Anomaly Detection Using Surprisals

Rob Hyndman () and David FrazierAuthor-Email:Â david.frazier@monash.eduÂ

No 3/26, Monash Econometrics and Business Statistics Working Papers from Monash University, Department of Econometrics and Business Statistics

Abstract: Anomaly detection methods are widely used but often rely on ad hoc rules or strong assumptions, and they often focus on tail events, missing inlier anomalies that occur in low-density gaps between modes. We propose a unified framework that defines an anomaly as an observation with unusually low probability under a possibly misspecified model. For each observation we compute its surprisal, defined as the negative log generalized density, and define an anomaly score as the probability of a surprisal at least as large as that observed. This reduces anomaly detection for complex univariate or multivariate data to estimating the upper tail of a univariate surprisal distribution. We develop two model-robust estimators of these tail probabilities: an empirical estimator based on the observed surprisal distribution and an extreme-value estimator that fits a Generalized Pareto Distribution above a high threshold. Simulations and applications to French mortality and Test-cricket data show the approach remains effective under substantial model misspecification.

Keywords: anomaly detection; surprisal; outlier detection; generalized Pareto distribution; extreme value theory; tail bounds (search for similar items in EconPapers)
JEL-codes: C10 C14 C46 (search for similar items in EconPapers)
Pages: 21
Date: 2026
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