Decision-making under statistical uncertainty

hypothesis testing;robust statistics;quickest change detection

Unnikrishnan, Jayakrishnan

关键词: hypothesis testing;    robust statistics;    quickest change detection;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/16767/Unnikrishnan_Jayakrishnan.pdf?sequence=3&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Statistical decision-making procedures are used in a wide rangeof contexts varying from communication receiver design toenvironment monitoring systems. Although such procedures havebeen studied for a long time, much of the focus has beenrestricted to systems where the underlying probabilistic modelis known accurately. In this thesis we consider the settingwhere there is some uncertainty about the probabilistic model.We focus on two different problems and present approaches todealing with statistical uncertainty in each of these cases.For the problem of universal hypothesis testing, we study teststhat improve upon the known optimal solution in two differentaspects. Firstly, we study the generalized likelihood ratiotest (GLRT) that exploits partial knowledge about the alternatedistribution to improve finite-sample performance over theHoeffding test. Although the Hoeffding test is universallyoptimal in an asymptotic sense, we show that it suffers fromhigh bias and variance which leads to a poor performance overfinite observation lengths. The performance degradation of theHoeffding test is particularly significant for the testing oflarge alphabet distributions. We also show that the teststatistic used in the GLRT is a relaxation of theKullback-Leibler divergence statistic used in the Hoeffdingtest. We present results on the asymptotic behavior of the twotest statistics to explain the advantage of the GLRT. We thenstudy robust procedures for universal hypothesis testing whenthere is uncertainty about the null hypothesis. We present newresults on the asymptotic behavior of the proposed teststatistic which can be used to obtain procedures for settingthresholds in these tests for a target false alarmrequirement.We also study the problem of quickest change detection understatistical uncertainty. We formulate a new problem in robustquickest change detection, in which one seeks to minimize theworst-case delay over all possible instances of the uncertaindistributions subject to false alarm constraints. We adoptHuber's robust approach and identify sufficient conditionsunder which change detection procedures designed for certainleast-favorable distributions are robust touncertainties in a minimax sense. These robust tests are simpleto implement and give significant performance improvement oversome benchmark procedures that are known to be optimal in anasymptotic sense.