We construct bootstrap confidence intervals for a monotone regression function. It has been shown that the ordinary nonparametric bootstrap, based on the nonparametric least squares estimator (LSE) $\hat f_n$ is inconsistent in this situation. We show, however, that a consistent bootstrap can be based on the smoothed $\hat f_n$, to be called the SLSE (Smoothed Least Squares Estimator). The asymptotic pointwise distribution of the SLSE is derived. The confidence intervals, based on the smoothed bootstrap, are compared to intervals based on the (not necessarily monotone) Nadaraya Watson estimator and the effect of Studentization is investigated. We also give a method for automatic bandwidth choice, correcting work in Sen and Xu (2015). The procedure is illustrated using a well known dataset related to climate change.
翻译:我们构建了一个单调回归函数的自助法置信区间。已经表明,基于非参数最小二乘估计器(LSE)$\hat {f_n}$的普通非参数自助法在这种情况下是不一致的。然而,我们展示了,基于平滑的 $\hat{f_n}$ 的一致性自助法可以被建立,并将其称为平滑最小二乘估计器(SLSE)。推导了 SLSE 的渐近点通分布。基于平滑自助法的置信区间与(不一定单调的)Nadaraya-Watson 估计器的区间进行了比较,并探讨了学生化的影响。我们还提供了一种自动带宽选择的方法,纠正了 Sen 和 Xu (2015) 的工作。该程序通过一个与气候变化相关的知名数据集进行演示。