#### Publish with Nova Science Publishers

We publish over 800 titles annually by leading researchers from around the world. Submit a Book Proposal Now!

$39.50

Oleksandr Masyutka^{1} and Mikhail Moklyachuk^{2
1}Department of Mathematics and Theoretical Radiophysics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

^{2}Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

**Part of the book: **Stochastic Processes: Fundamentals and Emerging Applications

**Chapter DOI: **https://doi.org/10.52305/VJZY7688

The problem of the mean-square optimal estimation of the linear functionals which depend on the unknown values of a stochastic stationary sequence is considered. Estimates are based on observations of the sequence with missing data. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimate of the functionals are derived under the condition of spectral certainty, where the spectral density of the sequence is exactly known. The minimax (robust) method of estimation is applied in the case where the spectral density of the sequence is not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities.

**Keywords: **stationary sequence, mean square error, minimax-robust estimate, least

favourable spectral density, minimax spectral characteristics

[1] Kolmogorov A. N., Selected works by Kolmogorov A. N., Vol. II: Probability theory

and mathematical statistics. Ed. by Shiryaev. mathematics and Applications. Soviev

Series. 26, Kluwer Academic Publishers, 597 p., 1992.

[2] Wiener N., Extrapolation, interpolation and smoothing of stationary time series. With

engineering applications, The M. I. T. Press, Massachusetts Institute of Technology,

Cambridge, Mass., 163 p., 1966.

[3] Yaglom A. M., Correlation theory of stationary and related random functions. Vol. 1:

Basic results, Springer Series in Statistics, Springer-Verlag, New York etc., 526 p.,

1987.

[4] Yaglom A. M., Correlation theory of stationary and related random functions. Vol. 2:

Supplementary notes and references, Springer Series in Statistics, Springer-Verlag,

New York etc., 258 p., 1987.

[5] Rozanov Yu. A., Stationary stochastic processes. 2nd rev. ed., Moskva: Nauka, 272 p.,

1990. (English transl. of 1st ed., Holden-Day, San Francisco, 211 p., 1967)

[6] Hannan E. J., Multiple time series, Wiley Series in Probability and Mathematical

Statistics. New York etc.: John Wiley & Sons, Inc. XI, 1970.

[7] Vastola K. S. and Poor H. V., An analysis of the effects of spectral uncertainty on

Wiener filtering, Automatica, vol. 28, pp. 289–293, 1983.

[8] Grenander U., A prediction problem in game theory, Arkiv f ¨or Matematik, vol. 3, pp.

371–379, 1957.

[9] Kassam S. A. and Poor H. V., Robust techniques for signal processing: A survey,

Proceedings of the IEEE, vol. 73, no. 3, pp. 433–481, 1985.

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

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

[12] Moklyachuk M. P., Minimax extrapolation and autoregressive-moving average processes, Theory of Probability and Mathematical Statistics, vol. 41, pp. 77–84, 1990.

[13] Moklyachuk M. P., Stochastic autoregressive sequences and minimax interpolation,

Theory of Probability and Mathematical Statistics, vol. 48, pp. 95–103, 1994.

[14] Moklyachuk M. P., Robust procedures in time series analysis, Theory of Stochastic

Processes, vol. 6, no. 3–4, pp. 127–147, 2000.

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

[16] Moklyachuk M. P., Robust estimations of functionals of stochastic processes, Ky¨ıv:

Vydavnychyj Tsentr “Ky¨ıvs’ky˘ı Universytet”, 320 p., 2008.

[17] Moklyachuk M. P., Minimax-robust estimation problems for stationary stochastic sequences, Statistics, Optimization and Information Computing, vol. 3, no. 4, pp. 348–

419, 2015.

[18] Masyutka O. Yu. and Moklyachuk M. P., On minimax interpolation of stationary sequences, Cybernetics and System Analysis, vol. 58, no. 2, pp. 268–279, 2022.

[19] Masyutka O. Yu., Golichenko I. I. and Moklyachuk M. P., On estimation problem

for continuous time stationary processes from observations in special sets of points,

Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 1, pp. 20–33, 2022.

[20] Moklyachuk M. P. and Masyutka O. Yu., Robust filtering of stochastic processes,

Theory of Stochastic Processes, vol. 13, no. 1-2, pp. 166–181, 2008.

[21] Moklyachuk M. P. and Masyutka O. Yu., Minimax prediction problem for multidimentional stationary stochastic sequence, Theory of Stochastic Processes, vol. 14, no.

3-4, pp. 89–103, 2008.

[22] Moklyachuk M. P. and Masyutka A. Yu., Minimax prediction problem for multidimensional stationary stochastic processes, Communications in Statistics–Theory and

Methods, vol. 40, no. 19-20, pp. 3700–3710, 2011.

[23] Moklyachuk M. and Masyutka O., Minimax-robust estimation technique for stationary stochastic processes, LAP Lambert Academic Publishing, 296 p. 2012.

[24] Dubovets’ka I. I. and Moklyachuk M. P., Filtration of linear functionals of periodically

correlated sequences, Theory of Probability and Mathematical Statistics, vol. 86, pp.

51–64, 2013.

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

[26] Golichenko I. I. and Moklyachuk M. P., Interpolation problem for periodically correlated stochastic sequences with missing observations, Statistics, Optimization and

Information Computing, vol. 8, no. 2, pp. 631-654, 2020.

[27] Moklyachuk M. P. and Golichenko I. I., Periodically correlated processes estimates,

LAP Lambert Academic Publishing, 308 p., 2016.

