Classical tests of fit typically reject a model for large enough real data samples. In contrast, often in statistical practice, a model offers a good description of the data even though it is not the ‘true’ random generator. We consider a more flexible approach based on contamination neighbourhoods: using trimming methods and the Kolmogorov metric, we introduce a functional statistic measuring departures from a contaminated model. We show how the plug-in estimator allows testing of fit for the (slightly) contaminated model vs sensible deviations from it, with uniformly exponentially small type I and type II error probabilities. We also address the asymptotic behaviour of the estimator showing that, under suitable regularity conditions, it asymptotically behaves as the supremum of a Gaussian process. As an application, we explore methods of comparison between descriptive models based on the paradigm of model falseness. We also include some connections of our approach with the false discovery rate setting, showing competitive behaviour when estimating the contamination level, and being applicable in a wider framework.

拟合的经典测试通常会拒绝用于足够大的实际数据样本的模型。相反，在统计实践中，即使模型不是“真正的”随机生成器，也可以很好地描述数据。我们考虑一种基于污染社区的更灵活的方法：使用修整方法和Kolmogorov度量，我们引入了一个功能统计量，用于测量与污染模型的偏离。我们展示了插入式估算器如何针对（轻微）受污染的模型进行拟合测试，以及相对于模型的合理偏差，I和II型错误概率呈指数规律地减小。我们还讨论了估计量的渐近行为，表明在适当的规律性条件下，它的渐近行为表现为高斯过程的极值。作为应用，我们探索基于模型错误范式的描述模型之间的比较方法。我们还将我们的方法与错误发现率设置联系起来，在估算污染水平时显示竞争行为，并适用于更广泛的框架。