Analysis of Double Single Index Models
提出一种双单指标模型，用于捕捉多变量响应与多变量协变量之间的非线性关联，并研究其估计、推断及渐近性质，适用于多元数据分析。
Motivated from problems in canonical correlation analysis, reduced rank regression and sufficient dimension reduction, we introduce a double dimension reduction model where a single index of the multivariate response is linked to the multivariate covariate through a single index of these covariates, hence the name double single index model. Since nonlinear association between two sets of multivariate variables can be arbitrarily complex and even intractable in general, we aim at seeking a principal one-dimensional association structure where a response index is fully characterized by a single predictor index. The functional relation between the two single-indices is left unspecified, allowing flexible exploration of any potential nonlinear association. We argue that such double single index association is meaningful and easy to interpret, and the rest of the multi-dimensional dependence structure can be treated as nuisance in model estimation. We investigate the estimation and inference of both indices and the regression function, and derive the asymptotic properties of our procedure. We illustrate the numerical performance in finite samples and demonstrate the usefulness of the modeling and estimation procedure in a multi-covariate multi-response problem concerning concrete.