The linear mixed-effects model (LMM) is widely used in the analysis of clustered or longitudinal data. This paper aims to address analytic challenges arising from estimation and selection in the application of the LMM to high-dimensional longitudinal data. We develop a doubly regularized approach in the LMM to simultaneously select fixed and random effects. On the theoretical front, we establish large sample properties for the proposed method under the high-dimensional setting, allowing both numbers of fixed effects and random effects to be much larger than the sample size. We present new regularity conditions for the diverging rates, under which the proposed method achieves both estimation and selection consistency. In addition, we propose a new algorithm that solves the related optimization problem effectively so that its computational cost is comparable with that of the Newton-Raphson algorithm for maximum likelihood estimator in the LMM. Through simulation studies we assess performances of the proposed regularized LMM in both aspects of variable selection and estimation. We also illustrate the proposed method by two data analysis examples.

中文翻译：

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高维纵向数据线性混合效应模型中的双正则化估计和选择

线性混合效应模型 (LMM) 广泛用于聚类或纵向数据的分析。本文旨在解决将 LMM 应用于高维纵向数据时估计和选择所产生的分析挑战。我们在 LMM 中开发了一种双重正则化方法，以同时选择固定和随机效应。在理论方面，我们在高维设置下为所提出的方法建立了大样本属性，允许固定效应和随机效应的数量远大于样本量。我们为发散率提出了新的规律性条件，在这种条件下，所提出的方法实现了估计和选择的一致性。此外，我们提出了一种有效解决相关优化问题的新算法，使其计算成本与 LMM 中最大似然估计的 Newton-Raphson 算法相当。通过模拟研究，我们评估了所提出的正则化 LMM 在变量选择和估计两个方面的性能。我们还通过两个数据分析示例说明了所提出的方法。