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Theory of Probability and Mathematical Statistics

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Nonparametric estimation for a compound Poisson process governed by a Markov chain

Author: O. A. Voĭna
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 85 (2011).
Journal: Theor. Probability and Math. Statist. 85 (2012), 41-52
MSC (2010): Primary 62M05; Secondary 60J99, 93E11
Published electronically: January 11, 2013
MathSciNet review: 2933701
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Abstract | References | Similar Articles | Additional Information

Abstract: The problem of nonparametric estimation for a Poisson process governed by a Markov chain with continuous time is considered in the case of incomplete observations. A method of the nonparametric estimation is proposed. The method is based on a representation of such a process in the form of a hidden Markov model. An example for the estimation of unknown distribution functions is considered in the case of incomplete observations after a compound Poisson process governed by a simple Markov regenerative process.

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  • 1. V. V. Anisimov, \cyr Sluchaĭnye protsessy s diskretnoĭ komponentoĭ, “Vishcha Shkola”, Kiev, 1988 (Russian). \cyr Predel′nye teoremy. [Limit theorems]. MR 955492
  • 2. A. A. Voĭna, Asymptotic optimization for stochastic models based on a compound Poisson process, Kibernetika i Sistemnyi Analiz (2011), no. 4, 11-23; English transl. in Cybernetics and System Analysis (2011), no. 4, 649-658.
  • 3. I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes, vol. 1, ``Nauka'', Moscow, 1971; vol. 2, ``Nauka'', Moscow, 1973; English transl., vol. 1, Springer-Verlag, New York-Heidelberg, 1974; vol. 2, Springer-Verlag, New York-Heidelberg, 1975.
  • 4. I. N. Kovalenko, Analysis of Rare Events for Estimating the Efficiency and Reliability of Systems, ``Sovetskoe Radio'', Moscow, 1980. (Russian)
  • 5. D. R. Cox and P. A. W. Lewis, The statistical analysis of series of events, Methuen & Co., Ltd., London; John Wiley & Sons, Inc., New York, 1966. MR 0199942
  • 6. Vladimir S. Korolyuk and Anatoly F. Turbin, Mathematical foundations of the state lumping of large systems, Mathematics and its Applications, vol. 264, Kluwer Academic Publishers Group, Dordrecht, 1993. Translated from the 1978 Russian original by V. V. Zayats and Y. A. Atanov and revised by the authors. MR 1281385
  • 7. D. S. Sīl′vestrov, \cyr Polumarkovskie protsessy s diskretnym mnozhestvom sostoyaniĭ, “Sovet. Radio”, Moscow, 1980 (Russian). \cyr Biblioteka Inzhenera po Nadezhnosti. [Library for the Engineer in Reliability]; \cyr Osnovy rascheta funktsional′nykh i nadezhnostnykh kharakteristik stokhasticheskikh sistem. [Principles of the calculation of functional and reliability characteristics of stochastic systems]. MR 612026
  • 8. M. Tim Jones, Al Application Programming, Charles River Media, Inc, Hingham, Massachusetts, 2003.
  • 9. L. R. Rabiner and B. H. Juang, An introduction to hidden Markov models, IEEE ASSP Mag. 3 (1986), no. 1, 4-16.
  • 10. A. A. Voĭna, Statistical estimation in a scheme of random variables on a Markov chain in the presence of incomplete observations, Teor. Veroyatnost. i Mat. Statist. 37 (1987), 16–26, 134 (Russian). MR 913905
  • 11. A. A. Voĭna and E. Czapla, An application of the correlation structure of a Markov chain for the estimation of shift parameters in queuing systems, Teor. Imovir. Mat. Stat. v71 (2004), 49-56; English transl. in Theor. Prob. Math. Statist. 71 (2005), 53-61.
  • 12. A. Wojna, Ryzyko w Procesach Finansowych Oraz Metody Badania Koniunktury, PK, Koszalin, 2009.

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Additional Information

O. A. Voĭna
Affiliation: Department of Economics and Management, Koszalin University of Technology, E. Kwiatkowskiego Street, 6E, Koszalin, 75-343, Poland

Keywords: Compound Poisson process, incomplete observations, nonparametric estimation, hidden Markov model, Cox model
Received by editor(s): September 9, 2010
Published electronically: January 11, 2013
Article copyright: © Copyright 2013 American Mathematical Society

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