In the analysis of longitudinal or functional data, a time-varying additive model (tvAM) has been introduced that is effective at avoiding the curse of dimensionality and capturing dynamic features. The present article focuses on the unified two-step estimators of a tvAM with sparse or dense longitudinal or functional data. It is proved that the two-step estimators have the same asymptotic distribution as that of oracle estimators. Furthermore, a unified convergence theory is established, based on which a unified inference is proposed without deciding whether the data are sparse or dense. Also, a testing statistic that can adapt to the sparse and dense cases in a unified framework is proposed to check whether the bivariate nonparametric functions are time varying, and the asymptotic distribution of the proposed test statistic is derived. Simulation studies are conducted to assess the finite-sample performance of the proposed model and methods, and two different types of data are considered to illustrate the proposed method.

中文翻译：

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具有时变加性结构的纵向/功能数据模型的有效推断

在纵向或功能数据分析中，引入了时变加性模型 (tvAM)，可有效避免维度灾难和捕获动态特征。本文重点关注具有稀疏或密集纵向或功能数据的 tvAM 的统一两步估计器。证明了两步估计量与预言机估计量具有相同的渐近分布。此外，建立了统一收敛理论，在此基础上提出统一推理，而不用决定数据是稀疏还是密集。此外，提出了一种可以在统一框架中适应稀疏和密集情况的测试统计量来检查二元非参数函数是否是时变的，并推导出所提出的测试统计量的渐近分布。