Filtration of linear functionals of periodically correlated sequences

Authors:
I. I. Dubovets′ka and M. P. Moklyachuk

Translated by:
N. Semenov

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **86** (2012).

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

**1.**W. R. Bennett,*Statistics of regenerative digital transmission*, Bell System Tech. J.**37**(1958), 1501–1542. MR**102138**, https://doi.org/10.1002/j.1538-7305.1958.tb01560.x**2.**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**, https://doi.org/10.1111/j.1467-9892.1984.tb00389.x**3.**Jürgen Franke,*Minimax-robust prediction of discrete time series*, Z. Wahrsch. Verw. Gebiete**68**(1985), no. 3, 337–364. MR**771471**, https://doi.org/10.1007/BF00532645**4.**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**, https://doi.org/10.1007/978-1-4615-7821-5_6**5.**E. G. Gladyšev,*Periodically correlated random sequences*, Dokl. Akad. Nauk SSSR**137**(1961), 1026–1029 (Russian). MR**0126873****6.**Ulf Grenander,*A prediction problem in game theory*, Ark. Mat.**3**(1957), 371–379. MR**90486**, https://doi.org/10.1007/BF02589429**7.**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****8.**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****9.**Andrzej Makagon,*Theoretical prediction of periodically correlated sequences*, Probab. Math. Statist.**19**(1999), no. 2, Acta Univ. Wratislav. No. 2198, 287–322. MR**1750905****10.**Andrzej Makagon,*Stationary sequences associated with a periodically correlated sequence*, Probab. Math. Statist.**31**(2011), no. 2, 263–283. MR**2853678****11.**M. P. Moklyachuk,*Estimates of stochastic processes from observations with noise*, Theory Stoch. Process.**3**(**19**) (1997), no. 3-4, 330-338.**12.**M. P. Moklyachuk,*Robust procedures in time series analysis*, Theory Stoch. Process.**6**(**22**) (2000), no. 3-4, 127-147.**13.**M. P. Moklyachuk,*Game theory and convex optimization methods in robust estimation problems*, Theory Stoch. Process.**7**(**23**) (2001), no. 1-2, 253-264.**14.**M. P. Moklyachuk,*Robust Estimates for Functionals of Stochastic Processes*, ``Kyivs'kyi Universytet'', Kyiv, 2008. (Ukrainian)**15.**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**, https://doi.org/10.1090/S0094-9000-08-00718-7**16.**B. N. Pshenichnyĭ,*Neobkhodimye usloviya èkstremuma*, Optimizatsiya i Issledovanie Operatsiĭ . [Optimization and Operations Research], “Nauka”, Moscow, 1982 (Russian). MR**686452****17.**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****18.**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****19.**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****20.**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**

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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

DOI:
https://doi.org/10.1090/S0094-9000-2013-00888-6

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