Automatic detection of statistical outliers is facilitated through knowledge of the source distribution of regular observations. Since the population distribution is often unknown in practice, one approach is to apply a transformation to Normality. However, the efficacy of transformation is hindered by the presence of outliers, which can have an outsized influence on transformation parameter(s) and lead to masking of outliers post-transformation. Robust Box-Cox and Yeo-Johnson transformations have been proposed but those transformations are only equipped to deal with skew. Here, we develop a novel robust method for transformation to Normality based on the highly flexible sinh-arcsinh (SHASH) family of distributions, which can accommodate skew, non-Gaussian tail weights, and combinations of both. A critical step is initializing outliers, given their potential influence on the highly flexible SHASH transformation. To this end, we consider conventional robust z-scoring and a novel anomaly detection approach. Through extensive simulation studies and real data analyses representing a wide variety of distribution shapes, we find that SHASH transformation outperforms existing methods, exhibiting high sensitivity to outliers even at heavy contamination levels (20-30\%). We illustrate the utility of SHASH transformation-based outlier detection in the context of noise reduction in functional neuroimaging data.

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