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

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

Journal Article (2018)

Author(s)

Hendrik P. Lopuhaä (TU Delft - Statistics)

Eni Musta (TU Delft - Statistics)

Research Group

Statistics

Grenander-type estimator Least concave majorant Brownian motion with parabolic drift Central limit theorem for L-distance Limit distribution

DOI related publication

https://doi.org/10.1016/j.spl.2018.04.001 Final published version

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https://resolver.tudelft.nl/uuid:b3476454-84c3-429f-80db-017ea64344b5

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Publication Year

2018

Language

English

Research Group

Statistics

Volume number

139

Pages (from-to)

119-128

Downloads counter

82

Abstract

We consider the process Λ̂_n−Λ_n, where Λ_n is a cadlag step estimator for the primitive Λ of a nonincreasing function λ on [0,1], and Λ̂_n is the least concave majorant of Λ_n. 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 Λ̂_n−Λ_n converges in distribution to the corresponding process for two-sided Brownian motion with parabolic drift and we establish a central limit theorem for the L_p-distance between Λ̂_n and Λ_n.