Litimein Ouahiba, Alshahrani Fatimah, Bouzebda Salim, Laksaci Ali, Mechab Boubaker
Laboratory of Statistics and Stochastic Processes, University of Djillali Liabes, BP 89, Sidi Bel Abbes 22000, Algeria.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia.
Entropy (Basel). 2023 Jul 24;25(7):1108. doi: 10.3390/e25071108.
The convergence rate for free-distribution functional data analyses is challenging. It requires some advanced pure mathematics functional analysis tools. This paper aims to bring several contributions to the existing functional data analysis literature. First, we prove in this work that Kolmogorov entropy is a fundamental tool in characterizing the convergence rate of the local linear estimation. Precisely, we use this tool to derive the uniform convergence rate of the local linear estimation of the conditional cumulative distribution function and the local linear estimation conditional quantile function. Second, a central limit theorem for the proposed estimators is established. These results are proved under general assumptions, allowing for the incomplete functional time series case to be covered. Specifically, we model the correlation using the ergodic assumption and assume that the response variable is collected with missing at random. Finally, we conduct Monte Carlo simulations to assess the finite sample performance of the proposed estimators.
自由分布函数数据分析的收敛速度具有挑战性。它需要一些先进的纯数学泛函分析工具。本文旨在为现有的函数数据分析文献做出若干贡献。首先，我们在这项工作中证明，柯尔莫哥洛夫熵是刻画局部线性估计收敛速度的一个基本工具。确切地说，我们使用这个工具来推导条件累积分布函数的局部线性估计和局部线性估计条件分位数函数的一致收敛速度。其次，为所提出的估计量建立了一个中心极限定理。这些结果是在一般假设下证明的，涵盖了不完全函数时间序列的情况。具体来说，我们使用遍历性假设对相关性进行建模，并假设响应变量是随机缺失的情况下收集的。最后，我们进行蒙特卡罗模拟以评估所提出估计量的有限样本性能。
