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

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.