Nonparametric composite outlier detection

Detection of the existence of data streams drawn from outlying distributions among data streams drawn from a typical distribution is investigated. It is assumed that the typical distribution is known and the outlying distribution is unknown. The generalized likelihood ratio test (GLRT) for this problem is constructed. With knowledge of the Kullback-Liebler divergence between the outlier and typical distributions, the GLRT is shown to be exponentially consistent (i.e, the error risk function decays exponentially fast). It is also shown that with knowledge of the Chernoff distance between the outlying and typical distributions, the same risk decay exponent as the parametric model can be achieved by using the GLRT. It is further shown that, without knowledge of the distance between the distributions, there does not exist an exponentially consistent test, although the GLRT with a diminishing threshold can still be consistent.