Universal Outlying sequence detection For Continuous Observations

The following detection problem is studied, in which there are $M$ sequences of samples out of which one outlier sequence needs to be detected. Each typical sequence contains $n$ independent and identically distributed (i.i.d.) continuous observations from a known distribution $π$, and the outlier sequence contains $n$ i.i.d. observations from an outlier distribution $μ$, which is distinct from $π$, but otherwise unknown. A universal test based on KL divergence is built to approximate the maximum likelihood test, with known $π$ and unknown $μ$. A data-dependent partitions based KL divergence estimator is employed. Such a KL divergence estimator is further shown to converge to its true value exponentially fast when the density ratio satisfies $0<K_1\leq \frac{dμ}{dπ}\leq K_2$, where $K_1$ and $K_2$ are positive constants, and this further implies that the test is exponentially consistent. The performance of the test is compared with that of a recently introduced test for this problem based on the machine learning approach of maximum mean discrepancy (MMD). We identify regimes in which the KL divergence based test is better than the MMD based test.