Tests on multiple correlation coefficient and multiple partial correlation coefficient

Let X1(1 - 1), X2(1 - p2), and X3(1 - p3) be three sets of random variables distributed jointly as a normal distribution. Let [varrho]1.23 and [varrho]1.2 be the multiple correlation coefficients between X1 and (X2, X3) and between X1 and X2, respectively. Invariant tests for the following four testing problems are considered: [varrho]1.23 = 0 vs [varrho]1.23 > 0, [varrho]1.23 = 0, vs [varrho]1.23 > 0, [varrho]1.23 = [varrho]1.2 vs [varrho]1.23 > [varrho]1.2, [varrho]1.2 = 0 vs [varrho]1.2 > 0. The joint distribution of the sample multiple correlation coefficients R1.23 and R1.2 is derived.