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Statistica Sinica 29 (2019), 2007-2033

FMEM: FUNCTIONAL MIXED-EFFECTS MODELS FOR
LONGITUDINAL FUNCTIONAL RESPONSES
Hongtu Zhu, Kehui Chen, Xinchao Luo, Ying Yuan and Jane-Ling Wang
The University of Texas MD Anderson Cancer Center, University of
Pittsburgh, Janssen R & D, LLC and University of California at Davis

Abstract: The aim of this study is to conduct a systematic and theoretical analysis of estimations and inferences for a class of functional mixed-effects models (FMEM). FMEMs consist of fixed effects that characterize the association between longitudinal functional responses and covariates of interest and random effects that capture the spatial-temporal correlations of longitudinal functional responses. We propose local linear estimates of refined fixed-effect functions and establish their weak convergence, along with a simultaneous confidence band for each fixed-effect function. We propose a global test for the linear hypotheses of varying coefficient functions and derive the associated asymptotic distribution under the null hypothesis and the asymptotic power under the alternative hypothesis. We also establish the convergence rates of the estimated spatial-temporal covariance operators and their associated eigenvalues and eigenfunctions. We conduct extensive simulations and apply our method to a white-matter fiber data set from a national database for autism research to examine the finite-sample performance of the proposed estimation and inference procedures.

Key words and phrases: Functional response, global test statistic, mixed effects, spatial-temporal correlation, weak convergence.

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