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Universal and Composite Hypothesis Testing via Mismatched Divergence

机译：通过不匹配的散度进行通用和复合假设检验

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摘要

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 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.
机译：对于通用假设检验问题，其目标是在已知零假设分布和某些其他未知分布之间做出决定，Hoeffding在19世纪60年代提出了通用检验。 Hoeffding的通用检验统计量可以用观测值的经验分布与原假设分布之间的Kullback-Leibler（K-L）差异来表示。在本文中，基于对K-L散度的松弛（称为不匹配散度），对Hoeffding检验进行了修改。对于备用分布位于以有限维参数为特征的参数分布族中的情况，结果是，失配检验显示为广义似然比检验（GLRT），即，它是相应复合假设的解决方案测试问题。对于交替分布的某些选择，表明在误差指数方面，Hoeffding检验和失配检验都具有相同的渐近性能。该结果的结果是，GLRT在区分特定分布与指数族中其他分布方面是最佳的。还表明，在涉及大字母分布的应用中，就有限的样本量性能而言，不匹配的测试比Hoeffding测试具有明显的优势。此优势是由于在原假设下两个检验统计量的渐近方差不同。

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