  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.
