Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

Interpolation of multidimensional stationary sequences


Authors: M. P. Moklyachuk and O. Yu. Masyutka
Translated by: M. P. Moklyachuk
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 73 (2005).
Journal: Theor. Probability and Math. Statist. 73 (2006), 125-133
MSC (2000): Primary 60G10; Secondary 60G35, 62M20, 93E10, 93E11
Published electronically: January 17, 2007
MathSciNet review: 2213847
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of optimal estimation is considered for the linear functional $ A_{N}\vec{\xi}=\sum_{j=0}^{N}\vec{a}(j)\vec {\xi }(j)$, where $ \{\vec{\xi}(j)\}$ and $ \{\vec{\eta}(j)\}$ are multidimensional stationary stochastic sequences. The estimation is based on observations of the sequence $ \vec {\xi} (j)+\vec {\eta} (j)$ for $ j \in Z\setminus {\left\{ {0,\dots, N} \right\}}$. We obtain formulas for calculating the mean-square error and spectral characteristic of the optimal estimate of the functional. The least favorable spectral densities and minimax spectral characteristics of the optimal estimates of the functional are found for some classes of spectral densities.


References [Enhancements On Off] (What's this?)

  • 1. 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 (86i:62192), http://dx.doi.org/10.1111/j.1467-9892.1984.tb00389.x
  • 2. Ulf Grenander, A prediction problem in game theory, Ark Mat. 3 (1957), 371–379. MR 0090486 (19,822g)
  • 3. E. J. Hannan, Multiple time series, John Wiley and Sons, Inc., New York-London-Sydney, 1970. MR 0279952 (43 #5673)
  • 4. Thomas Kailath, A view of three decades of linear filtering theory, IEEE Trans. Information Theory IT-20 (1974), 146–181. MR 0465437 (57 #5337)
  • 5. S. A. Kassam and H. V. Poor, Robust techniques for signal processing: A survey, Proc. IEEE 73 (1985), no. 3, 433-481.
  • 6. 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 (92j:01071)
  • 7. M. P. Moklyachuk, Stochastic autoregressive sequences and minimax interpolation, Teor. Ĭmovīr. Mat. Stat. 48 (1993), 135–146 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 48 (1994), 95–103. MR 1445234 (98b:60081)
  • 8. M. P. Moklyachuk, Interpolation of vector-valued stochastic processes, Exploring stochastic laws, VSP, Utrecht, 1995, pp. 329–341. MR 1714017 (2001d:60011)
  • 9. M. P. Moklyachuk, Robust procedures in time series analysis, Theory Stoch. Process. 6(22) (2000), no. 3-4, 127-147.
  • 10. M. P. Moklyachuk, Game theory and convex optimization methods in robust estimation problems, Theory Stoch. Process. 7(23) (2001), no. 1-2, 253-264.
  • 11. Mohsen Pourahmadi, Robust prediction of multivariate stationary processes, Sankhyā Ser. A 52 (1990), no. 1, 115–126. MR 1176280 (93e:60082)
  • 12. Yu. A. Rozanov, Statsionarnye sluchainye protsessy, 2nd ed., \cyr Teoriya Veroyatnosteĭ i Matematicheskaya Statistika [Probability Theory and Mathematical Statistics], vol. 42, “Nauka”, Moscow, 1990 (Russian). MR 1090826 (92d:60046)
  • 13. K. S. Vastola and H. V. Poor, An analysis of the effects of spectral uncertainty on Wiener filtering, Automatica J. IFAC 28 (1983), 289-293.
  • 14. Norbert Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series. With Engineering Applications, The Technology Press of the Massachusetts Institute of Technology, Cambridge, Mass, 1949. MR 0031213 (11,118j)
  • 15. 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 (89a:60105)
  • 16. 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 (89a:60106)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60G10, 60G35, 62M20, 93E10, 93E11

Retrieve articles in all journals with MSC (2000): 60G10, 60G35, 62M20, 93E10, 93E11


Additional Information

M. P. Moklyachuk
Affiliation: Department of Probability Theory and Mathematical Statistics, Mechanics and Mathematics Faculty, Kyiv National Taras Shevchenko University, Volodymyrs’ka Street 64, Kyiv 01033, Ukraine
Email: mmp@univ.kiev.ua

O. Yu. Masyutka
Affiliation: Department of Probability Theory and Mathematical Statistics, Mechanics and Mathematics Faculty, Kyiv National Taras Shevchenko University, Volodymyrs’ka Street 64, Kyiv 01033, Ukraine

DOI: http://dx.doi.org/10.1090/S0094-9000-07-00687-4
PII: S 0094-9000(07)00687-4
Keywords: Multidimensional stationary sequence, optimal linear estimate, mean-square error, spectral characteristic, least favorable spectral density, minimax-robust spectral characteristic
Received by editor(s): May 20, 2005
Published electronically: January 17, 2007
Dedicated: This paper is dedicated to our teacher Mykhaĭlo Ĭosypovych Yadrenko.
Article copyright: © Copyright 2007 American Mathematical Society