Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences

Maksym Luz, Mikhail Moklyachuk

Abstract


The problem of optimal estimation of the linear functionals $A {\xi}=\sum_{k=0}^{\infty}a (k)\xi(k)$ and $A_N{\xi}=\sum_{k=0}^{N}a (k)\xi(k)$ which depend on the unknown values of a stochastic sequence $\xi(m)$ with stationary $n$th increments is considered. Estimates are obtained which are based on observations of the sequence $\xi(m)+\eta(m)$ at points of time $m=-1,-2,\ldots$, where the sequence $\eta(m)$ is stationary and uncorrelated with the sequence $\xi(m)$. Formulas for calculating the mean-square errors and spectral characteristics of the optimal estimates of the functionals are derived in the case of spectral certainty, where spectral densities of the sequences $\xi(m)$ and $\eta(m)$ are exactly known. These results are applied for solving extrapolation problem for cointegrated sequences. In the case where spectral densities of the sequences are not known exactly, but sets of admissible spectral densities are given, the minimax-robust method of estimation is applied. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special classes of admissible densities. 


Keywords


Stochastic sequence with stationary increments; cointegrated sequence; minimax-robust estimate; mean square error; least favorable spectral density; minimax-robust spectral characteristic

References


W. Bell, Signal extraction for nonstationary time series, The Annals of Statistics, vol. 12, no. 2, pp. 646-664, 1984.

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

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

C. W. J. Granger, Cointegrated variables and error correction models, UCSD Discussion paper, 83-13a, 1983.

I. I. Dubovets’ka, O.Yu. Masyutka and M.P. Moklyachuk, Interpolation of periodically correlated stochastic sequences, Theory Probab. Math. Stat., vol. 84, pp. 43–56, 2012.

I. I. Dubovets’ka and M. P. Moklyachuk, Filtration of linear functionals of periodically correlated sequences, Theory Probab. Math. Stat., vol. 86, pp. 51–64, 2013.

I. I. Dubovets’ka and M. P. Moklyachuk, Extrapolation of periodically correlated processes from observations with noise, Theory Probab. Math. Stat., vol. 88, pp. 67–83, 2014.

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.

R. F. Engle and C. W. J. Granger, Co-integration and error correction: Representation, estimation and testing, Econometrica, vol. 55, pp. 251-276, 1987.

U. Grenander, A prediction problem in game theory, Arkiv för Matematik, vol. 3, pp. 371–379, 1957.

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.

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

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 Probab. Math. Stat., 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 Probab. Math. Stat., vol. 89, pp. 127–142, 2014.

M. Luz and M. Moklyachuk, Robust extrapolation problem for stochastic processes with stationary increments, Mathematics and Statistics, vol. 2, no. 2, pp. 78–88, 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 interpolation problem for random processes with stationary increments, Statistics, Optimization & Information Computing, vol. 3, no. 1, pp. 30–41, 2015.

M. Moklyachuk and M. Luz, Robust extrapolation problem for stochastic sequences with stationary increments, Contemporary Mathematics and Statistics, vol. 1, no. 3, pp. 123–150, 2013.

M. P. Moklyachuk, Minimax extrapolation and autoregressive-moving average processes, Theory Probab. Math. Stat., vol. 41, pp. 77–84, 1990.

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. Moklyachuk and A. Masyutka, Extrapolation of multidimensional stationary processes, Random Operators and Stochastic Equations, vol. 14, no. 3, pp.233–244, 2006.

M. Moklyachuk and A. Masyutka, Robust estimation problems for stochastic processes, Theory of Stochastic Processes, vol. 12, no. 3-4, pp. 88–113, 2006.

M. Moklyachuk and A. Masyutka, Robust filtering of stochastic processes, Theory of Stochastic Processes, vol. 13, no. 1-2, pp. 166–181, 2007.

M. Moklyachuk and A. Masyutka, Minimax prediction problem for multidimensional stationary stochastic sequences, Theory of Stochastic Processes, vol. 14, no. 3-4, pp. 89–103, 2008.

M. Moklyachuk and A. Masyutka, Minimax prediction problem for multidimensional stationary stochastic processes, Communications in Statistics – Theory and Methods., vol. 40, no. 19-20, pp. 3700–3710, 2001.

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

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

M. S. Pinsker and A. M. Yaglom, On linear extrapolation of random processes with nth stationary increments, Doklady Akademii Nauk SSSR, vol. 94, pp. 385–388, 1954.

M. S. Pinsker, The theory of curves with nth stationary increments in Hilber spaces, Izvestiya Akademii Nauk SSSR. Ser. Mat., vol. 19, no. 5, pp. 319–344, 1955.

B. N. Pshenichnyi, Necessary conditions of an extremum, “Nauka”, Moskva, 1982.

Yu. A. Rozanov, Stationary stochastic processes. 2nd rev. ed., “Nauka”, Moskva, 1990.

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.

A. M. Yaglom, Correlation theory of stationary and related random processes with stationary nth increments, Mat. Sbornik, vol. 37, no. 1, pp. 141–196, 1955.

A. M. Yaglom, Some classes of random fields in n-dimensional space related with random stationary processes, Teor. Veroyatn. Primen., vol. 11, no. 3, pp. 292–337, 1957.


Full Text: PDF

DOI: 10.19139/soic.v3i2.132

Refbacks

  • There are currently no refbacks.