测量误差回归的替代降维：一个不变定律
On surrogate dimension reduction for measurement error regression: An invariance law
摘要：
考虑了一般的非线性回归问题，其中预测变量包含测量误差。最近发现，几种著名的降维方法，如OLS，SIR和pHd，可以对替代回归问题进行降维，从而对包含未观测真预测量的原回归问题产生相合估计。在本文中，我们建立了代理和原始降维空间之间的一般不变性定律，这意味着，至少在总体水平上，这两个降维问题实际上是等价的。因此，我们可以将现有的降维方法应用于测量误差回归问题。等价性对多元正态预测变量完全成立，对任意预测变量近似成立。我们还刻画了代理降维估计的收敛速率。最后，我们将几种降维方法应用于包含测量误差的真实和模拟数据集，以比较它们的性能。
英文摘要：
We consider a general nonlinear regression problem where the predictors contain measurement error. It has been recently discovered that several well-known dimension reduction methods, such as OLS, SIR and pHd, can be performed on the surrogate regression problem to produce consistent estimates for the original regression problem involving the unobserved true predictor. In this paper we establish a general invariance law between the surrogate and the original dimension reduction spaces, which implies that, at least at the population level, the two dimension reduction problems are in fact equivalent. Consequently we can apply all existing dimension reduction methods to measurement error regression problems. The equivalence holds exactly for multivariate normal predictors, and approximately for arbitrary predictors. We also characterize the rate of convergence for the surrogate dimension reduction estimators. Finally, we apply several dimension reduction methods to real and simulated data sets involving measurement error to compare their performances.
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