On the Distance Between Cumulative Sum Diagram and Its Greatest Convex Minorant for Unequally Spaced Design Points

The supremum difference between the cumulative sum diagram, and its greatest convex minorant (GCM), in case of non-parametric isotonic regression is considered. When the regression function is strictly increasing, and the design points are unequally spaced, but approximate a positive density in even a slow rate ("n"-super- - 1/3), then the difference is shown to shrink in a very rapid (close to "n"-super- - 2/3) rate. The result is analogous to the corresponding result in case of a monotone density estimation established by Kiefer and Wolfowitz, but uses entirely different representation. The limit distribution of the GCM as a process on the unit interval is obtained when the design variables are i.i.d. with a positive density. Finally, a pointwise asymptotic normality result is proved for the smooth monotone estimator, obtained by the convolution of a kernel with the classical monotone estimator. Copyright 2006 Board of the Foundation of the Scandinavian Journal of Statistics..