Efficient computation of the cdf of the maximum distance between Brownian bridge and its concave majorant

In this paper, we describe two computational methods for calculating the cumulative distribution function and the upper quantiles of the maximal difference between a Brownian bridge and its concave majorant. The first method has two different variants that are both based on a Monte Carlo approach, whereas the second uses the Gaver–Stehfest (GS) algorithm for the numerical inversion of the Laplace transform. If the former method is straightforward to implement, it is very much outperformed by the GS algorithm, which provides a very accurate approximation of the cumulative distribution as well as its upper quantiles. Our numerical work has a direct application in statistics: the maximal difference between a Brownian bridge and its concave majorant arises in connection with a nonparametric test for monotonicity of a density or regression curve on [0,1]. Our results can be used to construct very accurate rejection region for this test at a given asymptotic level.