Previous works were dedicated to the functional k-Nearest Neighbors (kNN) and the local linearity method estimations of a regression operator. In this paper, a sequence pair of ${(X_i,Y_i)}_{i=1,\dots ,n}$ of functional mixing observations are considered. We treat the local linear estimation of the cumulative function of $Y_i$ given functional input variable $X_i$. Precisely, we combine the kNN method with the local linear algorithm to construct a new and fast efficiency estimator of the conditional distribution function. The main purpose of this paper is to prove the strong convergence of the constructed estimator under mixing conditions. An application to the functional times series prediction is used to compare our proposed estimator with the existing competitive estimators, and show its efficiency and superiority.

中文翻译：

---

条件分布函数的平滑kNN局部线性估计

先前的工作致力于函数k最近邻（k NN）和回归算子的局部线性方法估计。在本文中，一个序列对${（X_{\mathrm{一世}}，{ÿ}_{\mathrm{一世}}）}_{\mathrm{一世}=\mathrm{1个}，\dots ，ñ}$考虑了功能混合的观察。我们处理累积函数的局部线性估计${ÿ}_{\mathrm{一世}}$ 给定功能输入变量 $X_{\mathrm{一世}}$。精确地，我们将k NN方法与局部线性算法结合起来，构造了一个新的，有条件的分布函数的快速高效估计器。本文的主要目的是证明混合条件下构造的估计量的强收敛性。在功能时间序列预测中的一个应用程序用于将我们提出的估计量与现有的竞争估计量进行比较，并显示其效率和优越性。