  Universal hypothesis testing refers to the problem of deciding whether
samples come from a nominal distribution or an unknown distribution that is
different from the nominal distribution. Hoeffding's test, whose test statistic
is equivalent to the empirical Kullback-Leibler divergence (KLD), is known to
be asymptotically optimal for distributions defined on finite alphabets. With
continuous observations, however, the discontinuity of the KLD in the
distribution functions results in significant complications for universal
hypothesis testing. This paper introduces a robust version of the classical
KLD, defined as the KLD from a distribution to the L'evy ball of a known
distribution. This robust KLD is shown to be continuous in the underlying
distribution function with respect to the weak convergence. The continuity
property enables the development of a universal hypothesis test for continuous
observations that is shown to be asymptotically optimal for continuous
distributions in the same sense as that of the Hoeffding's test for discrete
distributions.
