The "best-estimate plus uncertainty"' (BEPU) methodology is the term used in the nuclear engineering community when dealing with uncertainty quantification issues in realistic numerical simulation models. One of the most critical hypothesis in these studies is the choice of the probability distributions of uncertain input variables which are propagated through the model. Bringing stringent justifications to the BEPU approach, especially in a safety study, requires quantifying the impact of potential uncertainty on the input variable distribution. To solve this problem, this paper deepens the robustness analysis based on the ``Perturbed Law-based sensitivity Indices'' (PLI). The PLI quantifies the impact of a perturbation of an input distribution on the quantity of interest (as a quantile of a model output or a safety margin) in the BEPU study. The mathematical formalism of the PLI is applied to two particular quantities of interest: the quantile and the superquantile. For both quantities, the PLI can be easily computed using a unique Monte-Carlo sample containing model inputs and output. Numerical tests are developed in order to define validity criteria of a PLI-based robustness analysis. The practical use of the method is illustrated on thermal-hydraulic computer experiments, simulating a cold leg Intermediate Break Loss Of Coolant Accident (IBLOCA) in a pressurized water nuclear reactor.