint test asks whether the population covariance agrees with that complete specification.

Under nonnormality, the ordinary sample covariance can be sensitive to a small number of extreme observations. A robust scatter estimator can reduce that sensitivity, but its sampling variability must also be handled correctly. The test below uses the joint asymptotic distribution of the estimator’s distinct entries; it does not treat those entries as independent unless their asymptotic covariances actually vanish.

Definition: Let \(X_1,\ldots,X_n\) be independent and identically distributed \(p\)-dimensional observations, and let \(\Sigma_0\) be a specified symmetric positive-definite matrix. A calibrated robust scatter estimator \(\hat{\Sigma}\) targets the population covariance matrix \(\Sigma\). The null hypothesis is \(H_0:\Sigma=\Sigma_0\), against \(H_1:\Sigma\ne\Sigma_0\). “Calibrated” means that the estimator’s population target is covariance, not merely an unscaled shape matrix.

Distinct Entries and Their Asymptotic Covariance

A symmetric \(p\times p\) matrix has \(k=p(p+1)/2\) distinct entries. Write \(\operatorname{vech}(A)\) for the vector formed by stacking the columns of a symmetric matrix \(A\), retaining the diagonal and lower-triangular entries. For \(p=3\), this convention gives \(\operatorname{vech}(A)=(A_{11},A_{21},A_{31},A_{22},A_{32},A_{33})^\mathsf{T}\). The same order must be used for the estimate, the null matrix, and the covariance matrix of the entries.

Suppose the robust estimator has an asymptotic linear representation under the null:

\sqrt{n}\{\operatorname{vech}(\hat{\Sigma})-\operatorname{vech}(\Sigma_0)\} = \frac{1}{\sqrt{n}}\sum_{i=1}^n \psi(X_i)+o_p(1), \qquad E[\psi(X_i)]=0, \qquad E[\psi(X_i)\psi(X_i)^\mathsf{T}]=\Omega.

Here \(\psi(X_i)\) is the \(k\)-vector influence contribution for the distinct scatter entries, and \(\Omega\) is their asymptotic covariance matrix. It includes the effects of estimating nuisance quantities, such as location, whenever those estimates are part of the scatter procedure. In particular, it is not generally enough to calculate separate variances for the diagonal and off-diagonal entries.

Result: If the observations are iid, the influence contributions have finite second moments, the representation above holds, and \(\Omega\) is positive definite, then \(\sqrt{n}\{\operatorname{vech}(\hat{\Sigma})-\operatorname{vech}(\Sigma_0)\}\xrightarrow{d}N_k(0,\Omega)\) under \(H_0\).

This covariance formula follows directly from the linear representation. Let \(Z_i=\psi(X_i)\). The \(Z_i\) are iid, have mean zero, and have finite covariance \(\Omega\). The multivariate central limit theorem therefore gives \(n^{-1/2}\sum_i Z_i\xrightarrow{d}N_k(0,\Omega)\). Adding a term that converges to zero in probability does not change this limiting distribution, by Slutsky’s theorem. Thus the displayed asymptotic normal law holds. \(\square\)

Worked Example: Counting the Jointly Tested Entries

Suppose the specified covariance matrix is \(4\times4\). The number of distinct entries is \(k=4(4+1)/2=10\). With the stated column-stacking convention, the tested vector is

\operatorname{vech}(\hat{\Sigma})= (\hat{\Sigma}_{11},\hat{\Sigma}_{21},\hat{\Sigma}_{31},\hat{\Sigma}_{41}, \hat{\Sigma}_{22},\hat{\Sigma}_{32},\hat{\Sigma}_{42}, \hat{\Sigma}_{33},\hat{\Sigma}_{43},\hat{\Sigma}_{44})^\mathsf{T}.

The joint covariance \(\Omega\) is therefore \(10\times10\), and the full-matrix null test will have 10 degrees of freedom when its regularity conditions hold. Testing only the four diagonal entries would instead test a different, incomplete hypothesis: it would leave the six covariances unspecified.

