In a recent article by Vexler [10], a test of bivariate normality is proposed. The statistic of that test is a function of the symmetric version of, so-called, sample configuration. Let \({\mathbf {X}}\) be a (p, n) data matrix with \(n> p\). Under the null hypothesis, the columns \({\mathbf {X}}_1,\dots , {\mathbf {X}}_n\) of \({\mathbf {X}}\) form a n-element i.i.d. sample from a p-variate normal distribution \(N_p({\mathbf {m}},\varvec{\Sigma })\), with \({\mathbf {m}}\in \mathbb {R}^p\) and a positive definite covariance matrix \(\varvec{\Sigma }\). Both \({\mathbf {m}}\) and \(\varvec{\Sigma }\) are considered unknown. Let \(\bar{{\mathbf {X}}}=n^{-1}\sum _{i=1}^n {\mathbf {X}}_i\) be the sample mean and \({\mathbf {S}}=n^{-1}({\mathbf {X}}-\bar{{\mathbf {X}}}{\mathbf {1}}_n^T)({\mathbf {X}}-\bar{{\mathbf {X}}}{\mathbf {1}}_n^T)^T\) be the sample covariance matrix, where \({\mathbf {1}}_n^T=(1,\dots , 1)\in \mathbb {R}^n\). The symmetric sample configuration, also called the matrix of scaled residuals, is defined as \({\mathbf {Z}}:={\mathbf {S}}^{-1/2}({\mathbf {X}}-\bar{{\mathbf {X}}}{\mathbf {1}}_n^T)\), where \({\mathbf {S}}^{-1/2}\) stands for the symmetric, positive definite square root of the inverse of \({\mathbf {S}}\) (with \(n > p\), \({\mathbf {S}}\) is almost surely non-singular, cf. [2]). On page 5 of [10], the author describes a Monte Carlo experiment aimed to “experimentally confirm” that the null distribution of the test statistic does not depend on \(({\mathbf {m}},\varvec{\Sigma })\), i.e., that \({\mathbf {Z}}\) is ancillary. In what follows, we show that, although \({\mathbf {Z}}\) is not invariant w.r.t. standard groups of data transformations usually considered in the context of testing multivariate normality, the null distribution of \({\mathbf {Z}}\) does not, indeed, depend on \(({\mathbf {m}},\varvec{\Sigma })\), so that the Monte Carlo study in [10] was not necessary.

More generally, a sample configuration may also be defined as \({\mathbf {C}}:={\mathbf {L}}^{-1}({\mathbf {X}}-\bar{{\mathbf {X}}}{\mathbf {1}}_n^T)\), with any matrix \({\mathbf {L}}\) such that \({\mathbf {S}}=\mathbf {LL}^T\). It is well known that such \({\mathbf {C}}\) is defined up to left multiplication by a rotation matrix. Although some authors seem to suggest that \({\mathbf {C}}\) is always ancillary (e.g., [4, Sect. 2]), it is not always true and the distribution and properties of \({\mathbf {C}}\) depend on the choice of \({\mathbf {L}}\).

In what follows, two groups of transformations will be of interest: the group \({\mathcal {G}}\) of affine transformations \({\mathbf {X}}\rightarrow \mathbf {AX}+{\mathbf {b}}{\mathbf {1}}_n^T\), with nonsingular (p, p) matrices \({\mathbf {A}}\) and with \({\mathbf {b}}\in \mathbb {R}^p\), and its subgroup \({\mathcal {G}}^*\), with \({\mathbf {A}}\in UT(p)\) - the group of upper triangular matrices with positive diagonal. Various questions related to invariant tests for multivariate normality were discussed in [7]. It was shown, in particular, that if \({\mathbf {S}}=\mathbf {LL}^T\) with \({\mathbf {L}}\in UT(p)\), then \({\mathbf {B}}:={\mathbf {L}}^{-1}({\mathbf {X}}-\bar{{\mathbf {X}}}{\mathbf {1}}_n^T)\) is a maximal invariant w.r.t. \({\mathcal {G}}^*\) and, hence, \({\mathbf {B}}\) is an ancillary statistic for \(({\mathbf {m}},\varvec{\Sigma })\). Additionally, the distribution of \({\mathbf {B}}\) is invariant w.r.t. left multiplication of \({\mathbf {B}}\) by fixed, orthogonal (p, p) matrices. This follows directly from [1]. It is shown there (in Sect. 5) that \({\mathbf {B}}={\mathbf {B}}_u{\mathbf {D}}\), where \({\mathbf {B}}_u={\mathbf {B}}_u({\mathbf {X}})\) is a \((p, n-1)\) random matrix, \({\mathbf {D}}\) is a specific, fixed \((n-1,n)\) matrix and the distribution of \({\mathbf {B}}_u\) is the Haar measure on the group \({\mathcal {SO}}(n-1)\) of \((n-1,n-1)\) orthogonal matrices, marginalized to the first p rows. This distribution is clearly invariant w.r.t. left multiplication of \({\mathbf {B}}_u\) by orthogonal (p, p) matrices, say \({\mathbf {R}}\), because this corresponds to left multiplication in \({\mathcal {SO}}(n-1)\) by orthogonal, block diagonal matrices \(\mathrm {diag}({\mathbf {R}}, {\mathbf {I}}_{n-1-p})\), where \({\mathbf {I}}_{n-1-p}\) stands for the identity matrix.

