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- Title
On the Distance Between Cumulative Sum Diagram and Its Greatest Convex Minorant for Unequally Spaced Design Points.
- Authors
PAL, JAYANTA KUMAR; WOODROOFE, MICHAEL
- Abstract
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−1/3), then the difference is shown to shrink in a very rapid (close to n−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.
- Subjects
REGRESSION analysis; ESTIMATION theory; STOCHASTIC processes; ASYMPTOTIC distribution; MATHEMATICAL convolutions; MATHEMATICAL statistics
- Publication
Scandinavian Journal of Statistics, 2006, Vol 33, Issue 2, p279
- ISSN
0303-6898
- Publication type
Article
- DOI
10.1111/j.1467-9469.2006.00461.x