# Filtering Problem for Functionals of Stationary Sequences

### Abstract

In this paper, we consider the problem of the mean-square optimal linear estimation of functionals which depend on the unknown values of a stationary stochastic sequence from observations with noise. In the case of spectral certainty in which the spectral densities of the sequences are exactly known, we propose formulas for calculating the spectral characteristic and value of the mean-square error of the estimate by using the Fourier coefficients of some functions from the spectral densities. When the spectral densities are not exactly known but a class of admissible spectral densities is given, the minimax-robust method of estimation is applied. Formulas for determining the least favourable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of the functionals are proposed for some specific classes of admissible spectral densities.### References

G. E. P. Box, G. M. Jenkins, and G. C. Reinsel, Time series analysis. Forecasting and control. 3rd ed., Englewood Cliffs, NJ: Prentice Hall, 1994.

I. I. Dubovets'ka, O.Yu. Masyutka and M.P. Moklyachuk, Interpolation of periodically correlated stochastic sequences, Theory of Probability and Mathematical Statistics, vol. 84, pp. 43-56, 2012.

I. I. Dubovets'ka and M. P. Moklyachuk, Filtration of linear functionals of periodically correlated sequences, Theory of Probability and Mathematical Statistics, vol. 86, pp. 51-64, 2013.

I. I. Dubovets'ka and M. P. Moklyachuk, Extrapolation of periodically correlated processes from observations with noise,Theory of Probability and Mathematical Statistics, vol. 88, pp. 43-55, 2013.

I. I. Dubovets'ka and M. P. Moklyachuk,Minimax estimation problem for periodically correlated stochastic processes,Journal of Mathematics and System Science, vol. 3, no. 1, pp. 26-30, 2013.

I. I. Dubovets'ka and M. P. Moklyachuk,

On minimax estimation problems for periodically correlated stochastic processes, Contemporary Mathematics and Statistics, vol.2, no. 1, pp. 123-150, 2014.

J. Franke, Minimax robust prediction of discrete time series, Z. Wahrscheinlichkeitstheor. Verw. Gebiete, vol. 68, pp. 337--364, 1985.

J. Franke and H. V. Poor, Minimax-robust filtering and finite-length robust predictors, Robust and Nonlinear Time Series Analysis. Lecture Notes in Statistics, Springer-Verlag, vol. 26, pp. 87--126, 1984.

I. I. Gikhman and A. V. Skorokhod, The theory of stochastic processes. I., Berlin: Springer, 2004.

I. I. Golichenko and M. P. Moklyachuk, Estimates of functionals of periodically correlated processes, Kyiv: NVP ”Interservis”, 2014.

U. Grenander, A prediction problem in game theory, Arkiv f"or Matematik, vol. 3, pp. 371--379, 1957.

E. J. Hannan, Multiple time series, Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons, Inc. XI, 1970.

K. Karhunen, "Uber lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae. Ser. A I, vol. 37, 1947.

S. A. Kassam and H. V. Poor, Robust techniques for signal processing: A survey, Proceedings of the IEEE, vol. 73, no. 3, pp. 433--481, 1985.

A. N. Kolmogorov, Selected works by A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics. Ed. by A.N. Shiryayev, Mathematics and Its Applications. Soviet Series. 26. Dordrecht etc. Kluwer Academic Publishers, 1992.

M. M. Luz and M. P. Moklyachuk, Interpolation of functionals of stochastic sequences with stationary increments, Theory of Probability and Mathematical Statistics, vol. 87, pp. 117-133, 2013.

M. M. Luz and M. P. Moklyachuk, Interpolation of functionals of stochastic sequences with stationary increments for observations with noise, Prykl. Stat., Aktuarna Finans. Mat., no. 2, pp. 131-148, 2012.

M. M. Luz and M. P. Moklyachuk, Minimax-robust filtering problem for stochastic sequence with stationary increments, Theory of Probability and Mathematical Statistics, vol. 89, pp. 127-142, 2014.

M. Luz and M. Moklyachuk, Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences, Statistics, Optimization & Information Computing, vol. 2, no. 3, pp. 176 - 199, 2014.

M. Luz and M. Moklyachuk, Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences, Statistics, Optimization & Information Computing, vol. 3, no. 2, pp. 160-188, 2015.

M. P. Moklyachuk, Minimax filtering of linear transformations of stationary sequences, Ukrainian Mathematical Journal, vol. 43, no. 1 pp. 92-99, 1991.

M. P. Moklyachuk, Stochastic autoregressive sequences and minimax interpolation, Theory of Probability and Mathematical Statistics, vol. 48, pp. 95-103, 1994.

M. P. Moklyachuk, Robust procedures in time series analysis, Theory of Stochastic Processes, vol. 6, no. 3-4, pp. 127-147, 2000.

M. P. Moklyachuk, Game theory and convex optimization methods in robust estimation problems, Theory of Stochastic Processes, vol. 7, no. 1-2, pp. 253--264, 2001.

M. P. Moklyachuk, Robust estimations of functionals of stochastic processes, Kyiv University, Kyiv, 2008.

M. P. Moklyachuk, Nonsmooth analysis and optimization, Kyiv University, Kyiv, 2008.

M. Moklyachuk and O. Masyutka, Minimax-robust estimation technique for stationary stochastic processes, LAP LAMBERT Academic Publishing, 2012.

B. N. Pshenichnyj, Necessary conditions of an extremum, Pure and Applied mathematics. 4. New York: Marcel Dekker, 1971.

Frigyes Riesz and B'ela Sz.--Nagy, Functional analysis, Dover Books on Advanced Mathematics. New York: Dover Publications, Inc. XII, 1990.

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1997.

Yu. A. Rozanov, Stationary stochastic processes, San Francisco-Cambridge-London-Amsterdam: Holden-Day, 1967.

N. Wiener, Extrapolation, interpolation and smoothing of stationary time series. With engineering applications, The M. I. T. Press, Massachusetts Institute of Technology, Cambridge, Mass., 1966.

A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. 1: Basic results, Springer Series in Statistics, Springer-Verlag, New York etc., 1987.

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 etc., 1987.

*Statistics, Optimization & Information Computing*,

*4*(1), 68-83. https://doi.org/10.19139/soic.v4i1.172

- 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).