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.

中文翻译：

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为数据不完整的回归函数构造置信带的简单方法

在渐近正确置信带的构建回归函数的长期存在的问题\（M（X）= E [Y | X = X] \） ，其中ÿ是由协变量的影响的响应变量X，涉及的情形其中Y值可能会随机丢失，并且选择概率，X的密度函数f（x）以及给定X的Y的条件方差完全未知。在非参数情况下，这尤其复杂。在本文中，我们提出了一种新的核型回归估计量，并研究了该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为渐近正确覆盖的基础回归曲线构造均匀的置信带。本文的重点是\（X \ in \ mathbb {R} \）中的情况。我们还进行了数值研究，以评估该方法的有限样本性能。本文讨论了方法的力学和理论有效性。