Interpolation problem for multidimensional harmonizable stable sequences
DOI:
https://doi.org/10.17721/1812-5409.2025/2.7Keywords:
harmonizable stable random sequence, periodically harmonizable stable random sequence, optimal linear estimate, minimax-robust estimate, least favorable spectral density, minimax spectral characteristicAbstract
The problem of estimation of unobserved values of stochastic processes is of constant interest in the theory and applications of stochastic processes. The problem of forecasting future values of economic and physical processes, the problem of restoring lost information, cleaning signals, or other data from observations with noise is magnified in an information-laden world. Therefore, the development of estimation methods is one of the main tasks of the modern theory of stochastic processes. In this paper, we consider the problem of optimal linear interpolation of a functional that depends on the unknown values of a vector-valued harmonizable symmetric alpha-stable random sequence from observations of the sequence with noise. We use the classical approach to derive formulas for computing values of the mean-square error and the spectral characteristic of the optimal linear estimate of the functional. The crucial assumption of this approach is that the spectral densities of the involved stochastic sequences are exactly known. However, in practice, complete information on the spectral densities is impossible in most cases. In this situation, one finds a parametric or non-parametric estimate of the unknown spectral density and then applies one of the traditional estimation methods provided that the selected density is the true one. This procedure can result in a significant increase in the value of the estimation error. To avoid this effect, one can search for the estimates that are optimal for all densities from a certain class of admissible spectral densities. These estimates are called minimax because they minimize the maximal values of the errors of estimates for all densities from a given class. Therefore, in the case of spectral uncertainty, we use the minimax approach and propose formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of the functional for some classes of admissible spectral densities.
Pages of the article in the issue: 47 - 59
Language of the article: English
References
Cambanis, S. (1983). Complex symmetric stable variables and processes. In Sen (Ed.), Contributions to statistics: Essays in honour of Norman L. Johnson (pp. 63–79). North-Holland, New York.
Cambanis, S., & Soltani, R. (1984). Prediction of stable processes: Spectral and moving average representations. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 66, 593–612. https://doi.org/10.1007/BF00531892
Franke, J. (1984). On the robust prediction and interpolation of time series in the presence of correlated noise. Journal of Time Series Analysis, 5(4), 227–244. https://doi.org/10.1111/J.1467-9892.1984.TB00389.X
Franke, J. (1985). Minimax robust prediction of discrete time series. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 68, 337–364. https://doi.org/10.1007/BF00532645
Franke, J., & Poor, H. V. (1984). Minimax-robust filtering and finite-length robust predictors. In Franke, Härdle, & Martin (Eds.), Robust and nonlinear time series analysis (pp. 87–126). Springer-Verlag. https://doi.org/10.1007/978-1-4615-7821-5_6
Grenander, U. (1957). A prediction problem in game theory. Arkiv för Matematik, 3(4), 371–379. https://doi.org/10.1007/BF02589429
Hosoya, Y. (1982). Harmonizable stable processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 60, 517–533. https://doi.org/10.1007/BF00535714
Ioffe, A. D., & Tihomirov, V. M. (1979). Theory of extremal problems. North–Holland Publishing Company.
Kassam, S., & Poor, H. (1985). Robust techniques for signal processing: A survey. Proceedings of the IEEE, 73(3), 433–481. https://doi.org/10.1109/PROC.1985.13167
Kolmogorov, A. (1992). Selected works of A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics. Dordrecht etc.: Kluwer Academic Publishers.
Masani, P., & Wiener, N. (1957). The prediction theory of multivariate stochastic processes. Acta Mathematica, 98, 111–150. https://doi.org/10.1007/BF02404472
Moklyachuk, M., & Golichenko, I. (2016). Periodically correlated processes estimates. LAP Lambert Academic Publishing.
Moklyachuk, M., & Masyutka, O. (2012). Minimax-robust estimation technique for stationary stochastic processes. LAP Lambert Academic Publishing.
Moklyachuk, M., & Ostapenko, V. (2016). Minimax interpolation of harmonizable sequences. Theory of Probability and Mathematical Statistics, 92, 135–146. https://doi.org/10.1090/TPMS%2F988
Pourahmadi, M. (1984). On minimality and interpolation of harmonizable stable processes. Siam Journal on Applied Mathematics, 44(5), 1023–1030. https://doi.org/10.1137/0144072
Pshenichnyj, B. (1971). Necessary conditions for an extremum. Marcel Dekker.
Rockafellar, R. T. (1970). Convex analysis. Princeton University Press.
Singer, I. (1970). Best approximation in normed linear spaces by elements of linear subspaces. Springer-Verlag.
Vastola, K. S., & Poor, H. V. (1983). An analysis of the effects of spectral uncertainty on Wiener filtering. Automatica, 19(3), 289–293. https://doi.org/10.1016/0005-1098%2883%2990105-X
Weron, A. (1985). Harmonizable stable processes on groups: spectral, ergodic and interpolation properties. Zeitschrift für Wahrschein-lichkeitstheorie und Verwandte Gebiete, 68, 473–491. https://doi.org/10.1007/BF00535340
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Oleksandr Masyutka, Mikhail Moklyachuk
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
