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

## TL;DR

This paper studies the asymptotic behavior of the difference between a step estimator and its least concave majorant, proving convergence to a Brownian motion process and establishing a CLT for their Lp-distance.

## Contribution

It extends previous results to a more general setting, providing new asymptotic distributional limits and a CLT for the estimator's deviation.

## Key findings

- Scaled difference converges to a Brownian motion with parabolic drift.
- Established a CLT for the Lp-distance between the estimator and its majorant.
- Generalized earlier results to broader conditions.

## Abstract

We consider the process $\widehat\Lambda_n-\Lambda_n$, where $\Lambda_n$ is a cadlag step estimator for the primitive $\Lambda$ of a nonincreasing function $\lambda$ on $[0,1]$, and $\widehat\Lambda_n$ is the least concave majorant of $\Lambda_n$. We extend the results in Kulikov and Lopuha\"a (2006, 2008) to the general setting considered in Durot (2007). Under this setting we prove that a suitably scaled version of $\widehat\Lambda_n-\Lambda_n$ 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 $\widehat\Lambda_n$ and $\Lambda_n$.