In order to further improve the efficiency of parameter estimation and reduce the number of estimated parameters, we adopt dimension reduction ideas of partial envelope model proposed by [Su and Cook, Partial envelopes for efficient estimation in multivariate linear regression, Biometrika 98 (2011) 133–146.] to center on some predictors of special interest. Based on the research results of [Cook et al., Envelopes and reduced-rank regression, Biometrika 102 (2015) 439–456.], we combine partial envelopes with reduced-rank regression to form reduced-rank partial envelope model which can reduce dimension efficiently. This method has the potential to perform better than both. Further, we demonstrate maximum likelihood estimators for the reduced-rank partial envelope model parameters, and exhibit asymptotic distribution and theoretical properties under normality. Meanwhile, we show selections of rank and partial envelope dimension. At last, under the normal and non-normal error distributions, simulation studies are carried out to compare our proposed reduced-rank partial envelope model with the other four methods, including ordinary least squares, reduced-rank regression, partial envelope model and reduced-rank envelope model. A real data analysis is also given to support the theoretic claims. The reduced-rank partial envelope estimators have shown promising performance in extensive simulation studies and real data analysis.

中文翻译：

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多元线性回归中降秩偏包络模型的有效估计

为了进一步提高参数估计的效率，减少估计参数的数量，我们采用了[Su和Cook，Partial envelopes for Effective Estimation in Multivariate Linear Regression，部分包络模型的降维思想，生物计量学 98(2011) 133–146.] 以一些特别感兴趣的预测为中心。基于[库克的研究结果等。, 包络和降秩回归,生物计量学 102(2015) 439–456.]，我们将部分包络与降秩回归相结合，形成降秩部分包络模型，可以有效地降维。这种方法有可能比两者都表现得更好。此外，我们展示了降秩部分包络模型参数的最大似然估计量，并展示了正态下的渐近分布和理论性质。同时，我们展示了等级和部分包络维度的选择。最后，在正态和非正态误差分布下，进行了仿真研究，将我们提出的降秩部分包络模型与普通最小二乘法、降秩回归、部分包络模型和降阶回归等四种方法进行了比较。秩信封模型。还给出了真实的数据分析来支持理论主张。