The distribution of the maximal difference between Brownian bridge and its concave majorant

TL;DR

This paper derives the distribution, density, and moments of the maximal difference between a Brownian bridge and its concave majorant, with applications in nonparametric statistical tests for monotonicity.

Contribution

It provides a novel representation and explicit formulas for the distribution and moments of this maximal difference, aiding statistical hypothesis testing.

Findings

Explicit distribution and density functions derived

Moments of the maximal difference calculated

Application in monotonicity testing in statistics

Abstract

We provide a representation of the maximal difference between a standard Brownian bridge and its concave majorant on the unit interval, from which we deduce expressions for the distribution and density functions and moments of this difference. This maximal difference has an application in nonparametric statistics where it arises in testing monotonicity of a density or regression curve.