Universal and Composite Hypothesis Testing via Mismatched Divergence

—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 Kullback–Leibler (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 hav...

• Type: Journal
• Year: 2011
• Where: TIT
• Authors: Jayakrishnan Unnikrishnan, Dayu Huang, Sean P. Meyn, Amit Surana, Venugopal V. Veeravalli
• Keywords: Education, Null Hypothesis, Test Statistics, TIT 2011, Universal Test