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：非参数回归问题中涉及单调性或凸性等定性约束在应用中十分常见。例如，在预测工厂产量与劳动工时关系时，条件均值函数的单调性是一个自然约束。我们可以使用无需调优参数的非参数最小二乘估计来估计单调条件均值函数。保序最小二乘估计具有若干有趣性质，包括收敛速度、自适应特性及逐点渐近分布。然而，我们认为文献中尚未充分探索渐近极限理论的全部丰富内涵，这正是本文的研究重点。此外，推断问题尚未完全得到解决。在本文中，我们提出了单调回归的一些新结果，包括将现有结果推广至三角阵列，并提供了在广泛分布类上一致有效的渐近置信区间。