In this paper, we study low rank high dimensional multivariate linear models (LRMLM) for high dimensional multi-response data. We propose an intuitively appealing estimation approach, and develop an algorithm for implementation purposes. Asymptotic properties are established in order to justify the estimation procedure theoretically. Intensive simulation studies are also conducted to demonstrate performance when the sample size is finite, and a comparison is made with some popular methods from the literature. The results show the proposed estimator outperforms all of the alternative methods under various circumstances. Finally, using our suggested estimation procedure we apply the LRMLM to analyse an environmental data set and predict concentrations of PM2.5 at the locations concerned. The results illustrate how the proposed method provides more accurate predictions than the alternative approaches.

在本文中，我们研究了用于高维多响应数据的低秩高维多元线性模型（LRMLM）。我们提出了一种直观吸引人的估算方法，并为实现目的开发了一种算法。建立渐近性质是为了从理论上证明估计程序的合理性。还进行了密集的模拟研究，以证明样本量有限时的性能，并与文献中的一些流行方法进行了比较。结果表明，所提出的估计器在各种情况下均优于所有其他方法。最后，使用建议的估算程序，我们应用LRMLM分析环境数据集并预测相关位置的PM2.5浓度。