Filtration of linear functionals of periodically correlated sequences
Authors:
I. I. Dubovets′ka and M. P. Moklyachuk
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 86 (2013), 51-64
MSC (2010):
Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11
DOI:
https://doi.org/10.1090/S0094-9000-2013-00888-6
Published electronically:
August 20, 2013
MathSciNet review:
2986449
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The problem of the optimal estimation is considered for the linear functional \[ A{\zeta }=\sum _{j=0}^\infty {a}(j){\zeta }(-j) \] that depends on unknown values of a periodically correlated stochastic sequence $\zeta (j)$; the estimator is constructed from observations of the sequence $\zeta (j)+\theta (j)$, $j\leq 0$, where $\theta (j)$ is a periodically correlated noise. We obtain the mean square error and spectral characteristic of the optimal linear estimate of the functional $A{\zeta }$ in the case where the spectral densities of the sequences that generate $\zeta (j)$ and $\theta (j)$ are known. For the case where these spectral densities are unknown but a set of admissible spectral densities is given, we find the least favorable spectral density and minimax spectral characteristic for the optimal estimate of the functional $A{\zeta }$.
References
- W. R. Bennett, Statistics of regenerative digital transmission, Bell System Tech. J. 37 (1958), 1501–1542. MR 102138, DOI https://doi.org/10.1002/j.1538-7305.1958.tb01560.x
- Jürgen Franke, On the robust prediction and interpolation of time series in the presence of correlated noise, J. Time Ser. Anal. 5 (1984), no. 4, 227–244. MR 782077, DOI https://doi.org/10.1111/j.1467-9892.1984.tb00389.x
- Jürgen Franke, Minimax-robust prediction of discrete time series, Z. Wahrsch. Verw. Gebiete 68 (1985), no. 3, 337–364. MR 771471, DOI https://doi.org/10.1007/BF00532645
- Jürgen Franke and H. Vincent Poor, Minimax-robust filtering and finite-length robust predictors, Robust and nonlinear time series analysis (Heidelberg, 1983) Lect. Notes Stat., vol. 26, Springer, New York, 1984, pp. 87–126. MR 786305, DOI https://doi.org/10.1007/978-1-4615-7821-5_6
- E. G. Gladyšev, Periodically correlated random sequences, Dokl. Akad. Nauk SSSR 137 (1961), 1026–1029 (Russian). MR 0126873
- Ulf Grenander, A prediction problem in game theory, Ark. Mat. 3 (1957), 371–379. MR 90486, DOI https://doi.org/10.1007/BF02589429
- Harry L. Hurd and Abolghassem Miamee, Periodically correlated random sequences, Wiley Series in Probability and Statistics, Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2007. Spectral theory and practice. MR 2348769
- A. N. Kolmogorov, Selected works. Vol. II, Mathematics and its Applications (Soviet Series), vol. 26, Kluwer Academic Publishers Group, Dordrecht, 1992. Probability theory and mathematical statistics; With a preface by P. S. Aleksandrov; Translated from the Russian by G. Lindquist; Translation edited by A. N. Shiryayev [A. N. Shiryaev]. MR 1153022
- Andrzej Makagon, Theoretical prediction of periodically correlated sequences, Probab. Math. Statist. 19 (1999), no. 2, Acta Univ. Wratislav. No. 2198, 287–322. MR 1750905
- Andrzej Makagon, Stationary sequences associated with a periodically correlated sequence, Probab. Math. Statist. 31 (2011), no. 2, 263–283. MR 2853678
- M. P. Moklyachuk, Estimates of stochastic processes from observations with noise, Theory Stoch. Process. 3(19) (1997), no. 3–4, 330–338.
- M. P. Moklyachuk, Robust procedures in time series analysis, Theory Stoch. Process. 6(22) (2000), no. 3–4, 127–147.
- M. P. Moklyachuk, Game theory and convex optimization methods in robust estimation problems, Theory Stoch. Process. 7(23) (2001), no. 1–2, 253–264.
