Universal Outlier Hypothesis Testing via Mean- and Median-Based Tests

Universal outlier hypothesis testing refers to a hypothesis testing problem where one observes a large number of length-$n$ sequences -- the majority of which are distributed according to the typical distribution $\pi$ and a small number are distributed according to the outlier distribution $\mu$ -- and one wishes to decide, which of these sequences are outliers without having knowledge of $\pi$ and $\mu$. In contrast to previous works, in this paper it is assumed that both the number of observation sequences and the number of outlier sequences grow with the sequence length. In this case, the typical distribution $\pi$ can be estimated by computing the mean over all observation sequences, provided that the number of outlier sequences is sublinear in the total number of sequences. It is demonstrated that, in this case, one can achieve the error exponent of the maximum likelihood test that has access to both $\pi$ and $\mu$. However, this mean-based test performs poorly when the number of outlier sequences is proportional to the total number of sequences. For this case, a median-based test is proposed that estimates $\pi$ as the median of all observation sequences. It is demonstrated that the median-based test achieves again the error exponent of the maximum likelihood test that has access to both $\pi$ and $\mu$, but only with probability approaching one. To formalize this case, the typical error exponent -- similar to the typical random coding exponent introduced in the context of random coding for channel coding -- is proposed.