标题： 一种构建不完整数据回归函数置信带的简单方法 显示英文标题

标题： A simple approach to construct confidence bands for a regression function with incomplete data

摘要： 构建回归函数$m(x)=E[Y|X=x]$的渐近正确置信带时，一个长期存在的问题涉及这样的情况，其中$Y$是受协变量$X$影响的响应变量，$Y$的值可能随机缺失，并且选择概率、$X$的密度函数$f(x)$以及在$X$给定下$Y$的条件方差都完全未知。 这在非参数情况下可能尤其复杂。 在本文中，我们提出了一种新的核类型回归估计量，并研究了所提出的估计量与真实回归曲线的最大偏差的适当归一化版本的极限分布。 所得的极限分布将用于构建具有渐近正确覆盖概率的底层回归曲线的统一置信带。 当前论文的重点是$X\in \mathbb{R}$的情况。 我们还进行了数值研究以评估所提方法的小样本性能。 在本文中，讨论了我们方法的机制和理论有效性。 显示英文摘要

摘要： A long-standing problem in the construction of asymptotically correct confidence bands for a regression function $m(x)=E[Y|X=x]$, where $Y$ is the response variable influenced by the covariate $X$, involves the situation where $Y$ values may be missing at random, and where the selection probability, the density function $f(x)$ of $X$, and the conditional variance of $Y$ given $X$ are all completely unknown. This can be particularly more complicated in nonparametric situations. In this paper we propose a new kernel-type regression estimator and study the limiting distribution of the properly normalized versions of the maximal deviation of the proposed estimator from the true regression curve. The resulting limiting distribution will be used to construct uniform confidence bands for the underlying regression curve with asymptotically correct coverages. The focus of the current paper is on the case where $X\in \mathbb{R}$. We also perform numerical studies to assess the finite-sample performance of the proposed method. In this paper, both mechanics and the theoretical validity of our methods are discussed.