On the powers of tests for homogeneity of regression coefficient vectors under synchronised order restrictions

We consider a multivariate multiple linear regression model and study the problem of testing homogeneity of regression coefficient vectors under synchronised order restrictions when the covariance matrices are common but unknown. Synchronised order restrictions are the generalisation of the multivariate isotonic order restrictions. Synchronised order restricted test could be applied to a situation where the values of some parameters increase, those of some other parameters decrease, and those of the rest of the parameters have no restriction, simultaneously. For this problem, some test statistics were proposed and some inequalities among their powers were obtained in the past. This showed that the proposed test statistics may equally be good in terms of their powers. In the present paper, we mathematically prove that the strict inequalities hold among the powers of the test statistics. Thus we attain an exact comparison among the powers of the test statistics indicating more accurate and stronger results than those obtained in the past.