Universal and composite hypothesis testing via mismatched divergence

Jayakrishnan Unnikrishnan, Dayu Huang, Sean P. Meyn, Amit Surana, Venugopal V. Veeravalli

Research output: Contribution to journalArticlepeer-review


For the universal hypothesis testing problem, where the goal is to decide between the known null hypothesis distribution and some other unknown distribution, Hoeffding proposed a universal test in the nineteen sixties. Hoeffding's universal test statistic can be written in terms of KullbackLeibler (K-L) divergence between the empirical distribution of the observations and the null hypothesis distribution. In this paper a modification of Hoeffding's test is considered based on a relaxation of the K-L divergence, referred to as the mismatched divergence. The resulting mismatched test is shown to be a generalized likelihood-ratio test (GLRT) for the case where the alternate distribution lies in a parametric family of distributions characterized by a finite-dimensional parameter, i.e., it is a solution to the corresponding composite hypothesis testing problem. For certain choices of the alternate distribution, it is shown that both the Hoeffding test and the mismatched test have the same asymptotic performance in terms of error exponents. A consequence of this result is that the GLRT is optimal in differentiating a particular distribution from others in an exponential family. It is also shown that the mismatched test has a significant advantage over the Hoeffding test in terms of finite sample size performance for applications involving large alphabet distributions. This advantage is due to the difference in the asymptotic variances of the two test statistics under the null hypothesis.

Original languageEnglish (US)
Article number5714276
Pages (from-to)1587-1603
Number of pages17
JournalIEEE Transactions on Information Theory
Issue number3
StatePublished - Mar 2011


  • Generalized likelihood-ratio test
  • KullbackLeibler (K-L) information
  • hypothesis testing
  • online detection

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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