Efficient computation of the cdf of the maximum distance between Brownian bridge and its concave majorant
Résumé
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 algorthim, which provides a very accurate approximation of the cumulative distribution as well as its upper quantiles. This numerical work has a direct application in statistics: The maximal difference between a Brownian bridge and its concave majorant arises in connection with testing monotonicity of a density or regression curve on the unit interval, and hence it is of great importance to accurately approximate its true upper quantiles.