This paper deals with a multivariate extension of Friedman's χ2r-test. A rank permutation distribution and the large sample properties of the criterion are studied. The asymptotic relative efficiency (A.R.E.) for a sequence of translation alternatives is studied and bounds are given for certain special cases. It is shown that, under specified conditions, the A.R.E. of this test with respect to the likelihood ratio test is largest when the block dispersion matrices differ and can be greater than unity when the differences are large.