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：我们构建了单调回归函数的自助法置信区间。已有研究表明，基于非参数最小二乘估计量（LSE）$\hat f_n$的普通非参数自助法在此情形下不一致。然而，我们证明，基于平滑后的$\hat f_n$（称为平滑最小二乘估计量SLSE）可以构建一致的自助法。推导了SLSE的渐近逐点分布。将基于平滑自助法的置信区间与基于（未必单调的）Nadaraya-Watson估计量的区间进行了比较，并研究了学生化处理的效果。此外，我们给出了一种自动带宽选择方法，修正了Sen与Xu（2015）的研究工作。该过程通过一个与气候变化相关的著名数据集进行了演示。