Box-Constrained Monotone Approximations to Lipschitz Regularizations, with Applications to Robust Testing

Eustasio del Barrio, Hristo Inouzhe and Carlos Matrán ()
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Eustasio del Barrio: Universidad de Valladolid
Hristo Inouzhe: Universidad de Valladolid
Carlos Matrán: Universidad de Valladolid

Journal of Optimization Theory and Applications, 2020, vol. 187, issue 1, No 4, 65-87

Abstract: Abstract Tests of fit to exact models in statistical analysis often lead to rejections even when the model is a useful approximate description of the random generator of the data. Among possible relaxations of a fixed model, the one defined by contamination neighbourhoods has received much attention, from its central role in Robust Statistics. For probabilities on the real line, consistent tests of fit to a contamination neighbourhood of a fixed model can be based on the minimal Kolmogorov distance between the model and the set of trimmings of the underlying random generator. We provide some alternative formulations for this functional in terms of a variational problem. As a consequence, a test of fit to contamination neighbourhoods can be effectively implemented. Also, we prove a result of directional differentiability giving the theoretical basis for the study of the asymptotic properties of such test.

Keywords: Contamination neighbourhoods; Kolmogorov distance; Lipschitz-continuous approximations; Distribution function; Trimmed probabilities; Pasch–Hausdorff envelopes; Lipschitz regularization; Robustness; Directional differentiability; 49J30; 26A16; 62G35; 41A29 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10957-020-01743-5

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