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

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  • JAYANTA KUMAR PAL
  • MICHAEL WOODROOFE

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.

Suggested Citation

  • Jayanta Kumar Pal & Michael Woodroofe, 2006. "On the Distance Between Cumulative Sum Diagram and Its Greatest Convex Minorant for Unequally Spaced Design Points," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 279-291, June.
    • Handle: RePEc:bla:scjsta:v:33:y:2006:i:2:p:279-291
    DOI: 10.1111/j.1467-9469.2006.00461.x
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    Cited by:

    1. Balabdaoui, Fadoua & Rufibach, Kaspar, 2008. "A second Marshall inequality in convex estimation," Statistics & Probability Letters, Elsevier, vol. 78(2), pages 118-126, February.

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