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On the Distance Between Cumulative Sum Diagram and Its Greatest Convex Minorant for Unequally Spaced Design Points
Author(s) -
PAL JAYANTA KUMAR,
WOODROOFE MICHAEL
Publication year - 2006
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/j.1467-9469.2006.00461.x
Subject(s) - mathematics , pointwise , estimator , monotone polygon , isotonic regression , infimum and supremum , combinatorics , mathematical analysis , statistics , geometry
. 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.