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

Journal article (2018)

Authors

H.P. Lopuhaä Statistics - EEMCS
E. Musta Statistics - EEMCS
Research Group
Statistics (EEMCS) (TU Delft)
Grenander-type estimator Least concave majorant Brownian motion with parabolic drift Central limit theorem for L-distance Limit distribution
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Published Date

2018

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Statistics

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.