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

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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.

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Acknowledgements

This research has been partially supported by FEDER, Spanish Ministerio de Economía y Competitividad, grant MTM2017-86061-C2-1-P and Junta de Castilla y León, grants VA005P17 and VA002G18.

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Authors and Affiliations

  1. Departamento de Estadística e Investigación Operativa and IMUVA, Universidad de Valladolid, Valladolid, Spain

    Eustasio del Barrio, Hristo Inouzhe & Carlos Matrán

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  1. Eustasio del Barrio
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  2. Hristo Inouzhe
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  3. Carlos Matrán
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Correspondence to Carlos Matrán.

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Barrio, E.d., Inouzhe, H. & Matrán, C. Box-Constrained Monotone Approximations to Lipschitz Regularizations, with Applications to Robust Testing. J Optim Theory Appl 187, 65–87 (2020). https://doi.org/10.1007/s10957-020-01743-5

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