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Unformatted text preview: Sequential Tests of Statistical Hypotheses Abstract A class of sequential tests for binary hypothesis testing is studied. We consider the problem in which the observations are a sequence of independent and identically distributed random variables. A Bayesian framework for this problem is introduced. Based on a dynamic program ming argument, the optimality of the sequential probability ratio test, as defined by Wald [7], is established. A detailed analysis of this problem for various cost assignments is presented. 1 Introduction A binary sequential detection problems is a procedure by which one of two possible hypotheses is selected based on a series of observations. At each time step, the detector can either admit one of the two hypotheses or wait and take an additional observation. Pioneer work in the area is due to Wald [7]. In his original paper, Wald conjecture the optimality of the sequential probability ratio test (SPRT), a stationary bithreshold policy. This conjecture was later proved by Wald and Wolfowitz [8]. In this report, we rework half of the steps necessary to prove the optimality of the SPRT. Specifically, we use the contemporary tool of Dynamic Programming to show that a stationary bithreshold policy is optimal for the Bayesian counterpart of the problem considered by Wald. The flow of the report and the structure of the proofs are akin to the material contained in Decen tralized Sequential Detection with a Fusion Center Performing the Sequential Test by Veeravalli, Ba sar, and Poor [6]. In the latter, the authors consider the more difficult problem of decentralized sequential detection. The aim of this report is to demonstrate a clear understanding of the problem considered by Wald [7], as well as a proficiency in using the tool of Dynamic Programming. 2 Problem Formulation We begin with a formal description of the sequential detection problem we wish to study. Let the state of nature H be a random variable drawn from the binary alphabet { H , H 1 } with prior probability and 1 , respectively. At discrete time t , the detector has access to observation Y t taking value in a measurable space ( Y , F ). We assume that the observations are conditionally independent and identically distributed, given H , with probability measure P ( P 1 ) under H ( H 1 ). The hypothesis testing problem consists of deciding, based on the information available at time t , whether the law generating the observation sequence Y 1 , . . . , Y t is P or P 1 . We define the information vector I t to be the set of past observations. At time t > 0, I t = ( y t , . . . , y t ) In this work, we focus on sequential tests of statistical hypotheses. A sequential test is a procedure whereby, at each stage of the experiment, the detector either admit one of the two hypotheses or wait and collect more observations....
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This note was uploaded on 03/30/2010 for the course ECEN 689 taught by Professor Enjeti during the Fall '07 term at Texas A&M.
 Fall '07
 Enjeti

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