Decision-making under statistical uncertainty

Statistical decision-making procedures are used in a wide range of contexts varying from communication receiver design to environment monitoring systems. Although such procedures have been studied for a long time, much of the focus has been restricted to systems where the underlying probabilistic model is known accurately. In this thesis we consider the setting where there is some uncertainty about the probabilistic model. We focus on two different problems and present approaches to dealing with statistical uncertainty in each of these cases. For the problem of universal hypothesis testing, we study tests that improve upon the known optimal solution in two different aspects. Firstly, we study the generalized likelihood ratio test (GLRT) that exploits partial knowledge about the alternate distribution to improve finite-sample performance over the Hoeffding test. Although the Hoeffding test is universally optimal in an asymptotic sense, we show that it suffers from high bias and variance which leads to a poor performance over finite observation lengths. The performance degradation of the Hoeffding test is particularly significant for the testing of large alphabet distributions. We also show that the test statistic used in the GLRT is a relaxation of the Kullback-Leibler divergence statistic used in the Hoeffding test. We present results on the asymptotic behavior of the two test statistics to explain the advantage of the GLRT. We then study robust procedures for universal hypothesis testing when there is uncertainty about the null hypothesis. We present new results on the asymptotic behavior of the proposed test statistic which can be used to obtain procedures for setting thresholds in these tests for a target false alarm requirement. We also study the problem of quickest change detection under statistical uncertainty. We formulate a new problem in robust quickest change detection, in which one seeks to minimize the worst-case delay over all possible instances of the uncertain distributions subject to false alarm constraints. We adopt Huber's robust approach and identify sufficient conditions under which change detection procedures designed for certain least-favorable distributions are robust to uncertainties in a minimax sense. These robust tests are simple to implement and give significant performance improvement over some benchmark procedures that are known to be optimal in an asymptotic sense.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Electrical & Computer Engr

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2010

Author dc:creator

• Unnikrishnan, Jayakrishnan

Contributors dc:contributor

• Meyn, Sean P.
• Veeravalli, Venugopal V.
• Hajek, Bruce
• Viswanath, Pramod

• hypothesis testing
• robust statistics
• quickest change detection

Statement dc:rights

• Copyright 2010 Jayakrishnan Unnikrishnan

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/16767

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/16767