Robust Sphericity and Shape Tests

Shape Hypotheses Separate Geometry from Scale

A scatter matrix describes both the orientation and relative spread of a multivariate distribution, as well as its overall magnitude. A test of shape asks about the first two features while allowing the overall scale to vary. Sphericity is the special case in which all directions have equal scatter. A specified-shape test asks whether the scatter has the geometry of a given positive-definite matrix, up to a positive multiplier.

In robust inference, the object being tested is a robust scatter functional: a population matrix defined by a chosen robust procedure, such as an M-estimator functional. The test therefore concerns that functional, not automatically the ordinary covariance matrix. Its asymptotic covariance must also be the one appropriate to the robust estimator used.

The scale \(c\) is a nuisance parameter: its value is not part of the claim about shape. For the specified-shape hypothesis, define the whitened scatter \(A=S^{-1/2}\Sigma(F)S^{-1/2}\), where \(S^{-1/2}\) is the symmetric inverse square root of \(S\). Then \(\Sigma(F)=cS\) exactly when \(A=cI_p\). Sphericity is obtained by taking \(S=I_p\).

This formulation makes the geometry explicit. Whitening maps the proposed shape to a sphere, so the test checks whether the whitened scatter has no directional differences. Multiplying \(\Sigma(F)\) by any positive constant changes its overall size but not whether it is spherical.

Independent Shape Restrictions

A symmetric \(p\times p\) matrix has \(p(p+1)/2\) free entries. Requiring it to be a multiple of the identity imposes \(p(p+1)/2-1\) independent restrictions: the diagonal entries must agree, and the off-diagonal entries must be zero. The one dimension left unrestricted is the overall scale.

For a convenient formal representation, use the Frobenius inner product \(\langle U,V\rangle=\operatorname{tr}(U^\mathsf{T}V)\) on symmetric matrices. Choose matrices \(B_1,\ldots,B_q\) forming an orthonormal basis of the traceless symmetric matrices, where \(q=p(p+1)/2-1\). Thus each \(B_a\) is symmetric, \(\operatorname{tr}(B_a)=0\), and \(\operatorname{tr}(B_aB_b)\) equals 1 if \(a=b\) and 0 otherwise. Define the shape coordinates

These coordinates record all departures from a scalar matrix. To see why, every symmetric \(A\) has a unique orthogonal decomposition into a scalar part and a traceless part:

Consequently, \(A=cI_p\) for some \(c>0\) if and only if every \(\beta_a=0\). The number of restrictions is \(q\), not \(p(p+1)/2\): the trace, which determines the scale, is deliberately not tested.

The Asymptotic Wald Test

Let \(\widehat{\Sigma}\) estimate the robust scatter functional, and set \(\widehat{A}=S^{-1/2}\widehat{\Sigma}S^{-1/2}\). Define \(\widehat{\beta}_a=\operatorname{tr}(B_a\widehat{A})\), and collect these coordinates in the \(q\)-vector \(\widehat{\beta}\). The covariance needed for inference is the asymptotic covariance of these robust-estimator coordinates, not the scatter matrix itself.

The theorem requires the stated asymptotic normality and a consistent covariance estimate; it does not assert that every robust scatter procedure has those properties without conditions. In applications, they are justified by the estimator’s regularity theory. For example, asymptotic linearization typically requires suitable differentiability of the functional, finite second moments for its influence function, and a nonsingular limiting covariance. The estimated covariance must account for estimating location as well if the scatter procedure estimates location jointly.

Proof. Under the null, the shape coordinates are zero. By the assumed convergence, \(Z_n=\sqrt{n}\widehat{\beta}\xrightarrow{d}Z\), where \(Z\sim N_q(0,\Omega)\). Consistency of \(\widehat{\Omega}\) and positive definiteness of \(\Omega\) imply \(\widehat{\Omega}^{-1}\xrightarrow{p}\Omega^{-1}\). Therefore, by Slutsky’s theorem and continuity of the quadratic-form map,

The vector \(\Omega^{-1/2}Z\) has the \(N_q(0,I_q)\) distribution. Its squared Euclidean length is the sum of squares of \(q\) independent standard normal variables, which has a chi-square distribution with \(q\) degrees of freedom. Hence \(W\xrightarrow{d}\chi_q^2\). \(\square\)

A large value of \(W\) is evidence against the shape hypothesis. For an approximate level-\(\alpha\) test, reject when \(W\) exceeds the upper \(\alpha\) critical value of \(\chi_q^2\). This is an asymptotic calibration; its quality in a particular sample depends on the estimator and the data-generating distribution.

Testing a Specified Shape

For a specified \(S\), the same construction applies after whitening. Under the null, \(\widehat{A}=S^{-1/2}\widehat{\Sigma}S^{-1/2}\) estimates a scalar multiple of \(I_p\). The coordinates \(\widehat{\beta}_a=\operatorname{tr}(B_a\widehat{A})\) are then tested jointly. Their estimated covariance must be calculated for this transformed estimator, including the effect of the whitening transformation.

An equivalent way to describe the target is to normalize the whitened scatter by its average eigenvalue: under the null, \(A/\{\operatorname{tr}(A)/p\}=I_p\). This representation makes the removal of scale visible, but the Wald test uses the uncertainty in the estimated shape coordinates. Simply comparing a normalized point estimate with \(I_p\) does not account for that uncertainty.

Interpretation and Common Pitfalls

These tests are useful when the scientific question concerns relative directional spread rather than the absolute scale of measurements. A prespecified \(S\) can encode a scientifically meaningful geometric pattern; the test asks whether the robust scatter follows that pattern, allowing its magnitude to differ. A sphericity test asks the more restrictive geometric question of equal spread in every direction.

• Counting scale as a restriction. Treating all \(p(p+1)/2\) matrix entries as constrained gives the wrong degrees of freedom. Scale is free, so the correct number is \(p(p+1)/2-1\).
• Testing ordinary covariance by accident. Substituting a sample covariance estimate changes the estimand and can change sensitivity to outliers. The test concerns the robust scatter functional actually used.
• Using the scatter estimate as its own standard error. \(\widehat{\Sigma}\) describes estimated spread in the observations; \(\widehat{\Omega}\) describes sampling uncertainty in the estimated shape coordinates. They are different matrices.
• Ignoring uncertainty in the target. The theorem treats \(S\) as fixed. If the target shape is estimated from data, its uncertainty and dependence on \(\widehat{\Sigma}\) must be included in the joint inference; treating it as fixed is not justified automatically.

The null hypothesis is about the robust functional, not an unrestricted claim about the entire distribution. A distribution may have nonnormal tails and still yield a spherical robust scatter functional. Conversely, a rejection identifies a discrepancy in that functional’s shape; it does not by itself identify which directions, observations, or distributional features produced it.