LetX=(Xij)=(X1, ...,Xn)’,X’i=(Xi1, ...,Xip)’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ2=ϱ
be the squared multiple correlation coefficient between the first and the remainingp2+p3=p−1 components of eachXi. We have considered here the problem of testingH0:ϱ2=0 against the alternativesH1:ϱ
>0 on the basis ofX andn1 additional observationsY1 (n1×1) on the first component,n2 observationsY2(n2×p2) on the followingp2 components andn3 additional observationsY3(n3×p3) on the lastp3 components and we have derived here the locally minimax test ofH0 againstH1 when ϱ
→0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.