Gradient‐based approach to sufficient dimension reduction with functional or longitudinal covariates
针对响应变量为实数值、协变量为函数型或纵向数据的回归问题，提出一种基于条件分布函数梯度的充分降维新方法，仅需光滑性条件，通过函数主成分分析即可实现估计。
Abstract In this paper, we focus on the sufficient dimension reduction problem in regression analysis with real‐valued response and functional or longitudinal covariates. We propose a new method based on gradients of the conditional distribution function to estimate the sufficient dimension reduction subspace. While existing inverse‐regression‐type methods relies on a linearity condition, our method is based on the gradient of conditional distribution function and its validity only requires smoothness conditions on the population parameters. Practically, the proposed estimator can be obtained by standard algorithm of functional principal component analysis. The proposed method is demonstrated through extensive simulations and two empirical examples.