Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations
Abstract
This paper focuses on the problem of the mean square optimal estimation of linear functionals which dependon the unknown values of a multidimensional stationary stochastic sequence. Estimates are based on observations of these quence with an additive stationary noise sequence. The aim of the paper is to develop methods of finding the optimal estimates of the functionals in the case of missing observations. The problem is investigated in the case of spectral certainty where the spectral densities of the sequences are exactly known. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of functionals are derived under the condition of spectral certainty.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 of the optimal estimates of functionals are proposed for some special sets of admissible densities.References
P. Bondon, Influence of missing values on the prediction of a stationary time series, Journal of Time Series Analysis, vol. 26, no.4, pp. 519–525, 2005.
P. Bondon, Prediction with incomplete past of a stationary process, Stochastic Processes and their Applications. vol. 98, pp. 67–76,2002.
G. E. P. Box, G. M. Jenkins, G. C. Reinsel, and G. M. Ljung, Time series analysis. Forecasting and control. 5rd ed., Wiley, 2016.
P. J. Brockwell, and R. A. Davis, Time series: Theory and methods. 2nd ed.. New York: Springer, 1998.
R. Cheng, A. G. Miamee, and M. Pourahmadi, Some extremal problems in Lp(w), Proceedings of the American Mathematical Society, vol. 126, pp. 2333–2340, 1998.
R. Cheng, and M. Pourahmadi, Prediction with incomplete past and interpolation of missing values, Statistics & Probability Letters, vol. 33, pp. 341–346, 1996.
J. Franke, On the robust prediction and interpolation of time series in the presence of correlated noise. Journal of Time Series Analysis, vol. 5, no. 4, pp. 227–244, 1984.
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.
U. Grenander, A prediction problem in game theory, Arkiv för 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.
A. D. Ioffe, and V. M. Tihomirov, Theory of extremal problems, Studies in Mathematics and its Applications, Vol. 6. Amsterdam, New York, Oxford: North-Holland Publishing Company. XII, 1979.14. K. Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae. Ser. A I, vol.37, 1947.
Y. Kasahara, M. Pourahmadi, and A. Inoue, Duals of random vectors and processes with applications to prediction problems with missing values, Statistics & Probability Letters, vol. 79, no. 14, pp. 1637–1646, 2009.
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, 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. M. Luz, and M. P. Moklyachuk, Minimax prediction of stochastic processes with stationary increments from observations with stationary noise, Cogent Mathematics, vol. 3:1133219, pp. 1–17, 2016.
M. M. Luz, and M. P. Moklyachuk, Estimates of functionals from processes with stationary increments and cointegrated sequences, Kyiv: NVP ”Interservis”, 2016.
M. P. Moklyachuk, Nonsmooth analysis and optimization, Kyiv University, Kyiv, 2008.
M. P. Moklyachuk, Robust estimations of functionals of stochastic processes, Kyiv University, Kyiv, 2008.
M. P. Moklyachuk, Minimax-robust estimation problems for stationary stochastic sequences, Statistics, Optimization & Information Computing, vol. 3, no. 4, pp. 348–419, 2015.
M. P. Moklyachuk, and I. I. Golichenko, Periodically correlated processes estimates, LAP Lambert Academic Publishing, 2016.
M. Moklyachuk, and O. Masyutka, Minimax prediction problem for multidimensional stationary stochastic sequences, Theory of Stochastic Processes, vol. 14, no. 3-4, pp. 89–103, 2008.
M. P. Moklyachuk, and O. Yu. Masyutka, Minimax prediction problem for multidimensional stationary stochastic processes, Communications in Statistics – Theory and Methods, vol. 40, no. 19-20, pp. 3700–3710, 2011.
M. Moklyachuk, and O. Masyutka, Minimax-robust estimation technique for stationary stochastic processes, LAP Lambert Academic Publishing, 2012.
M. P. Moklyachuk, and M. I. Sidei, Interpolation of stationary sequences observed with the noise, Theory of Probability and Mathematical Statistics, vol. 93, pp. 143–156, 2015.
M. P. Moklyachuk, and M. I. Sidei,Interpolation problem for stationary sequences with missing observations,Statistics, Optimization & Information Computing, vol. 3, no. 3, pp. 259–275, 2015.
M. P. Moklyachuk, and M. I. Sidei, Filtering problem for stationary sequences with missing observations. Statistics, Optimization & Information Computing, vol. 4, no. 4, pp. 308–325, 2016.
M. P. Moklyachuk, and M. I. Sidei,Extrapolation problem for stationary sequences with missing observations,Statistics, Optimization & Information Computing, vol. 5, no. 3, pp. 212–233, 2017.
M. M. Pelagatti, Time series modelling with unobserved components, New York: CRC Press, 2015.
M. Pourahmadi, A. Inoue, and Y. Kasahara A prediction problem in L2(w). Proceedings of the American Mathematical Society,Vol. 135, No. 4, pp. 1233–1239, 2007.
B. N. Pshenichnyj, Necessary conditions of an extremum, Pure and Applied mathematics. 4. New York: Marcel Dekker, 1971.
R. T. Rockafellar, Convex Analysis, Princeton University Press, 1997.
Yu. A. Rozanov, Stationary stochastic processes, San Francisco Cambridge-London-Amsterdam: Holden-Day, 1967.
H. Salehi, Algorithms for linear interpolator and interpolation error for minimal stationary stochastic processes, The Annals of Probability, Vol. 7, No. 5, pp. 840–846, 1979.
K. S. Vastola, and H. V. Poor, An analysis of the effects of spectral uncertainty on Wiener filtering, Automatica, vol. 28, pp. 289–293,1983.
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, SpringerSeries in Statistics, Springer-Verlag, New York etc., 1987.
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