Nonparametric regression problems with qualitative constraints such as monotonicity or convexity are ubiquitous in applications. For example, in predicting the yield of a factory in terms of the number of labor hours, the monotonicity of the conditional mean function is a natural constraint. One can estimate a monotone conditional mean function using nonparametric least squares estimation, which involves no tuning parameters. Several interesting properties of the isotonic LSE are known including its rate of convergence, adaptivity properties, and pointwise asymptotic distribution. However, we believe that the full richness of the asymptotic limit theory has not been explored in the literature which we do in this paper. Moreover, the inference problem is not fully settled. In this paper, we present some new results for monotone regression including an extension of existing results to triangular arrays, and provide asymptotically valid confidence intervals that are uniformly valid over a large class of distributions.

翻译：具有单调性或凸性等定性约束的非参数回归问题在应用中普遍存在。例如，在预测工厂产量与劳动时间的关系时，条件均值函数的单调性是一个自然约束。我们可以通过不涉及调节参数的非参数最小二乘估计来估计单调条件均值函数。等渗LSE的若干有趣性质已被揭示，包括其收敛速度、自适应性质以及逐点渐近分布。然而，我们认为现有文献尚未充分探索渐近极限理论的完备性，这正是本文的研究重点。此外，推断问题尚未得到完全解决。本文针对单调回归提出了一些新结果，包括将现有结果推广至三角阵列，并提供了在广泛分布类中一致有效的渐近有效置信区间。