Box-Constrained Monotone Approximations to Lipschitz Regularizations, with Applications to Robust Testing
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Box-Constrained Monotone Approximations to Lipschitz Regularizations, with Applications to Robust Testing Eustasio del Barrio1 · Hristo Inouzhe1 · Carlos Matrán1 Received: 7 January 2020 / Accepted: 31 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 Mathematics Subject Classification 49J30 · 26A16 · 62G35 · 41A29
1 Introduction A repeated joker phrase in Statistics says that all models are wrong, but some are useful. This celebrated aphorism, attributed to the statistician G. Box, on the one
Communicated by Gabriel Peyré.
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Carlos Matrán [email protected] Departamento de Estadística e Investigación Operativa and IMUVA, Universidad de Valladolid, Valladolid, Spain
123
Journal of Optimization Theory and Applications
hand cautions that all models are approximations, while, on the other, stresses the usefulness of good approximate models. Here, approximation should be interpreted, in words of Davies [1], as “some formal admission of the fact that the statistical models are not true representations of the data”. From this perspective, within the research objectives of Mathematical Statistics, it becomes natural the permanent interest in the design and analysis of well-behaved procedures under small variations in the model. This includes the reconsideration of excessively restrictive concepts in Statistics, such as exact fit to models (say in homogeneity, regression or time series settings). The interest is not exact equality, but only “similarity” or, alternatively, to find a “relevant” difference. Also notice that this concept is of great relevance in some applications, such as bioequivalence in Biostatistics (see, for example, [2]). Some recent references sharing this spirit are ([3–9]). That is also the perspective of our recent work [10]. The present paper provides the mathematical bases giving support to the approach in this la
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