  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 numerical
inversion of 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.
