Browsing by Subject "Poisson disorder problem"
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Item Open Access Compound Poisson disorder problem with uniformly distributed disorder time(2019-07) Ürü, ÇağınSuppose that arrival rate and jump distribution of a compound Poisson process change suddenly at an unknown and unobservable time. The problem of detecting the change (disorder) as soon as it occurs is known as compound Poisson disorder. In practice, an unfavorable regime shift may require immediate action, and a quickest detection rule can allow the decision maker to react to the change and take the necessary countermeasures in a timely manner. Dayanık and Sezer [Compound Poisson disorder problem, Math. Oper. Res., vol. 31, no. 4, pp. 649-672, 2006] completely solve the compound Poisson disorder problem assuming a change-point with an exponential prior distribution. Although the exponential prior is convenient when solving the problem, it has aws when expressing reality due to the memoryless property. Besides, as an informative prior, it fails to represent the case when the decision maker has no prior information on the change-point. Considering these defects, we assume a uniformly distributed change-point instead in our study. Unlike the exponential prior, the uniform prior has a memory and can be used when the decision maker does not have a strong belief on the change-point. We reformulate the quickest detection problem as a nite-horizon optimal stopping problem for a piecewisedeterministic and Markovian sufficient statistic. With Monte Carlo simulation and Chebyshev interpolation, we calculate the value function numerically via successive approximations. Studying the sample-paths of the sufficient statistic, we describe an explicit quickest detection rule and provide numerical examples for our solution method.Item Open Access Compound Poisson disorder problem with uniformly distributed disorder time(Bernoulli Society for Mathematical Statistics and Probability, 2023-08) Uru, C.; Dayanık, Savaş; Sezer, Semih O.Suppose that the arrival rate and the jump distribution of a compound Poisson process change suddenly at an unknown and unobservable time. We want to detect the change as quickly as possible to take counteractions, e.g., to assure top quality of products in a production system, or to stop credit card fraud in a banking system. If we have no prior information about future disorder time, then we typically assume that the disorder is equally likely to happen any time – or has uniform distribution – over a long but finite time horizon. We solve this so-called compound Poisson disorder problem for the practically important case of unknown, unobserved, but uniformly distributed disorder time. The solution hinges on the complete separation of information flow from the hard time horizon constraint, by describing the former with an autonomous time-homogeneous one-dimensional Markov process in terms of which the detection problem translates into a finite horizon optimal stopping problem. For any given finite horizon, the solution is two-dimensional. For cases where the horizon is large and one is unwilling to set a fixed value for it, we give a one-dimensional approximation. Also, we discuss an extension where the disorder may not happen on the given interval with a positive probability. In this extended model, if no detection decision is made by the end of the horizon, then a second level hypothesis testing problem is solved to determine the local parameters of the observed process.Item Open Access Poisson disorder problem with control on costly observations(2012) Kadiyala, BharadwajA Poisson process Xt changes its rate at an unknown and unobservable time θ from λ0 to λ1. Detecting the change time as quickly as possible in an optimal way is described in literature as the Poisson disorder problem. We provide a more realistic generalization of the disorder problem for Poisson process by introducing fixed and continuous costs for being able to observe the arrival process. As a result, in addition to finding the optimal alarm time, we also characterize an optimal way of observing the arrival process. We illustrate the structure of the solution spaces with the help of some numerical examples.