Multisource Bayesian sequential change detection

Suppose that local characteristics of several independent compound Poisson and Wiener processes change suddenly and simultaneously at some unobservable disorder time. The problem is to detect the disorder time as quickly as possible after it happens and minimize the rate of false alarms at the same time. These problems arise, for example, from managing product quality in manufacturing systems and preventing the spread of infectious diseases. The promptness and accuracy of detection rules improve greatly if multiple independent information sources are available. Earlier work on sequential change detection in continuous time does not provide optimal rules for situations in which several marked count data and continuously changing signals are simultaneously observable. In this paper, optimal Bayesian sequential detection rules are developed for such problems when the marked count data is in the form of independent compound Poisson processes, and the continuously changing signals form a multi-dimensional Wiener process. An auxiliary optimal stopping problem for a jump-diffusion process is solved by transforming it first into a sequence of optimal stopping problems for a pure diffusion by means of a jump operator. This method is new and can be very useful in other applications as well, because it allows the use of the powerful optimal stopping theory for diffusions.Comment: Published in at http://dx.doi.org/10.1214/07-AAP463 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Sequential testing of simple hypotheses about compound Poisson processes

One of two simple hypotheses for the unknown arrival rate and jump distribution of a compound Poisson process is correct. We start observing the process, and the problem is to decide on the correct hypothesis as soon as possible and with the smallest probability of wrong decision. We find a Bayes-optimal sequential decision rule and describe completely how to calculate its parameters without any restrictions on the arrival rate and the jump distribution.Sequential hypothesis testing Sequential analysis Compound Poisson processes Optimal stopping

SEQUENTIAL TESTING OF SIMPLE HYPOTHESES ABOUT COMPOUND POISSON PROCESSES

Abstract. One of two simple hypotheses is correct about the unknown arrival rate and jump distribution of a compound Poisson process. We start observing the process, and the problem is to decide on the correct hypothesis as soon as possible and with the smallest probability of wrong decision. We find a Bayes-optimal sequential decision rule and describe completely how to calculate its parameters without any restrictions on the arrival rate and the jump distribution. 1

Submitted to the Annals of Applied Probability arXiv: math.PR/0000000 MULTISOURCE BAYESIAN SEQUENTIAL CHANGE DETECTION

Suppose that local characteristics of several independent compound Poisson and Wiener processes change suddenly and simultaneously at some unobservable disorder time. The problem is to detect the disorder time as quickly as possible after it happens and minimize the rate of false alarms at the same time. These problems arise, for example, from managing product quality in manufacturing systems and preventing the spread of infectious diseases. The promptness and accuracy of detection rules improve greatly if multiple independent information sources are available. Earlier work on sequential change detection in continuous time does not provide optimal rules for situations in which several marked count data and continuously changing signals are simultaneously observable. In this paper, optimal Bayesian sequential detection rules are developed for such problems when the marked count data is in the form of independent compound Poisson processes, and the continuously changing signals form a multidimensional Wiener process. An auxiliary optimal stopping problem for a jump-diffusion process is solved by transforming it first into a sequence of optimal stopping problems for a pure diffusion by means of a jump operator. This method is new and can be very useful in other applications as well, because it allows the use of the powerful optimal stopping theory for diffusions. 1. Introduction. Suppos

An optimal stopping approach for the end-of-life inventory problem

Sezer, Semih Onur/0000-0003-4215-7703We consider the end-of-life inventory problem for the supplier of a product in its final phase of the service life cycle. This phase starts when the production of the items stops and continues until the warranty of the last sold item expires. At the beginning of this phase the supplier places a final order for spare parts to serve customers coming with defective items. At any time during the final phase the supplier may also decide to switch to an alternative and more cost effective service policy. This alternative policy may be in the form of replacing defective items with substitutable products or offering discounts/rebates on the new generation ones. In this setup, the objective is to find a final order quantity and a time to switch to an alternative policy which will minimize the total expected discounted costs of the supplier. The switching time is a stopping time and is based on the realization of the arrival process of defective items. In this paper, we study this problem under a general cost structure in a continuous-time framework where the arrival of customers is given by a non-homogeneous Poisson process. We show in detail how to compute the value function, and illustrate our approach on numerical examples