The envelope model provides substantial efficiency gains over the standard multivariate linear regression by identifying the material part of the response to the model and by excluding the immaterial part. In this paper, we propose the enhanced response envelope by incorporating a novel envelope regularization term in its formulation. It is shown that the enhanced response envelope can yield better prediction risk than the original envelope estimator. The enhanced response envelope naturally handles high-dimensional data for which the original response envelope is not serviceable without necessary remedies. In an asymptotic high-dimensional regime where the ratio of the number of predictors over the number of samples converges to a non-zero constant, we characterize the risk function and reveal an interesting double descent phenomenon for the first time for the envelope model. A simulation study confirms our main theoretical findings. Simulations and real data applications demonstrate that the enhanced response envelope does have significantly improved prediction performance over the original envelope method.

翻译：信封模型通过确定对模型反应的物质部分和排除非物质部分,为标准的多变量线性回归提供了巨大的效率收益。在本文中,我们建议通过将新的信封正规化术语纳入设计中来增加响应封封包,表明增强响应封包比原信封估计值可以产生更好的预测风险。增强响应封包自然处理高维数据,而原始反应信封无需必要补救就无法使用这些数据。在一个无药可救的高维系统中,预测器数量与样本数量之比接近于非零常数,我们描述风险功能,并首次为信封模型揭示出有趣的双向下降现象。模拟研究证实了我们的主要理论结论。模拟和真实数据应用表明,增强反应信封确实大大改善了原信封方法的预测性能。