Any configuration \({\mathbf {C}}\) is related to \({\mathbf {B}}\) through \({\mathbf {C}}=\mathbf {MB}\), with an orthogonal matrix \({\mathbf {M}}\). Invariance and ancillarity of \({\mathbf {C}}\) depend on the way \({\mathbf {M}}\) is defined as a function of \({\mathbf {X}}\).

If \({\mathbf {M}}\) is stochastically independent of \({\mathbf {B}}\), then the invariance of the distribution of \({\mathbf {B}}\) w.r.t. left multiplication by fixed, orthogonal matrices leads via a standard conditioning argument to the conclusion, that the distributions of \({\mathbf {C}}\) and \({\mathbf {B}}\) are identical. As an example, with \({\mathbf {L}}\in UT(p)\), since \({\mathbf {S}}=\mathbf {LL}^T={\mathbf {S}}^{1/2}{\mathbf {S}}^{1/2}\), the symmetric configuration satisfies \({\mathbf {Z}}=\mathbf {MB}\), with an orthogonal matrix \({\mathbf {M}}={\mathbf {S}}^{-1/2}{\mathbf {L}}\). Since \({\mathbf {M}}\) is a function of \({\mathbf {S}}\) only (because both \({\mathbf {L}}\) and \({\mathbf {S}}^{1/2}\) are) and \((\bar{{\mathbf {X}}}, {\mathbf {S}})\) is sufficient and complete in the Gaussian model, \({\mathbf {M}}\) and \({\mathbf {B}}\) are stochastically independent by the Basu theorem, and \({\mathbf {Z}}\) has the same distribution as \({\mathbf {B}}\) and is, hence, ancillary. The same conclusion holds true, if \({\mathbf {M}}\) is any function of \((\bar{{\mathbf {X}}},{\mathbf {S}})\).

If \({\mathbf {M}}\) is a function of \({\mathbf {B}}\) only, then \({\mathbf {C}}=\mathbf {MB}\) is clearly ancillary as a function of the ancillary statistic \({\mathbf {B}}\), with a distribution possibly different from that of \({\mathbf {B}}\). \({\mathbf {C}}\) is then invariant w.r.t. \({\mathcal {G}}^*\), but not necessarily w.r.t. \({\mathcal {G}}\).

If, moreover, \({\mathbf {M}}\) is a function of \(\mathbf {BB}^T\), i.e., of the Mahalanobis distances and angles, which is a maximal invariant w.r.t. \({\mathcal {G}}\) (see, e.g., [3]), then \({\mathbf {C}}=\mathbf {MB}\) is also invariant w.r.t. \({\mathcal {G}}\). For an example, see, e.g., [7, Th. 3]. As one of the columns of the configuration \({\mathbf {C}}\) constructed there is always proportional to \((1,0,\dots ,0)^T\), its distribution is clearly different from that of \({\mathbf {B}}\).

Finally, if \({\mathbf {M}}\) depends on \({\mathbf {X}}\) in an arbitrary way, i.e., not necessarily through \({\mathbf {B}}\) only, then \({\mathbf {C}}=\mathbf {MB}\) does not have to be ancillary. \({\mathbf {M}}\) can be, e.g., a rotation that makes the first column \({\mathbf {B}}_1\) of \({\mathbf {B}}\) parallel to \({\mathbf {X}}_1\), randomly chosen from the Haar distribution on \({\mathcal {SO}}(p)\), conditionally on \(\mathbf {MB}_1\) being parallel to \({\mathbf {X}}_1\). Then, clearly, the expectation of the first column of \({\mathbf {C}}\) is proportional to \({\mathbf {m}}\), and \({\mathbf {C}}\) is not ancillary.

It should be noted that many tests for multivariate normality are based on the symmetric sample configuration \({\mathbf {Z}}\) and not all of them are \({\mathcal {G}}\) invariant, even if a strong case for this property is sometimes made, e.g., at the beginning of Sect. 2 in [5]. Invariance w.r.t. \({\mathcal {G}}^*\) only may also be of interest, if directed tests for multivariate normality against some restricted alternatives are considered, as in [6, 7], or if the goal is maximin testing between some neighborhoods of transformation families of distributions (see, e.g., [8]).

If the sample covariance matrix \({\mathbf {S}}\) is replaced in the definition of the sample configuration with another affine equivariant estimator \({\mathbf {V}}({\mathbf {X}})\) that satisfies \({\mathbf {V}}(\mathbf {AX}+{\mathbf {b}}{\mathbf {1}}_n^T)=\mathbf {AV}({\mathbf {X}}){\mathbf {A}}^T\) for nonsingular \({\mathbf {A}}\) and \({\mathbf {b}}\in \mathbb {R}^p\), e.g., a robust estimator studied in [9], then \({\mathbf {B}}\) remains a maximal invariant w.r.t. \({\mathcal {G}}^*\), but the rows of \({\mathbf {B}}\) are not orthogonal, as it was the case with \({\mathbf {S}}\). As the distribution of \({\mathbf {B}}\) does not have to be, in that case, invariant under left multiplication by orthogonal matrices, the discussion of the distributional issues does not carry over to that more general case.