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首页> 外文期刊>IEEE Transactions on Information Theory >Robust Kullback-Leibler Divergence and Universal Hypothesis Testing for Continuous Distributions
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Robust Kullback-Leibler Divergence and Universal Hypothesis Testing for Continuous Distributions

机译：连续分布的鲁棒Kullback-Leibler散度和通用假设检验

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摘要

Universal hypothesis testing (UHT) 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 (KL divergence), is known to be asymptotically optimal for distributions defined on finite alphabets. With continuous observations, however, the discontinuity of the KL divergence in the distribution functions results in significant complications for UHT. This paper introduces a robust version of the classical KL divergence, defined as the KL divergence from a distribution to the Levy ball of a known distribution. This robust KL divergence is shown to be continuous in the underlying distribution function with respect to the weak convergence. The continuity property enables the development of an asymptotically optimal test for the university hypothesis testing problem with continuous observations. The optimality is in the same sense as that of the Hoeffding's test and stronger than that of Zeitouni and Gutman. Perhaps more importantly, the developed test statistic can be computed through convex programs, making it much more meaningful in practice. Numerical experiments are also conducted to evaluate its performance as compared with some kernel based goodness of fit test that has been proposed recently.
机译：通用假设检验（UHT）是指确定样本是来自标称分布还是不同于标称分布的未知分布的问题。霍夫丁检验的检验统计量等同于经验Kullback-Leibler散度（KL散度），对于有限字母上定义的分布，它是渐近最优的。然而，通过连续观察，分布函数中KL散度的不连续性会导致UHT的严重并发症。本文介绍了经典KL散度的可靠版本，定义为从分布到已知分布的Levy球的KL散度。相对于弱收敛，这种强大的KL散度在基础分布函数中显示为连续的。连续性使得能够针对具有连续观测的大学假设检验问题开发渐近最优检验。最优性与霍夫丁检验的最优性相同，并且比Zeitouni和Gutman的最优性强。也许更重要的是，可以通过凸程序来计算已开发的测试统计量，从而使其在实践中更有意义。与最近提出的一些基于核的拟合优度测试相比，还进行了数值实验以评估其性能。

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