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Universal hypothesis testing refers to the problem of deciding whethersamples come from a nominal distribution or an unknown distribution that isdifferent from the nominal distribution. Hoeffding's test, whose test statisticis equivalent to the empirical Kullback-Leibler divergence (KLD), is known tobe asymptotically optimal for distributions defined on finite alphabets. Withcontinuous observations, however, the discontinuity of the KLD in thedistribution functions results in significant complications for universalhypothesis testing. This paper introduces a robust version of the classicalKLD, defined as the KLD from a distribution to the L'evy ball of a knowndistribution. This robust KLD is shown to be continuous in the underlyingdistribution function with respect to the weak convergence. The continuityproperty enables the development of a universal hypothesis test for continuousobservations that is shown to be asymptotically optimal for continuousdistributions in the same sense as that of the Hoeffding's test for discretedistributions.

Robust Kullback-Leibler Divergence and Universal Hypothesis Testing for Continuous Distributions