- M. P. Moklyachuk, Robust Estimates for Functionals of Stochastic Processes, “Kyivs′kyi Universytet”, Kyiv, 2008. (Ukrainian)
- M. P. Moklyachuk and O. Yu. Masyutka, On the problem of filtering vector stationary sequences, Teor. Ĭmovīr. Mat. Stat. 75 (2006), 95–104 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 75 (2007), 109–119. MR 2321185, DOI https://doi.org/10.1090/S0094-9000-08-00718-7
- B. N. Pshenichnyĭ, Neobkhodimye usloviya èkstremuma, Optimizatsiya i Issledovanie Operatsiĭ. [Optimization and Operations Research], “Nauka”, Moscow, 1982 (Russian). MR 686452
- Yu. A. Rozanov, Statsionarnye sluchaĭ nye protsessy, 2nd ed., Teoriya Veroyatnosteĭ i Matematicheskaya Statistika [Probability Theory and Mathematical Statistics], vol. 42, “Nauka”, Moscow, 1990 (Russian). MR 1090826
- Norbert Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications, The Technology Press of the Massachusetts Institute of Technology, Cambridge, Mass; John Wiley & Sons, Inc., New York, N. Y.; Chapman & Hall, Ltd., London, 1949. MR 0031213
- A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. I, Springer Series in Statistics, Springer-Verlag, New York, 1987. Basic results. MR 893393
- A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. II, Springer Series in Statistics, Springer-Verlag, New York, 1987. Supplementary notes and references. MR 915557
References
- W. R. Bennett, Statistics of regenerative digital transmission, Bell Syst. Tech. 37 (1958), 1501–1542. MR 0102138 (21:932)
- J. Franke, On the robust prediction and interpolation of time series in the presence of correlated noise, J. Time Series Anal. 5 (1984), 227–244. MR 782077 (86i:62192)
- J. Franke, Minimax robust prediction of discrete time series, Z. Wahrsch. Verw. Gebiete 68 (1985), 337–364. MR 771471 (86f:62164)
- J. Franke and H. V. Poor, Minimax-robust filtering and finite-length robust predictors, Robust and Nonlinear Time Series Analysis, Lecture Notes in Statistics, vol. 26, Springer-Verlag, 1984. MR 786305 (86i:93058)
- E. G. Gladyshev, Periodically correlated random sequences, Doklady Akad. Nauk SSSR 137 (1961), no. 5, 1026–1029; English transl. in Soviet Math. Dokl. 2 (1961), 385–388. MR 0126873 (23:A4167)
- U. Grenander, A prediction problem in game theory, Ark. Mat. 3 (1957), 371–379. MR 0090486 (19:822g)
- H. L. Hurd and A. Miamee, Periodically Correlated Random Sequences: Spectral Theory and Practice, John Wiley & Sons, 2007. MR 2348769 (2009e:62002)
- A. N. Kolmogorov, Probability Theory and Mathematical Statistics, Collection of problems, “Nauka”, Moscow, 1986; English transl., Selected works, vol. II, Mathematics and its Applications (Soviet Series), vol. 26, Kluwer Academic Publishers Group, Dordrecht, 1992; translated from the Russian by G. Lindquist; translation edited by A. N. Shiryayev. MR 1153022 (92j:01071)
- A. Makagon, Theoretical prediction of periodically correlated sequences, Probab. Math. Statist. 19 (1999), 287–322. MR 1750905 (2001m:60093)
- A. Makagon, Stationary sequences associated with a periodically correlated sequence, Probab. Math. Statist. 31 (2011), 263–283. MR 2853678 (2012j:60091)
- M. P. Moklyachuk, Estimates of stochastic processes from observations with noise, Theory Stoch. Process. 3(19) (1997), no. 3–4, 330–338.
- M. P. Moklyachuk, Robust procedures in time series analysis, Theory Stoch. Process. 6(22) (2000), no. 3–4, 127–147.
- M. P. Moklyachuk, Game theory and convex optimization methods in robust estimation problems, Theory Stoch. Process. 7(23) (2001), no. 1–2, 253–264.
- M. P. Moklyachuk, Robust Estimates for Functionals of Stochastic Processes, “Kyivs′kyi Universytet”, Kyiv, 2008. (Ukrainian)
- M. P. Moklyachuk and O. Yu. Masyutka, On the problem of filtration for stationary vector sequences, Teor. Imovir. Matem. Statyst. 75 (2007), 95–104; English transl. in Theor. Probability Math. Statist. 75 (2007), 109–119. MR 2321185
- B. N. Pshenichnyĭ, Necessary Conditions for an Extremum, “Nauka”, Moscow, 1982; English transl., Marcel Dekker, Inc., New York, 1971. MR 686452 (84c:49003)
- Yu. A. Rozanov, Stationary Random Processes, 2nd ed., “Nauka”, Moscow, 1990. (Russian) MR 1090826 (92d:60046)
- N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series. With Engineering Applications, The M. I. T. Press, Massachusetts Institute of Technology, Cambridge, Mass., 1966. MR 0031213 (11:118j)
- A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions. Vol. 1: Basic Results, Springer Series in Statistics, Springer-Verlag, New York, etc., 1987. MR 893393 (89a:60105)
- A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions. Vol. 2: Supplementary Notes and References, Springer Series in Statistics, Springer-Verlag, New York, etc., 1987. MR 915557 (89a:60106)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
60G10,
60G25,
60G35,
62M20,
93E10,
93E11
Retrieve articles in all journals
with MSC (2010):
60G10,
60G25,
60G35,
62M20,
93E10,
93E11
Additional Information
I. I. Dubovets′ka
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4-E, Kiev 03127, Ukraine
Email:
idubovetska@gmail.com
M. P. Moklyachuk
Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4-E, Kiev 03127, Ukraine
Email:
mmp@univ.kiev.ua
Keywords:
Periodically correlated sequence,
robust estimate,
mean square error,
least favorable spectral density,
minimax spectral characteristic
Received by editor(s):
November 21, 2011
Published electronically:
August 20, 2013
Article copyright:
© Copyright 2013
American Mathematical Society