Abstract
We consider the problem of identification of the position of some source by observations of K detectors receiving signals from this source. The time of arriving of the signal to the k-th detector depends of the distance between this detector and the source. The signals are observed in the presence of small Gaussian noise. The properties of the MLE and Bayesian estimators are studied in the asymptotic of small noise.
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References
Almagbile, A., Wang, J., Ding, W. (2010). Evaluating the performances of adaptive Kalman filter methods in GPS/INS integration. Journal of Global Positioning Systems, 9(1), 33–40.
Arakelyan, N., Kutoyants, Y. A. (2019). On identification of the source emission on the plane. Proceedings of Yerevan State University, 53(2), 75–81.
Arato, M. (1983). Linear stochastic systems with constant coefficients a statistical approach. Lecture notes in control and information sciences, Vol. 45. New York: Springer.
Bickel, P. J., Ritov, Y., Rydén, T. (1998). Asymptotic normality of the maximum likelihood estimator for general hidden Markov models. Annals of Statistics, 26(4), 1614–1635.
Cappé, O., Moulines, E., Rydén, T. (2005). Inference in hidden Markov models. New York: Springer.
Chernoyarov, O. V., Kutoyants, Y. A. (2020). Poisson source localization: The smooth case. Metrika, 83(4), 411–435.
Chernoyarov, O. V., Dachian, S., Kutoyants, Y. A. (2020). Poisson source localization on the plane: Cusp case. Annals of the Institute of Statistical Mathematics, 72, 1137–1157.
Chigansky, P. (2009). Maximum likelihood estimation for hidden Markov models in continuous time. Statistical Inference for Stochastic Processes, 12(2), 139–163.
Elliott, R. J., Aggoun, L., Moor, J. B. (1995). Hidden Markov models. New York: Springer.
Ephraim, Y., Mehrav, N. (2002). Hidden Markov processes. IEEE Transactions on Informormation Theory, 48(6), 1518–1569.
Farinetto, C., Kutoyants, Y. A., Top, A. (2020). Radioactive source localization: Change-point case. Annals of the Institute of Statistical Mathematics, 72(3), 675–698.
Gustaffson, F. (2000). Adaptive filtering and change detection. New York: Wiley.
Hutchinson, C. E. (1984). The Kalman filter applied to aerospace and electronic systems. IEEE Transactions on Aerospace and Electronic Systems, 20(4), 500–504.
Ibragimov, I. A., Khasminskii, R. Z. (1981). Statistical estimation—Asymptotic theory. New York: Springer.
Kallianpur, G., Selukar, R. S. (1991). Parameter estimation in linear filtering. Journal of Multivariate Analysis, 39, 284–304.
Kalman, R. E., Bucy, R. S. (1961). New results in linear filtering and prediction theory. Transactions ASME, 83D, 95–100.
Khasminskii, R. Z., Kutoyants, Y. A. (2018). On parameter estimation of hidden telegraph process. Bernoulli, 24(3), 2064–2090.
Konecny, F. (1990). Maximum likelihood estimation of a drift parameter from partially observed diffusion in the case of small measurement noise. Statistics and Decisions, 8, 115–130.
Kutoyants, Y. A. (1984). Parameter estimation for stochastic processes. Berlin: Heldermann.
Kutoyants, Y. A. (1994). Identification of dynamical systems with small noise. Dordrecht: Kluwer.
Kutoyants, Y. A. (2019a). On parameter estimation of hidden ergodic Ornstein–Uhlenbeck process. Electronic Journal of Statistics, 13, 4508–4526.
Kutoyants, Y. A. (2019b). On parameter estimation of the hidden Ornstein–Uhlenbeck process. Journal of Multivariate Analysis, 169(1), 248–269.
Kutoyants, Yu, A., Zhou, L. (2019). On parameter estimation of the hidden Gaussian process in perturbed SDE. Submitted (arXiv:1904.09750)
Liptser, R. S., Shiryayev, A. N. (2001). Statistics of random processes, I. General theory2nd ed. New York: Springer.
Luo, X. (2013). GPS stochastic modeling. New York: Springer.
Wang, W., Liu, Z., Xie, R.-R. (2006). Quadratic extended Kalman filter approach for GPS/INS integration. Aerospace Science and Technology, 10(8), 709–713.
Acknowledgements
I am grateful to the Reviewer for very useful comments. This research was supported by RSF Project No. 20-61-47043.
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Kutoyants, Y.A. On localization of source by hidden Gaussian processes with small noise. Ann Inst Stat Math 73, 671–702 (2021). https://doi.org/10.1007/s10463-020-00763-2
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DOI: https://doi.org/10.1007/s10463-020-00763-2