Nonparametric framework for the definition, adaptive detection and probabilistic interpretation of outliers

RESUMO

Outlier detection is a fundamental challenge in data processing, with critical implications for robustness across statistical modeling, machine learning and exploratory data analysis. However, existing proposals rarely offer a universal, domain-agnostic definition of an outlier, often relying on heuristic trimming quotas that lack a statistical interpretation. To address this, we propose a nonparametric framework built on a pseudo-isolation outlier score. This score enables a formal, probabilistic definition of an anomaly tied to a false-alarm rate $\alpha$, which extends into a rigorous geometric classification of internal and external outliers. We show that this mechanism seamlessly embeds into any objective-based clustering framework to identify cluster-specific outliers. Here, we integrate it into $K$-means to create ODK-means. The inferential capabilities and topological properties of this framework are explored both theoretically through formal propositions and empirically through extensive simulations and methodological tutorials, highlighting the practical actionability and intuitive appeal of the proposed outlier detection logic.