Browsing by Subject "Quickest detection"

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Open Access
Compound Poisson disorder problem with uniformly distributed disorder time
(2019-07) Ürü, Çağın
Suppose 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.
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.
Open Access
Compound Poisson disorder with general prior and misspecified Wiener disorder problem
(2024-07) Şahin, Deniz
For a system modeled with a compound Poisson or a Wiener process, let us assume that the underlying model parameters change at an unknown and unobservable time. For a compound Poisson process, these are arrival rate and mark distribution while for a Wiener process, it is the drift parameter. Suppose the decision maker knows the pre- and post-disorder process parameters, as well as the prior density of the disorder time. In this case, finding a stopping rule that optimizes a Bayesian penalty function is called the compound Poisson and Wiener disorder problem, respectively. For the compound Poisson problem, we consider a general prior distribution where the decision maker has more general knowledge about the disorder time than exponential and uniform priors which were addressed in the previous studies. For the Wiener problem, we revisit the asset selling problem with an exponential prior, where the decision maker specifies problem parameters incorrectly. In both cases, the original problems reduce to optimal stopping problems. We use time discretization and successive approximation methods for the first case and Markov chain approximation and Monte Carlo simulations for the second case. We provide the quickest detection rules and discuss various numerical examples.