In the main text the true probability measure, P, is the nuisance parameter. In this supplementary material we examine which contiguous perturbations of the original fixed P preserve or do not preserve the estimation and coverage properties of the regions constructed in the main text. A useful feature of the local approach is that the conditions for the robustness of the estimation and coverage properties do not depend on the way the consistent critical values are generated (e.g. bootstrap or other means). The conditions are simple to check and apply to any consistent method of estimating a critical value.