The Robust Wald Test

Let \(\hat{\Omega}\) be a consistent estimator of \(\Omega\), and define \(d=\operatorname{vech}(\hat{\Sigma})-\operatorname{vech}(\Sigma_0)\). A common estimator of \(\Omega\) is the empirical covariance of estimated influence contributions:

\hat{\Omega} = \frac{1}{n}\sum_{i=1}^n (\hat{\psi}_i-\bar{\psi})(\hat{\psi}_i-\bar{\psi})^\mathsf{T}, \qquad \bar{\psi}=\frac{1}{n}\sum_{i=1}^n\hat{\psi}_i.

This formula is appropriate when the fitted influence contributions consistently estimate the contributions in the asymptotic representation and the required moment and regularity conditions hold. Some estimators instead provide a sandwich estimate of the same asymptotic covariance. Either way, \(\hat{\Omega}\) must be on the scale of the covariance of one influence contribution, not already divided by \(n\).

Test statistic: For the null \(H_0:\Sigma=\Sigma_0\), the robust Wald statistic is \[ W=n\,d^\mathsf{T}\hat{\Omega}^{-1}d. \] Under \(H_0\), if the asymptotic linear representation holds, \(\hat{\Omega}\xrightarrow{p}\Omega\), and \(\Omega\) is positive definite, \(W\xrightarrow{d}\chi^2_k\), where \(k=p(p+1)/2\).

To justify the calibration, first suppose \(\Omega\) were known. Under the null, the vector \(\sqrt n\,d\) converges in distribution to \(Z\sim N_k(0,\Omega)\). Since \(\Omega\) is positive definite, \(\Omega^{-1/2}Z\sim N_k(0,I_k)\). Its squared Euclidean length is the sum of squares of \(k\) independent standard normal variables, and so has a \(\chi^2_k\) distribution. Equivalently,

n\,d^\mathsf{T}\Omega^{-1}d = (\sqrt{n}\,d)^\mathsf{T}\Omega^{-1}(\sqrt{n}\,d) \xrightarrow{d}\chi^2_k.

Replacing \(\Omega\) by a consistent \(\hat{\Omega}\) preserves this limit by Slutsky’s theorem: matrix inversion is continuous at a positive-definite matrix, so \(\hat{\Omega}^{-1}\xrightarrow{p}\Omega^{-1}\). The asymptotic reference distribution is therefore \(\chi^2_k\). At significance level \(\alpha\), reject when \(W\) exceeds the \(1-\alpha\) quantile of that distribution. \(\square\)

Worked Example: A Diagonal Influence Covariance

Consider a two-dimensional test with \(\Sigma_0=\begin{pmatrix}4&1\\1&2\end{pmatrix}\), \(n=30\), and robust estimate \(\hat{\Sigma}=\begin{pmatrix}4.6&0.7\\0.7&2.4\end{pmatrix}\). Suppose the estimated asymptotic covariance of the ordered distinct entries is \(\hat{\Omega}=\operatorname{diag}(1,0.5,1.5)\). These are hypothetical values from the estimator’s influence-function covariance calculation.

The ordering is \((11,21,22)\), so \(d=(4.6-4,\;0.7-1,\;2.4-2)^\mathsf{T}=(0.6,-0.3,0.4)^\mathsf{T}\). Since \(\hat{\Omega}\) is diagonal, its inverse is \(\operatorname{diag}(1,2,2/3)\). Hence

W=30\left(0.6^2+\frac{(-0.3)^2}{0.5}+\frac{0.4^2}{1.5}\right) =30(0.36+0.18+0.1067) \approx 19.40.

Here \(k=3\). At the 5% level the \(\chi^2_3\) critical value is approximately \(7.815\), so this test rejects the specified covariance matrix. This conclusion uses the joint robust covariance estimate, not a separate unadjusted test for each entry.

Worked Example: Accounting for Correlation Between Entries

Keep \(p=2\), so \(k=3\), and suppose \(n=20\) and the ordered discrepancy is \(d=(0.4,0.3,-0.2)^\mathsf{T}\). This time let the estimated asymptotic covariance be

\hat{\Omega}= \begin{pmatrix} 1&0.2&0\\ 0.2&1&0\\ 0&0&0.5 \end{pmatrix}.

