Geodesic
Robust filtering of stochastic
Teoriâ slučajnyh processov, Tome 13 (2007) no. 2, pp. 166-181.

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The considered problem is estimation of the unknown value of the functional $A\vec{\xi}=\int^\infty_0\vec{a}(t)\vec{\xi}(-t)dt$ which depends on the unknown values of a multidimensional stationary stochastic process $\vec{\xi}(t)$ based on observations of the process $\vec{\xi}(t)+ vec{\eta}(t)$ for $t\leq0.$ Formulas are obtained for calculation the mean square error and the spectral characteristic of the optimal estimate of the functional under the condition that the spectral density matrix $F(\lambda)$ of the signal process $\vec{\xi}(t)$ and the spectral density matrix $G(\lambda)$ of the noise process $\vec{\eta}(t)$ are known. The least favorable spectral densities and the minimax-robust spectral characteristic of the optimal estimate of the functional $A\vec{\xi}$ are found for concrete classes $D = D_F\times D_G$ of spectral densities under the condition that spectral density matrices $F(\lambda)$ and $G(\lambda)$ are not known, but classes $D = D_F\times D_G$ of admissible spectral densities are given.
Keywords: Stationary stochastic process, filtering, robust estimate, observations with noise, mean square error, least favorable spectral densities, minimax-robust spectral characteristic. 
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Mikhail Moklyachuk; Aleksandr Masyutka. Robust filtering of stochastic. Teoriâ slučajnyh processov, Tome 13 (2007) no. 2, pp. 166-181. http://geodesic.mathdoc.fr/item/THSP_2007_13_2_a1/

[1] J. Franke, “On the robust prediction and interpolation of time series in the presence of correlated noise”, J. Time Series Analysis, 5:4 (1984), 227–244

[2] J. Franke, “Minimax robust prediction of discrete time series”, Z.Wahrsch. Verw. Gebiete., 68 (1985), 337–364

[3] J. Franke, H. V. Poor, “Minimax robust filtering and finitelength robust predictors”, Robust and Nonlinear Time Series Analysis (Heidelberg, 1983), Lecture Notes in Statistics, 26, Springer-Verlag, 1984, 87–126

[4] J. Franke, “A general version of Breiman minimax filter”, Note di Matematica, 11 (1991), 157–175

[5] U. Grenander, “A prediction problem in game theory”, Ark.Mat., 3 (1957), 371–379

[6] T. Kailath, “A view of three decades of linear filtering theory”, IEEE Trans. on Inform. Theory, 20:2 (1974), 146–181

[7] S. A. Kassam, H. V. Poor, “Robust techniques for signal processing: A survey”, Proc. IEEE, 73:3 (1985), 433–481

[8] A. N. Kolmogorov, Selected works of A. N. Kolmogorov, v. II, Mathematics and Its Applications. Soviet Series, 26., Probability theory and mathematical statistics, ed. A. N. Shiryayev, Kluwer Academic Publishers, Dordrecht etc., 1992

[9] M. P. Moklyachuk, “Stochastic autoregressive sequence and minimax interpolation”, Theor. Probab. and Math. Stat., 48 (1994), 95–103

[10] M. P. Moklyachuk, “Estimates of stochastic processes from observations with noise”, Theory Stoch. Process., 3(19):3-4 (1997), 330–338

[11] M. P. Moklyachuk, “Extrapolation of stationary sequences from observations with noise”, Theor. Probab. and Math. Stat., 57 (1998), 133–141

[12] M. P. Moklyachuk, “Robust procedures in time series analysis”, Theory Stoch. Process., 6(22):3-4 (2000), 127–147

[13] M. P. Moklyachuk, “Game theory and convex optimization methods in robust estimation problems”, Theory Stoch. Process., 7(23):1-2 (2001), 253–264

[14] M. P. Moklyachuk, A. Yu. Masyutka, “Interpolation of vector-valued stationary sequences”, Theor. Probab. and Math. Stat., 73 (2005), 112–119

[15] M. P. Moklyachuk, A. Yu. Masyutka, “Extrapolation of vector-valued stationary sequences”, Visn., Ser. Fiz.-Mat. Nauky, Kyiv. Univ. Im. Tarasa Shevchenka, 3 (2005), 60–70

[16] M. P. Moklyachuk, A. Yu. Masyutka, “Extrapolation of multidimensional stationary processes”, Random Oper. and Stoch. Equ., 14 (2006), 233–244

[17] B. N. Pshenichnyi, Necessary conditions for an extremum, 2nd ed., Nauka, Moscow, 1982

[18] Holden-Day, San Francisco, 1967

[19] K. S. Vastola, H. V. Poor, “An analysis of the e?ects of spectral uncertainty on Wiener filtering”, Automatica, 28 (1983), 289–293

[20] 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

[21] A. M. Yaglom, Correlation theory of stationary and related random functions, v. I, Springer Series in Statistics, Basic results, Springer-Verlag, New York etc., 1987

[22] A. M. Yaglom, Correlation theory of stationary and related random functions, v. II, Springer Series in Statistics, Supplementary notes and references, Springer-Verlag, New York etc., 1987