Chapter

Advances in Stochastic Models for Reliability, Quality and Safety

Part of the series Statistics for Industry and Technology pp 167-181

Optimal Sequential Estimation for Markov-Additive Processes

  • Ryszard MagieraAffiliated withTechnical University of Wroclaw

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Abstract

The problem of estimating the parameters of a Markov-additive process from data observed up to a random stopping time is considered. Markov-additive processes are a class of Markov processes which have important applications to queueing and data communication models. They have been used to model queueing-reliability systems, arrival processes in telecommunication networks, environmental data, neural impulses etc. The problem of obtaining optimal sequential estimation procedures, i.e., optimal stopping times and the corresponding estimators, in estimating functions of the unknown parameters of Markov-additive processes is considered. The parametric functions and sequential procedures which admit minimum variance unbiased estimators are characterized. In the main, the problem of finding optimal sequential procedures is considered in the case where the loss incurred is due not only to the error of estimation, but also to the cost of observing the process. Using a weighted squared error loss and assuming the cost is a function of the additive component of a Markov-additive process (for example, the cost depending on arrivals at a queueing system up to the moment of stopping), a class of minimax sequential procedures is derived for estimating the ratios between transition intensities of the embedded Markov chain and the mean value parameter of the additive part of the Markov-additive process considered.

Keywords and phrases

Sequential estimation procedure minimax estimation efficient estimation stopping time Markov-additive process