Theory of Probability and Mathematical Statistics
Minimax interpolation of harmonizable sequences
M. P. Moklyachuk, V. I. Ostapenko
Download PDF
Abstract: The problem of estimation of the functional A_Nξ=∑_{j=0}^{N}a_jξ_j that depends on unknown values ξ_j, j=0,1,...,N, of a harmonizable symmetric α-stable random sequence ξ_n, n∈Z, by using observations of the sequence at the points n∈Z\{0,1,...,N} is studied under one of the conditions, either a condition of spectral certainty or a condition of spectral uncertainty. Expressions for calculating the value of the error and spectral characteristic of the optimal linear estimator of the functional are obtained under the condition of spectral certainty in the case where the spectral density of a sequence is known. In the case of spectral uncertainty where the spectral density of a sequence is not known but a class of admissible spectral densities is given, we propose relations to determine the least favorable spectral density and the minimax spectral characteristic.
Keywords: Harmonizable sequence, robust estimator, least favorable spectral density, minimax spectral characteristic
Bibliography: 1. S. Cambanis, Complex symmetric stable variables and processes, Contributions to Statistics, North-Holland, Amsterdam, 1983, pp. 63-79.
2. S. Cambanis and R. Soltani, Prediction of stable processes: spectral and moving average representations, Z. Wahrsch. Verw. Gebiete 66 (1984), 593-612.
3. I. I. Dubovets'ka, O. Yu. Masyutka, and M. P. Moklyachuk, Interpolation of periodically correlated stochastic sequences, Teor. Ĭmovir. Mat. Stat. 84 (2011), 43-56; English transl in. Theory Probab. Math. Stat. 84 (2012), 43-56.
4. I. I. Dubovets'ka and M. P. Moklyachuk, On minimax estimation problems for periodically correlated stochastic processes, Contemporary Math. Statist. 2 (2014), no. 1, 1-24.
5. J. Franke, Minimax robust prediction of discrete time series, Z. Wahrsch. Verw. Gebiete 68 (1985), 337-364.
6. I. I. Golichenko and M. P. Moklyachuk, Estimators for Functionals of Periodically Correlated Stochastic Processes, ''Interservis'', Kyiv, 2014. (Ukrainian)
7. U. Grenander, A prediction problem in game theory, Ark. Mat. 3 (1957), 371-379.
8. E. J. Hannan, Multiple time series, John Wiley & Sons, Inc., New York-London-Sydney, 1970.
9. Y. Hosoya, Harmonizable stable processes, Z. Wahrsch. Verw. Gebiete 60 (1982), 517-533.
10. A. D. Ioffe and V. M. Tihomirov, Theory of extremal problems, Studies in Mathematics and its Applications, vol. 6, North-Holland Publishing Company, Amsterdam-New York-Oxford, 1979.
11. S. A. Kassam and H. V. Poor, Robust techniques for signal processing: A survey, Proc. IEEE 73 (1985), 433-481.
12. 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, 1992.
13. M. G. Kreĭn and A. A. Nudel'man, The Markov moment problem and extremal problems, ''Nauka'', Moscow, 1973; English transl., American Mathematical Society, Providence, R.I., 1977.
14. M. M. Luz and M. P. Moklyachuk, Interpolation for functionals of random sequences with stationary increments constructed from observations with the noise, Appl. Stat. Actuar. Finance Math. 2 (2012), 131-148. (Ukrainian)
15. M. M. Luz and M. P. Moklyachuk, Interpolation of functionals of stochastic sequences with stationary increments, Teor. Ĭmovir. Mat. Stat. 87 (2012), 105-119; English transl. in Theory Probab. Math. Stat. 87 (2013), 117-133.
16. M. M. Luz and M. P. Moklyachuk, Minimax interpolation problem for random processes with stationary increments, Stat. Optim. Inf. Comput. 3 (2015), 30-41.
17. M. P. Moklyachuk, Robust procedures in time series analysis, Theory Stoch. Process. 6 (2000), no. 3-4, 127-147.
18. M. P. Moklyachuk, Game theory and convex optimization methods in robust estimation problems, Theory Stoch. Process. 7 (2001), no. 1-2, 253-264.
19. M. P. Moklyachuk, Robust Estimators for Functionals of Stochastic Processes, ''Kyiv University'', Kyiv, 2008. (Ukrainian)
20. M. P. Moklyachuk, Nonsmooth Analysis and Optimization, ''Kyiv University'', Kyiv, 2008. (Ukrainian)
21. M. P. Moklyachuk and I. I. Dubovets'ka, Minimax interpolation of periodically correlated processes, Nauk. Visnyk Uzhgorod Univ. Ser. Mat. Inform. 23 (2012), no. 2, 51-62. (Ukrainian)
22. M. P. Moklyachuk and O. Yu. Masyutka, Interpolation of multidimensional stationary sequences, Teor. Ĭmovir. Mat. Stat. 73 (2005), 112-119; English transl in. Theory Probab. Math. Stat. 73 (2006), 125-133.
23. M. P. Moklyachuk and O. Yu. Masyutka, Robust estimation problems for stochastic processes, Theory Stoch. Process. 12 (2006), no. 3-4, 88-113.
24. M. Moklyachuk and O. Masyutka, Minimax-Robust Estimation Technique for Stationary Stochastic Processes, LAP LAMBERT Academic Publishing, 2012.
25. M. Pourahmadi, On minimality and interpolation of harmonizable stable processes, SIAM J. Appl. Math. 44 (1984), no. 5, 1023-1030.
26. I. Singer, Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer, Berlin-Heidelberg, 1970.
27. K. S. Vastola and H. V. Poor, An analysis of the effects of spectral uncertainty on Wiener filtering, Automatica 28 (1983), 289-293.
28. A. Weron, Harmonizable stable processes on groups: spectral, ergodic and interpolation properties, Z. Wahrsch. Verw. Gebiete 68 (1985), no. 4, 473-491.
29. N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series. With Engineering Applications, M. I. T. Press, Massachusetts Institute of Technology, Cambridge, Mass., 1966.
30. A. M. Yaglom, Correlation Theory of Stationary and Related Random Functions, vol. 1: Basic Results, Springer Series in Statistics, Springer-Verlag, New York, 1987.
31. 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, 1987.
