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Tests of fit to exact models in statistical analysis often lead to rejectionseven when the model is a useful approximate description of the random generatorof the data. Among possible relaxations of a fixed model, the one defined bycontamination neighbourhoods, namely,$\mathcal{V}_\alpha(P_0)=\{(1-\alpha)P_0+\alpha Q: Q \in \mathcal{P}\}$, where$\mathcal{P}$ is the set of all probabilities in the sample space, has receivedmuch attention, from its central role in Robust Statistics. For probabilitieson the real line, consistent tests of fit to $\mathcal{V}_\alpha(P_0)$ can bebased on $d_K(P_0,R_\alpha(P))$, the minimal Kolmogorov distance between $P_0$and the set of trimmings of $P$, $R_\alpha(P)=\big\{\tildeP\in\mathcal{P}:\tilde P\ll P,\,{\textstyle \frac{d\tildeP}{dP}\leq\frac{1}{1-\alpha}}\, P\text{-a.s.}\big\}$. We show that thisfunctional admits equivalent formulations in terms of, either bestapproximation in uniform norm by $L$-Lipschitz functions satisfying a boxconstraint, or as the best monotone approximation in uniform norm to the$L$-Lipschitz regularization, which is seen to be expressable in terms of theaverage of the Pasch-Hausdorff envelopes. This representation for the solutionof the variational problem allows to obtain results showing stability of thefunctional $d_K(P_0,R_\alpha(P))$, as well as directional differentiability,providing the basis for a Central Limit Theorem for that functional.

Box-constrained monotone $L_\infty$-approximations to Lipschitz regularizations, with applications to robust testing