Summary: Let \(X\) and \(Y\) be \(d\)-dimensional random vectors having elliptically symmetric distributions. Call \(X\) and \(Y\) affinely equivalent if \(Y\) has the same distribution as \(AX+b\) for some nonsingular \(d\times d\)-matrix \(A\) and some \(b \in \mathbb R^d\). This paper studies a class of affine invariant tests for affine equivalence under certain moment restrictions. The test statistics are measures of discrepancy between the empirical distributions of the norm of suitably standardized data.