Optimal Sequential Estimation for Markov-Additive Processes

  • Ryszard Magiera
Part of the Statistics for Industry and Technology book series (SIT)


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 


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  1. 1.
    Barndorff-Nielsen, O. (1978). Information and Exponential Families, New York: John Wiley & Sons.zbMATHGoogle Scholar
  2. 2.
    Brown, L. (1986). Fundamentals of Statistical Exponential Families, Hay-ward, CA: IMS.zbMATHGoogle Scholar
  3. 3.
    Çinlar, E. (1972). Markov additive processes. I. Z. Wahrschein, verw. Gebiete, 24, 85–93.zbMATHCrossRefGoogle Scholar
  4. 4.
    Ezhov, I. and Skorohod, A. (1969). Markov processes with homogeneous second component. I. Teor. Verojatn. Primen., 14, 3–14 (in Russian).zbMATHGoogle Scholar
  5. 5.
    Franz, J. and Magiera, R. (1997). On information inequalities in sequential estimation for stochastic processes, Mathematical Methods of Operations Research, 46, 1–27.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Fygenson, M. (1991). Optimal sequential estimation for semi-Markov and Markov renewal processes, Communications in Statistics—Theory and Methods, 20, 1427–1444.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gill, R. (1980). Nonparametric estimation based on censored observation of a Markov renewal process, Z. Wahrschein, verw. Gebiete, 53, 97–116.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Karr, A. (1991). Point Processes and their Statistical Inference, Second edition, New York: Marcel Dekker.zbMATHGoogle Scholar
  9. 9.
    Magiera, R. and Stefanov, V. T. (1989). Sequential estimation in exponential-type processes under random initial conditions, Sequential Analysis, 8, 147–167.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Magiera, R. and Wilczyński, M. (1991). Conjugate priors for exponential-type processes, Statistics and Probability Letters, 12, 379–384.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Neuts, M. (1992). Models based on the Markovian arrival process, IEICE Transactions and Communications, E75-B, 1255–1265.Google Scholar
  12. 12.
    Pacheco, A. and Prabhu, N. (1995). Markov-additive processes of arrivals, In Advances in Queueing: Theory, Methods, and Open Problems (Ed., J. H. Dshalalow), pp. 167–194, Boca Raton: CRC Press.Google Scholar
  13. 13.
    Phelan, M. (1990a). Bayes estimation from a Markov renewal process, Annals in Statistics, 18, 603–616.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Phelan, M. (1990b). Estimating the transition probabilities from censored Markov renewal processes, Statistics & Probability Letters, 10, 43–47.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Prabhu, N. (1991). Markov renewal and Markov-additive processes, Technical Report 984, College of Engineering, Cornell University.Google Scholar
  16. 16.
    Rydén, T. (1994). Parameter estimation for Markov modulated Poisson processes, Communications in Statistics—Stochastic Models, 10, 795–829.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Stefanov, V. (1986). Efficient sequential estimation in exponential-type processes, Annals in Statistics, 14, 1606–1611.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Stefanov, V. (1995). Explicit limit results for minimal sufficient statistics and maximum likelihood estimators in some Markov processes: exponential families approach, Annals in Statistics, 23, 1073–1101.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Try bula, S. (1982). Sequential estimation in finite-state Markov processes, Zastos. Matem., 17, 227–248.MathSciNetzbMATHGoogle Scholar
  20. 20.
    Wilczyński, M. (1985). Minimax sequential estimation for the multinomial and gamma processes, Zastos. Matem., 18, 577–595.zbMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 1998

Authors and Affiliations

  • Ryszard Magiera
    • 1
  1. 1.Technical University of WroclawWroclawPoland

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