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

Author

Listed:
  • Eustasio del Barrio

    (Universidad de Valladolid)

  • Hristo Inouzhe

    (Universidad de Valladolid)

  • Carlos Matrán

    (Universidad de Valladolid)

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.

Suggested Citation

  • Eustasio del Barrio & Hristo Inouzhe & Carlos Matrán, 2020. "Box-Constrained Monotone Approximations to Lipschitz Regularizations, with Applications to Robust Testing," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 65-87, October.
    • Handle: RePEc:spr:joptap:v:187:y:2020:i:1:d:10.1007_s10957-020-01743-5
    DOI: 10.1007/s10957-020-01743-5
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    References listed on IDEAS

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    1. P. C. Álvarez-Esteban & E. del Barrio & J. A. Cuesta-Albertos & C. Matrán, 2016. "A contamination model for the stochastic order," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(4), pages 751-774, December.
    2. Holger Dette & Kathrin Möllenhoff & Stanislav Volgushev & Frank Bretz, 2018. "Equivalence of Regression Curves," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 711-729, April.
    3. Axel Munk & Claudia Czado, 1998. "Nonparametric validation of similar distributions and assessment of goodness of fit," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(1), pages 223-241.
    4. Holger Dette & Dominik Wied, 2016. "Detecting relevant changes in time series models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(2), pages 371-394, March.
    5. Alvarez-Esteban, Pedro Cesar & del Barrio, Eustasio & Cuesta-Albertos, Juan Antonio & Matran, Carlos, 2008. "Trimmed Comparison of Distributions," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 697-704, June.
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