Accounting for Time Series Errors in Partially Linear Model With Single- or Multiple-Runs

Abstract

This article concerns statistical estimation of the partially linear model (PLM) for time course measurements, which are temporally correlated and allow multiple-runs for repeated measurements to enhance experimental accuracy without extending the number of time points within each trial. Such features arise naturally from biomedical data, for example, in brain fMRI, and call for special treatment beyond classical methods in either a purely nonparametric regression model or a PLM with independent errors. We develop a stepwise procedure for estimating the parametric and nonparametric components of the multiple-run PLM and making inference for parameters of interest, adaptive to either single- or multiple-run, in the presence of error temporal dependence. Simulation study and real fMRI data applications illustrate the computational simplicity and effectiveness of the proposed methods. Supplementary material for this article is available online.

Key Words:

• Autocorrelation matrix
• Difference-based method
• fMRI
• Matrix inverse
• Multiple testing
• Semiparametric model

Acknowledgments

The authors thank the Associate Editor and an anonymous referee for insightful comments. Zhang’s research is supported by the NSF grants DMS–1106586, DMS–1308872, and DMS--1521761, and Wisconsin Alumni Research Foundation. Han’s research is supported by the Scientific Research Foundation of Northeast Dianli University (No. BSJXM-201216).

Additional information

Notes on contributors

Chunming Zhang

Chunming Zhang, School of Mathematical Sciences, Nankai University, Tianjin 300071, China; Department of Statistics, University of Wisconsin, Madison, WI 53706 (E-mail: [email protected]). Yu Han, School of Science, Northeast Dianli University, Jilin, Jilin 132013, P.R. China (E-mail: [email protected]). Shengji Jia, Department of Statistics, University of Wisconsin, Madison, WI 53706 (E-mail: [email protected]).

Yu Han

Chunming Zhang, School of Mathematical Sciences, Nankai University, Tianjin 300071, China; Department of Statistics, University of Wisconsin, Madison, WI 53706 (E-mail: [email protected]). Yu Han, School of Science, Northeast Dianli University, Jilin, Jilin 132013, P.R. China (E-mail: [email protected]). Shengji Jia, Department of Statistics, University of Wisconsin, Madison, WI 53706 (E-mail: [email protected]).

Shengji Jia

Chunming Zhang, School of Mathematical Sciences, Nankai University, Tianjin 300071, China; Department of Statistics, University of Wisconsin, Madison, WI 53706 (E-mail: [email protected]). Yu Han, School of Science, Northeast Dianli University, Jilin, Jilin 132013, P.R. China (E-mail: [email protected]). Shengji Jia, Department of Statistics, University of Wisconsin, Madison, WI 53706 (E-mail: [email protected]).