The upper-left block has determinant \(1-0.2^2=0.96\), so its inverse is \(\frac{1}{0.96}\begin{pmatrix}1&-0.2\\-0.2&1\end{pmatrix}\). The third diagonal entry of the inverse is \(2\). Therefore,

d^\mathsf{T}\hat{\Omega}^{-1}d = \frac{0.4^2-2(0.2)(0.4)(0.3)+0.3^2}{0.96} +\frac{(-0.2)^2}{0.5} = \frac{0.202}{0.96}+0.08 \approx 0.2904.

Thus \(W=20(0.2904)\approx5.81\), below the 5% critical value \(7.815\) for \(\chi^2_3\). The joint test does not reject at that level. The off-diagonal covariance \(0.2\) matters: the joint quadratic form weights combinations of entry discrepancies, rather than simply summing three standardized squares.

Worked Example: Matching the Covariance Scale

Suppose a procedure reports \(\hat{V}\), an estimate of the covariance of \(\operatorname{vech}(\hat{\Sigma})\) itself, with \(\hat{V}=\hat{\Omega}/n\). For a discrepancy vector \(d\), the Wald statistic can be written either as \(n\,d^\mathsf{T}\hat{\Omega}^{-1}d\) or as \(d^\mathsf{T}\hat{V}^{-1}d\). Indeed, \(\hat{V}^{-1}=n\hat{\Omega}^{-1}\), so the two expressions agree exactly.

For example, if \(n=30\) and \(\hat{\Omega}\) is the diagonal matrix in the first numerical example, then \(\hat{V}=\operatorname{diag}(1/30,0.5/30,1.5/30)\). Using \(\hat{V}\) in \(n\,d^\mathsf{T}\hat{V}^{-1}d\) would multiply by \(n\) twice and produce a statistic 30 times too large. The covariance scale and the leading factor in the test must be paired consistently.

Assumptions, Interpretation, and Common Pitfalls

This is an asymptotic test: its calibration is justified as \(n\) grows with \(p\) fixed and the stated regularity conditions satisfied. Robustness of the scatter estimate does not make the chi-square approximation exact in small samples, nor does it make the test distribution-free. The influence-function covariance must reflect the sampling distribution under the null model and the actual estimator, including any estimated location, tuning constants, and consistency corrections.

A particularly important distinction is between shape and covariance. Some robust procedures estimate scatter only up to an unknown scale, or their target under a nonnormal model is not the population covariance. A test of a specified covariance matrix requires a procedure calibrated to target covariance under the assumptions being used. Applying the Wald formula to an uncalibrated shape estimate does not silently turn it into a covariance estimate.

Ignoring dependence among entries. The \(k\) distinct entries can have correlated influence contributions. Use the full \(\hat{\Omega}\), not only its diagonal, unless diagonality is justified.
Using the wrong ordering. If \(d\) is arranged in one order but \(\hat{\Omega}\) in another, the quadratic form tests mismatched quantities. Fix the \(\operatorname{vech}\) convention and use it consistently.
Confusing covariance scales. If the matrix estimates the covariance of one influence contribution, multiply the quadratic form by \(n\). If it estimates the covariance of the estimator itself, do not multiply by \(n\) again.
Testing a different target. A shape test or a test of only variances is not the full null \(H_0:\Sigma=\Sigma_0\). State which entries and which population target are being tested.

Key takeaway: A robust covariance-matrix test is a joint Wald test of the \(p(p+1)/2\) distinct entries. Its validity depends on a covariance-calibrated scatter estimate, a correct joint asymptotic covariance for those entries, consistent estimation of that covariance, and careful scale and ordering conventions.

Check Your Understanding

Answer each question by referring to the joint asymptotic distribution and the test construction.

How many distinct entries are tested when \(p=5\), and what is the corresponding asymptotic chi-square degrees of freedom?
Why does the covariance matrix \(\Omega\) generally need off-diagonal entries?
If \(\hat{\Omega}\) estimates the covariance of one influence contribution, where does the factor \(n\) appear in the Wald statistic?
What is the difference between testing a covariance matrix and testing an unscaled shape matrix?
State two assumptions needed for the chi-square calibration of the robust Wald statistic.