Universal outlier detection

The following outlier detection problem is studied in a universal setting. Vector observations are collected each with M coordinates. When the i-th coordinate is the outlier, the observations in that coordinate are assumed to be distributed according to the “outlier” distribution, distinct from the common “typical” distribution governing the observations in all the other coordinates. Nothing is known about the outlier and the typical distributions except that they are distinct and have full supports. The goal is to design a universal detector to best discern the outlier coordinate. A universal detector is proposed and is shown to be universally exponentially consistent, and a single-letter characterization of the exponent for a symmetric error criterion achievable by this detector is derived. An upper bound for the error exponent that applies to any universal detector is also derived. For the special case of M = 3, a tighter upper bound is derived that quantifies the loss in the exponent when the knowledge of the outlier and typical distributions is absent, from when they are known.