D-optimal designs for multi-response linear models with two groups

In recent years, multi-response linear models have gained significant popularity in various statistical applications. However, the design aspects of multi-response linear models with group-wise considerations have received limited attention in the literature. This paper aims to thoroughly investigate -optimal designs for such models. Specifically, we focus on scenarios involving two groups, where the proportions of observations for each group can be arbitrarily selected or not. Two equivalence theorems are presented to elaborate the characterization of -optimal designs. Additionally, we delve into the admissibility of approximate designs and establish necessary conditions for a design to be deemed admissible. Several illustrative examples are addressed to demonstrate the application of the derived theoretical results.