Theory of Probability and Mathematical Statistics
Estimates of functionals of stochastic sequences with periodically stationary increments
P. S. Kozak, M. P. Moklyachuk
Download PDF
Abstract: The problem of optimal estimation of the linear functional depending on the unknown values of a stochastic sequence with periodically stationary increments from observations of the sequence at points Z\{0,1,...,N} is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed in the case where spectral density is exactly known. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for the given sets of admissible spectral densities.
Keywords: Sequence with periodically stationary increments, robust estimate, mean square error, least favorable spectral density, minimax spectral characteristics
Bibliography: 1. I. I. Gikhman, A. V. Skorokhod, The theory of stochastic processes, vol. 1, Springer, Berlin, 2004.
2. U. Grenander, A prediction problem in game theory, Ark. Mat., 3 (1957), 371-379.
3. J. Franke, Minimax robust prediction of discrete time series, Z. Wahrsch. Verw. Gebiete, 68 (1985), 337-364.
4. S. A. Kassam, H. V. Poor, Robust techniques for signal processing: A survey, Proc. IEEE, 73 (1985), 433-481.
5. A. N. Kolmogorov, Selected works by A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics (A. N. Shiryayev, ed.), Mathematics and Its Applications, Soviet Series, vol. 26, Kluwer Academic Publishers, Dordrecht etc., 1992.
6. M. M. Luz, M. P. Moklyachuk, Interpolation of functionals of stochactic sequenses with stationary increments, Theory Probab. Math. Statist., 87 (2012), 117-133.
7. M. M. Luz, M. P. Moklyachuk, Minimax-robust ltering problem for stochastic sequences with stationary increments, Theory Probab. Math. Statist., 89 (2014), 127-142.
8. M. M. Luz, M. P. Moklyachuk, Minimax-robust ltering problem for stochastic sequences with stationary increments and cointegrated sequences, Stat. Optim. Inf. Comput., 2 (2014), no. 3, 176-199.
9. M. M. Luz, M. P. Moklyachuk, Minimax interpolation problem for random processes with stationary increments, Stat. Optim. Inf. Comput., 3 (2015), no. 1, 30-41.
10. M. M. Luz, M. P. Moklyachuk, Filtering problem for functionals of stationary sequences, Stat.Optim. Inf. Comput., 4 (2016), no. 1, 68-83.
11. M. M. Luz, M. P. Moklyachuk, Minimax interpolation of stochastic processes with stationary increments from observations with noise, Theory Probab. Math. Statist., 94 (2017), 121-135.
12. M. M. Luz, M. P. Moklyachuk, Estimates of functionals of random processes with stationary increments and cointegrated sequences, Interservice, Kyiv, 2016. (Ukrainian)
13. M. P. Moklyachuk, Robust procedures in time series analysis, Theory Stoch. Process., 6(22) (2000), no. 34, 127-147.
14. M. P. Moklyachuk, Game theory and convex optimization methods in robust estimation problems, Theory Stoch. Process. 7(23) (2001), no. 12, 253-264.
15. M. P. Moklyachuk, Robust Estimations of Functionals of Stochastic Processes, Kyivskyi Universytet, Kyiv, 2008. (Ukrainian)
16. M. P. Moklyachuk, Minimax-robust estimation problems for stationary stochastic sequences, Stat. Optim. Inf. Comput., 3 (2015), no. 4, 348-419.
17. M. P. Moklyachuk, M. I. Sidei, Interpolation problem for stationary sequences with missing observations, Stat. Optim. Inf. Comput., 3 (2015), no. 3, 259-275.
18. M. P. Moklyachuk, M. I. Sidei, Filtering problem for stationary sequences with missing observations, Stat. Optim. Inf. Comput., 4 (2016), no. 4, 308-325.
19. M. P. Moklyachuk, M. I. Sidei, Extrapolation problem for stationary sequences with missing observations, Stat. Optim. Inf. Comput., 5 (2017), no. 1, 212-233.
20. M. P. Moklyachuk, I. I. Golichenko, Periodically correlated processes estimates, Lambert Academic Publishing, 2016.
21. M. P. Moklyachuk, A. Yu. Masyutka, Minimax-robust estimation technique for stationary stochastic processes, Lambert Academic Publishing, 2012.
22. M. S. Pinsker, A. M. Yaglom, On linear extrapolation of random processes with nth stationary increments, Dokl. Akad. Nauk SSSR, 94 (1954), no. 3, 385-388. (Russian)
23. M. S. Pinsker, The theory of curves with nth stationary increments in Hilbert spaces, Izv. Akad.Nauk SSSR. Ser. Mat., 19 (1955), no. 3, 319-344. (Russian)
24. B. N. Pshenichnyi, Necessary conditions for an extremum, Pure and Applied mathematics,vol. 4, Marcel Dekker, New York, 1971.
25. Yu. A. Rozanov, Stationary stochastic processes, Holden-Day, San Francisco-Cambridge-London-Amsterdam, 1967.
26. N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series. Whith Engineering Applications, Massachusetts Institute of Technology, Cambridge, 1966.
27. A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. 1: Basicresults, Springer Series in Statistics, Springer-Verlag, New York etc., 1987.
28. A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. 2: Suplementary notes and references, Springer Series in Statistics, Springer-Verlag, New York etc., 1987.
29. A. M. Yaglom, Correlation theory of processes with random stationary nth increments, Mat.Sb., 37(79) (1955), no. 1, 141-196. (Russian)
