# Extrapolation Problem for Stationary Sequences with Missing Observations

### Abstract

In this paper, we consider the problem of the mean square optimal estimation of linear functionals which depend on unknown values of a stationary stochastic sequence based on observations of the sequence with a stationary noise. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functional are derived under the condition of spectral certainty, where spectral densities of the sequences are exactly known. The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the sequences are not known exactly while sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some special sets of admissible densities.### References

Bondon, P. (2002). Prediction with incomplete past of a stationary process.

Stochastic Process and their Applications, 98, 67-76.

Bondon, P. (2005). Influence of missing values on the prediction of a stationary time series. Journal of Time Series Analysis, 26(4), 519-525.

Brockwell, P. J. and Davis, R. A. (1991). Time series: Theory and methods (2nd ed.). New York: Springer.

Cheng, R., Miamee, A.G., and Pourahmadi, M. (1998). Some extremal problems in $L^p(w)$. Proc. Am. Math. Soc., 126, 2333-2340.

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

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

Gantmacher, F. R. (1959). Applications of the theory of matrices.

Interscience publishers, inc., New York.

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

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

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

Ioffe, A. D. and Tihomirov, V. M. (1979). Theory of extremal problems. Amsterdam, New York, Oxford: North--Holland Publishing Company.

Karhunen, K. (1947). Uber lineare Methoden in der Wahrscheinlichkeitsrechnung. Annales Academiae Scientiarum Fennicae. Series A I. Mathematica 37, 3-79.

Kasahara, Y., Pourahmadi, M. and Inoue, A. (2009). Duals of random vectors and processes with applications to prediction problems with missing values.

Statistics & Probability Letters, 79(14), 1637-1646.

Kassam, S. A. and Poor, H. V. (1985). Robust techniques for signal processing: A survey. Proceedings of the IEEE, 73(3), 433-481.

Kolmogorov, A.N. (1992). Selected works of A. N. Kolmogorov. Volume II: Probability theory and mathematical statistics. Edited by A. N. Shiryayev. Transl. from the Russian by G. Lindquist. Dordrecht etc.: Kluwer Academic Publishers.

Krein, M. G. and Nudelman, A. A. (1977). The Markov moment problem and extremal problems. Ideas and problems of P.L.~Cebysev and A.A.~Markov and their further development. Translations of Mathematical Monographs. Vol. 50. Providence, R.I.: American Mathematical Society (AMS).

Luz, M. and Moklyachuk, M. (2014). Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences.

Statistics, Optimization & Information Computing, 2(3), 176-199.

Luz, M. and Moklyachuk, M. (2016). Filtering problem for functionals of stationary sequences. Statistics, Optimization & Information Computing, 4(1), 68-83.

Luz, M. and Moklyachuk, M. (2016). Estimates of functionals from processes with stationary increments and cointegrated sequences. NVP "Interservis"

Moklyachuk, M. P. (2000). Robust procedures in time series analysis. Theory Stoch. Process. 6(3-4), 127-147.

Moklyachuk, M. P. (2001). Game theory and convex optimization methods in robust estimation problems. Theory of Stochastic Processes, 7(1-2), 253-264.

Moklyachuk, M. P. (2008). Robust estimates for functionals of stochastic processes. Kyiv: Kyiv University Publishing.

Moklyachuk, M. (2015). Minimax-robust estimation problems for stationary stochastic sequences. Statistics, Optimization & Information Computing, 3(4), 348-419.

Moklyachuk, M. and Golichenko, I. (2016). Periodically correlated processes estimates. LAP Lambert Academic Publishing.

Moklyachuk, M. P. and Masyutka, O. Yu. (2012). Minimax-robust estimation technique for stationary stochastic processes. LAP Lambert Academic Publishing.

Moklyachuk, M. and Sidei, M. (2015a). Interpolation of stationary sequences observed with the noise. Theory of Probability and Mathematical Statistics, 93, 143-156.

Moklyachuk, M. P. and Sidei, M. I. (2015b). Interpolation problem for stationary sequences with missing observations. Statistics, Optimization & Information Computing, 3(3), 259-275.

Moklyachuk, M. P. and Sidei, M. I. (2016a). Extrapolation problem for functionals of stationary processes with missing observations. Bukovinian Mathematical Journal, 4(1-2), 122-129.

Moklyachuk, M. P. and Sidei, M. I. (2016b). Interpolation of functionals of stationary processes with missing observations. Bulletin of Taras Shevchenko National University of Kyiv, 1, 24-30.

Moklyachuk, M. P. and Sidei, M. I. (2016c). Filtering problem for stationary sequences with missing observations. Statistics, Optimization & Information Computing, 4(4), 308 - 325.

Moklyachuk, M. P. and Sidei, M. I. (2016d). Filtering Problem for functionals of stationary processes with missing observations. Communications in Optimization Theory, 2016, pp.1-18, Article ID 21.

Pelagatti, M. M. (2015). Time series modelling with unobserved components, New York: CRC Press.

Pourahmadi, M. (2001). Foundations of time series analysis and prediction

theory. New York: Wiley.

Pourahmadi, M., Inoue, A. and Kasahara, Y. (2007). A prediction problem in $L^2(w)$. Proceedings of the American Mathematical Society, 135(4), 1233-1239.

Pshenichnyi, B. N. (1971). Necessary conditions for an extremum. Pure and Applied mathematics. 4.; New York: Marcel Dekker, Inc. XVIII.

Rockafellar, R. T. (1997). Convex analysis. Princeton, NJ: Princeton University Press.

Rozanov, Yu. A. (1967). Stationary stochastic processes. Holden-Day, San Francisco.

Vastola, K. S. and Poor, H. V. (1983). An analysis of the effects of spectral uncertainty on Wiener filtering. Automatica, 28, 289-293.

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

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

Yaglom, A. M. (1987b). Correlation theory of stationary and related random functions. Vol. 2: Supplementary notes and references. Springer Series in Statistics, Springer-Verlag, New York etc.

*Statistics, Optimization & Information Computing*,

*5*(3), 212-233. https://doi.org/10.19139/soic.v5i3.284

- 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).