Recently, Su and Cook proposed a dimension reduction technique called the inner envelope which can be substantially more efficient than the original envelope or existing dimension reduction techniques for multivariate regression. However, their technique relied on a linear model with normally distributed error, which may be violated in practice. In this work, we propose a semiparametric variant of the inner envelope that does not rely on the linear model nor the normality assumption. We show that our proposal leads to globally and locally efficient estimators of the inner envelope spaces. We also present a computationally tractable algorithm to estimate the inner envelope. Our simulations and real data analysis show that our method is both robust and efficient compared to existing dimension reduction methods in a diverse array of settings.

翻译：最近,Su和Cook提出了一种叫“内封”的减少维度技术,它比原先的封套或现有的多变量回归的减少维度技术效率要高得多。然而,它们的技术依赖于一个通常分布错误的线性模型,在实践中可能会被违反。在这项工作中,我们提出了一个不依赖线性模型或正常假设的内部封套的半参数变量。我们表明,我们的提议导致对内封空间进行全球和地方高效的测算。我们还提出了一个可计算可移植的算法来估计内封。我们的模拟和真实数据分析表明,我们的方法与多种环境中的现有减少维度方法相比是稳健和有效的。