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This article investigates techniques for the assessment of model error in the context of insurance risk analysis. The methodology is based on finding robust estimates for actuarial quantities of interest, which are obtained by solving optimization problems over the unknown probabilistic models, with constraints capturing potential nonparametric misspecification of the true model. We demonstrate the solution techniques and the interpretations of these optimization problems, and illustrate several examples, including calculating loss probabilities and conditional value-at-risk.