not maximal invariant for G $_{p}$, as xx ′ and ( xP ) ( xP $_{)}$′ = xPP ′ x ′ = xx ′ , where P is an arbitrary n × n orthogonal matrix, share the same λ ( x ) = λ ( xP ) although, in general, there is no p × p orthogonal matrix Q such that xP = Qx. Now, the joint density of the elements of X is By the factorization theorem, XX ′ is a sufficient statistic, and it is legitimate to restrict attention to XX ′ -measurable inference procedures. Left-hand orthogonal multiplications Qx of x yields, for xx ′ , a transformation of the form Qxx ′ Q ′ . When Q range over the family O p of p × p orthogonal matrices, those transformations also form a group, ˜ G $_{p}$, say, now acting on the space of p × p symmetric positive semidefinite real matrices of rank m . Clearly, λ ( x ) is maximal invariant for ˜ G $_{p}$, as xx ′ and yy ′ share the same eigenvalues if and only if yy ′ = Qxx ′ Q ′ for some p × p orthogonal matrix Q. Combining the principles of sufficiency and invariance thus leads to considering λ -measurable tests only. A similar reasoning applies in the case of unspecified σ $^{2}$, with a larger group combining multiplication by an arbitrary non-zero constant with the p × p left orthogonal transformations. Sufficiency and invariance then lead to restricting attention to µ -measurable tests. Proof of Proposition 5. With the same notation as above, write T = T ( X ) = XX ′ for the sufficient statistic. Consider an arbitrary invariant (under the group G p of left orthogonal transformations of R p × $^{n}$) test φ ( X ), and define ψ ( t ) = E ( φ ( X ) | T = t ). Then ψ ( T ) is a T -measurable test with the same size and power function as φ ( X ) . It follows from the proof of Theorem 6.5.3 (i) in Lehmann and Romano (2005) that 39 f ( n ) Σ ( x ) = (2 π $_{)}$− np/ 2 | Σ | − n/ $^{2}$exp { − 1 2 tr ( $_{Σ}$− 1 $_{xx}$′ ) } , x ∈ R p × $^{n}$.