# On approximate validation of models: A Kolmogorov-Smirnov based approach

## TL;DR

This paper introduces a Kolmogorov-Smirnov based method for approximate model validation that effectively tests for model fit in the presence of contamination, with strong error control and asymptotic properties.

## Contribution

It proposes a novel contamination-aware testing approach using trimming and the Kolmogorov metric, extending classical goodness-of-fit tests to more realistic scenarios.

## Key findings

- Estimator achieves exponentially small error probabilities.
- Method effectively detects model deviations with contamination.
- Connections to false discovery rate enhance applicability.

## Abstract

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 around a model. Using trimming methods and the Kolmogorov metric we introduce a functional statistic measuring departures from a contaminated model and the associated estimator corresponding to its sample version. We show how this 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 behavior 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 behavior when estimating the contamination level, although applicable in a wider framework.