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无处不在，局部线性平滑技术是最常用的方法。Zhang与Wang（2016）探讨了该估计的不同类型渐近性质，揭示了基于每个个体平均采样频率$T$与个体数量$n$的相对顺序所产生的有趣相变现象，将数据分为三类：“稀疏”、“半密集”和“超密集”。在日益常见的高维场景中，函数变量数量$p$相对于$n$较大时，我们通过推导局部线性平滑器的全面浓度不等式，从非渐近视角重新审视这一开放问题。除了本身具有理论价值外，我们的非渐近结果还推导出$L_2$收敛的逐元最大速率和一致收敛速率，这些结果在$p$随$n$以及可能的$T$呈指数增长时，成为进一步收敛分析的基础性重要工具。考虑高维效应中额外的$\log p$项后，我们进一步研究了从稀疏到半密集再到超密集的高维函数型数据中的标度化相变现象及相应的逐元最大速率。最后，通过数值实验验证了所建立的理论性质。