This paper is concerned with the determination of whether two multivariate normal mean vectors differ by a multiplicative factor or by a multiplicative factor plus a displacement along a specified direction. These considerations arise, for example, in the evaluation of drugs, where two drugs affecting the same symptoms are considered equivalent if they can be made equal by a change in dosage. Maximum likelihood estimators and likelihood ratio tests for the relevant parameters are considered.
"Estimation and Testing for Differences in Magnitude or Displacement in the Mean Vectors of Two Multivariate Normal Populations." Ann. Math. Statist. 43 (2) 455 - 467, April, 1972. https://doi.org/10.1214/aoms/1177692626