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The distance between a naive cumulative estimator and its least concave majorant

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  • Lopuhaä, Hendrik P.
  • Musta, Eni

Abstract

We consider the process Λ̂n−Λn, where Λn is a cadlag step estimator for the primitive Λ of a nonincreasing function λ on [0,1], and Λ̂n is the least concave majorant of Λn. We extend the results in Kulikov and Lopuhaä (2006, 2008) to the general setting considered in Durot (2007). Under this setting we prove that a suitably scaled version of Λ̂n−Λn converges in distribution to the corresponding process for two-sided Brownian motion with parabolic drift and we establish a central limit theorem for the Lp-distance between Λ̂n and Λn.

Suggested Citation

  • Lopuhaä, Hendrik P. & Musta, Eni, 2018. "The distance between a naive cumulative estimator and its least concave majorant," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 119-128.
    • Handle: RePEc:eee:stapro:v:139:y:2018:i:c:p:119-128
    DOI: 10.1016/j.spl.2018.04.001
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    References listed on IDEAS

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    1. Hendrik P. Lopuhaä & Gabriela F. Nane, 2013. "Shape Constrained Non-parametric Estimators of the Baseline Distribution in Cox Proportional Hazards Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(3), pages 619-646, September.
    2. Wang, Yazhen, 1994. "The limit distribution of the concave majorant of an empirical distribution function," Statistics & Probability Letters, Elsevier, vol. 20(1), pages 81-84, May.
    3. Hendrik P. Lopuhaä & Eni Musta, 2017. "Smooth estimation of a monotone hazard and a monotone density under random censoring," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 71(1), pages 58-82, January.
    4. Kulikov, Vladimir N. & Lopuhaä, Hendrik P., 2006. "The limit process of the difference between the empirical distribution function and its concave majorant," Statistics & Probability Letters, Elsevier, vol. 76(16), pages 1781-1786, October.
    5. Cécile Durot & Piet Groeneboom & Hendrik P. Lopuhaä, 2013. "Testing equality of functions under monotonicity constraints," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 25(4), pages 939-970, December.
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