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      Wiener disorder problem with observations at fixed discrete time epochs

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      Author
      Dayanik, S.
      Date
      2010
      Source Title
      Mathematics of Operations Research
      Print ISSN
      0364-765X
      Electronic ISSN
      1526-5471
      Publisher
      Institute for Operations Research and the Management Sciences (I N F O R M S)
      Volume
      35
      Issue
      4
      Pages
      756 - 785
      Language
      English
      Type
      Article
      Item Usage Stats
      123
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      90
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      Abstract
      Suppose that a Wiener process gains a known drift rate at some unobservable disorder time with some zero-modified exponential distribution. The process is observed only at known fixed discrete time epochs, which may not always be spaced in equal distances. The problem is to detect the disorder time as quickly as possible by means of an alarm that depends only on the observations of Wiener process at those discrete time epochs. We show that Bayes optimal alarm times, which minimize expected total cost of frequent false alarms and detection delay time, always exist. Optimal alarms may in general sound between observation times and when the space-time process of the odds that disorder happened in the past hits a set with a nontrivial boundary. The optimal stopping boundary is piecewise-continuous and explodes as time approaches from left to each observation time. On each observation interval, if the boundary is not strictly increasing everywhere, then it irst decreases and then increases. It is strictly monotone wherever it does not vanish. Its decreasing portion always coincides with some explicit function. We develop numerical algorithms to calculate nearly optimal detection algorithms and their Bayes risks, and we illustrate their use on numerical examples. The solution of Wiener disorder problem with discretely spaced observation times will help reduce risks and costs associated with disease outbreak and production quality control, where the observations are often collected and/or inspected periodically.
      Keywords
      Optimal stopping
      Sequential change detection
      Wiener disorder problem
      Discrete time epochs
      Disease outbreaks
      Exponential distributions
      Numerical algorithms
      Numerical example
      Observation interval
      Optimal detection algorithm
      Optimal stopping
      Piecewise-continuous
      Production quality
      Sequential change detection
      Wiener disorder problem
      Alarm systems
      Algorithms
      Disease control
      Signal detection
      Optimization
      Permalink
      http://hdl.handle.net/11693/22158
      Published Version (Please cite this version)
      http://dx.doi.org/10.1287/moor.1100.0471
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      • Department of Industrial Engineering 692
      • Department of Mathematics 639
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