Nonparametric estimation of the mean and covariance functions is ubiquitous in functional data analysis and local linear smoothing techniques are most frequently used. Zhang and Wang (2016) explored different types of asymptotic properties of the estimation, which reveal interesting phase transition phenomena based on the relative order of the average sampling frequency per subject $T$ to the number of subjects $n$, partitioning the data into three categories: ``sparse'', ``semi-dense'' and ``ultra-dense''. In an increasingly available high-dimensional scenario, where the number of functional variables $p$ is large in relation to $n$, we revisit this open problem from a non-asymptotic perspective by deriving comprehensive concentration inequalities for the local linear smoothers. Besides being of interest by themselves, our non-asymptotic results lead to elementwise maximum rates of $L_2$ convergence and uniform convergence serving as a fundamentally important tool for further convergence analysis when $p$ grows exponentially with $n$ and possibly $T$. With the presence of extra $\log p$ terms to account for the high-dimensional effect, we then investigate the scaled phase transitions and the corresponding elementwise maximum rates from sparse to semi-dense to ultra-dense functional data in high dimensions. Finally, numerical studies are carried out to confirm our established theoretical properties.

翻译：均值函数与协方差函数的非参数估计在函数型数据分析中普遍存在，局部线性平滑技术是最常用的方法。张和王（2016）探讨了该估计的不同类型渐近性质，基于每个受试者平均采样频率$T$与受试者数量$n$的相对阶数揭示了有趣的相变现象，将数据划分为三类："稀疏"、"半稠密"和"超稠密"。在日益常见的高维场景中，即函数变量数量$p$相对于$n$较大时，我们从非渐近视角重新审视这一开放问题，通过推导局部线性平滑器的全面浓度不等式展开研究。除了其自身的研究价值，我们的非渐近结果还导出了$L_2$收敛和一致收敛的逐元素最大速率，这为当$p$随$n$及可能$T$呈指数增长时的进一步收敛分析提供了基础重要工具。在引入额外$\log p$项以考虑高维效应后，我们进一步研究了从稀疏到半稠密再到超稠密高维函数型数据的尺度化相变及相应的逐元素最大速率。最后，通过数值研究验证了所建立的理论性质。