  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 $\pi$, and the outlier
sequence contains $n$ i.i.d. observations from an outlier distribution $\mu$,
which is distinct from $\pi$, but otherwise unknown. A universal test based on
KL divergence is built to approximate the maximum likelihood test, with known
$\pi$ and unknown $\mu$. 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
$0<K_1\leq \frac{d\mu}{d\pi}\leq K_2$, 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.
