Large Deviations for Outlier Hypothesis Testing of Continuous Sequences

In outlier hypothesis testing, one aims to detect outlying sequences among a given set of sequences, where most sequences are generated i.i.d. from a nominal distribution while outlying sequences (outliers) are generated i.i.d. from a different anomalous distribution. Most existing studies focus on discrete-valued sequences, where each data sample takes values in a finite set. To account for practical scenarios where data sequences usually take real values, we study outlier hypothesis testing for continuous sequences when both the nominal and anomalous distributions are unknown. Specifically, we propose distribution free test and prove that the probabilities of misclassification error, false reject and false alarm decay exponentially fast for the proposed test, where one fixes the sample size of each observed sequence. In this work, we mainly consider the case with multiple but unknown number of outliers.