The aim of this paper is to conduct a systematic and theoretical analysis of estimation and inference for a class of functional mixed effects models (FMEM). Such 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 are derived. 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.

中文翻译：

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FMEM：纵向功能响应的功能混合效应模型

本文的目的是对一类功能混合效应模型 (FMEM) 的估计和推理进行系统和理论分析。此类 FMEM 由表征纵向功能响应与感兴趣的协变量之间关联的固定效应和捕获纵向功能响应的时空相关性的随机效应组成。我们提出了细化固定效应函数的局部线性估计，并建立了它们的弱收敛性以及每个固定效应函数的同时置信区间。我们提出了对不同系数函数的线性假设的全局检验，并导出了原假设下的相关渐近分布，并导出了备择假设下的渐近幂。我们还建立了估计的时空协方差算子及其相关特征值和特征函数的收敛率。我们进行了广泛的模拟，并将我们的方法应用于来自国家自闭症研究数据库的白质纤维数据集，以检查所提出的估计和推理程序的有限样本性能。