INVERSE REGRESSION FOR LONGITUDINAL DATA

• Published in: The Annals of statistics 2014-04, Vol.42 (2), p.563-591
• Main Authors: Jiang, Ci-Ren, Yu, Wei, Wang, Jane-Ling
• Format: Article
• Language: English
• Subjects: 62G05; 62G08; 62G20; Asymptotic methods; Covariance; Covariance operator; Data analysis; Data smoothing; dimension reduction; Dimensionality reduction; Eigenfunctions; Eigenvalues; Estimating techniques; Estimation methods; Fecundity; functional data analysis; Inverse problems; Linear regression; local polynomial smoothing; Longitudinal data; Mathematical models; Multivariate analysis; Regression analysis; regularization; sparse data; Studies
• ISSN: 0090-5364; 2168-8966
• DOI: 10.1214/13-AOS1193

Sliced inverse regression (Duan and Li [Ann. Statist. 19 (1991) 505-530], Li [J. Amer. Statist. Assoc. 86 (1991) 316-342]) is an appealing dimension reduction method for regression models with multivariate covariates. It has been extended by Ferré and Yao [Statistics 37 (2003) 475-488, Statist. Sinica 15 (2005) 665-683] and Hsing and Ren [Ann. Statist. 37 (2009) 726-755] to functional covariates where the whole trajectories of random functional covariates are completely observed. The focus of this paper is to develop sliced inverse regression for intermittently and sparsely measured longitudinal covariates. We develop asymptotic theory for the new procedure and show, under some regularity conditions, that the estimated directions attain the optimal rate of convergence. Simulation studies and data analysis are also provided to demonstrate the performance of our method.