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Anomaly detection is a critical and evolving field in Machine Learning, with applications targeting different domains such as cybersecurity, finance, healthcare, manufacturing and IoT (Internet of Things) systems. Traditionally, anomaly detection algorithms have been designed using both supervised and unsupervised learning paradigms. The fundamental challenge in real-world anomaly detection scenarios is related to the inherent class imbalance (anomalies are typically rare) and, for supervised methods, to the scarcity of labelled anomalous data. Indeed, labelling is both expensive and time-consuming. Conversely unsupervised methods do not require labelling, but may suffer from high false positive rates when deployed in safety-critical applications. In this work we introduce a novel unsupervised algorithm for anomaly detection in time series based on the Haar discrete wavelet and a suitably designed $t$-test. We establish the theoretical foundation of the proposed $t$-test and, through extensive experimentation across 343 datasets, demonstrate that our algorithm outperforms state-of-the-art unsupervised and self-supervised benchmarks.