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

[29] Golichenko I. I., Masyutka O. Yu and Moklyachuk M. P., Extrapolation problem

for periodically correlated stochastic sequences with missing observations, Bulletin

of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics,

no. 2, pp. 39–52, 2021.

[30] Luz M. and Moklyachuk M., Interpolation of functionals of stochastic sequences with

stationary increments, Theory Probability and Mathematical Statistics, vol. 87, pp.

117–133, 2013.

[31] Luz M. M. and Moklyachuk M. P., Minimax-robust filtering problem for stochastic sequence with stationary increments, Theory of Probability and Mathematical Statistics,

vol. 89, pp. 127–142, 2014.

[32] Luz M. and Moklyachuk M., Minimax interpolation problem for random processes

with stationary increments, Statistics, Optimization and Information Computing,

vol. 3, no. 1, pp. 30–41, 2015.

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

[34] Luz M. and Moklyachuk M., Filtering problem for functionals of stationary sequences, Statistics, Optimization and Information Computing, vol. 4, no. 1, pp. 68-83,

2016

[35] Luz M. and Moklyachuk M., Minimax-robust forecasting of sequences with periodically stationary long memory multiple seasonal increments, Statistics, Optimization

and Information Computing, vol. 8, no. 3, pp. 684–721, 2020.

[36] Luz M. and Moklyachuk M., Minimax-robust estimation problems for sequences with

periodically stationary increments observed with noise, Bulletin of Taras Shevchenko

National University of Kyiv. Series: Physics and Mathematics, no. 3, pp. 68–83, 2020.

[37] Luz M. and Moklyachuk M., Robust filtering of sequences with periodically stationary

multiplicative seasonal increments, Statistics, Optimization and Information Computing, vol. 9, no. 4, pp. 1010–1030, 2021.

[38] Luz M. and Moklyachuk M., Minimax filtering of sequences with periodically stationary increments, Cybernetics and System Analysis, vol. 58, no. 1, pp. 126–143,

2022.

[39] Luz M. and Moklyachuk M., Robust interpolation of sequences with periodically stationary multiplicative seasonal increments, Carpathian Math. Publ., vol.14, no.1 pp.

105 – 126, 2022.

[40] Luz M. and Moklyachuk M., Robust forecasting of sequences with periodically stationary long memory multiplicative seasonal increments observed with noise and cointegrated sequences, Statistics, Optimization and Information Computing, vol.10, no.2

pp. 295 – 338, 2022.

[41] Luz M. and Moklyachuk M., Estimation of stochastic processes with stationary increments and cointegrated sequences, London: ISTE; Hoboken, NJ: John Wiley & Sons,

282 p., 2019.

[42] Kozak P. S. and Moklyachuk M. P., Estimates of functionals constructed from random

sequences with periodically stationary increments, Theory Probability and Mathematical Statistics, vol. 97, pp. 85–98, 2018.

[43] Kozak P. S., Luz M. M. and Moklyachuk M. P., Minimax prediction of sequences with

periodically stationary increments, Carpathian Math. Publ., Vol.13, Iss.2 pp. 352–

376, 2021

[44] Moklyachuk M. and Sidei M., Interpolation of stationary sequences observed with the

noise, Theory of Probability and Mathematical Statistics, vol. 93, pp. 143–156, 2015.

[45] Moklyachuk M. and Sidei M., Interpolation problem for stationary sequences with

missing observations, Statistics, Optimization and Information Computing, vol. 3, no.

3, pp. 259–275, 2015.

[46] Moklyachuk M. and Sidei M., Filtering problem for stationary sequences with missing

observations, Statistics, Optimization and Information Computing, vol. 4, no. 4, pp.

308–325, 2016.

[47] Moklyachuk M. and Sidei M., Filtering problem for functionals of stationary processes with missing observations, Communication in Optimization Theory, Article ID

21, 18 p., 2016.

[48] Moklyachuk M. and Sidei M., Extrapolation problem for functionals of stationary processes with missing observations, Bukovynskyi Mathematical Journal, Vol. 4, No.1–2,

pp. 122–129, 2016.

[49] Moklyachuk M. and Sidei M., Extrapolation problem for stationary sequences with

missing observations, Statistics, Optimization and Information Computing, vol. 5, no.

3, pp. 212–233, 2017.

[50] Masyutka O. Yu., Moklyachuk M. P. and Sidei M. I., Minimax extrapolation of multidimensional stationary processes with missing observations, International Journal of

Mathematical Models and Methods in Applied Sciences, vol. 12, pp. 94–105, 2018.

[51] Masyutka O. Yu., Moklyachuk M. P. and Sidei M. I., Extrapolation problem for multidimensional stationary sequences with missing observations, Statistics, Optimization

and Information Computing, vol. 7, no.1, pp. 97–117, 2019.

[52] Masyutka O. Yu., Moklyachuk M. P. and Sidei M. I., Interpolation problem for multidimensional stationary processes with missing observations, Statistics, Optimization

and Information Computing, vol. 7, no.1, pp. 118–132, 2019.

[53] Masyutka O. Yu., Moklyachuk M. P. and Sidei M. I., Signal extraction for nonstationary multivariate time series with illustrations for trend inflation, Universal Journal of Mathematics and Applications, vol. 2, no.1, pp. 24–35, 2019.

[54] Masyutka O. Yu., Moklyachuk M. P. and Sidei M. I., Filtering of multidimensional

stationary sequences with missing observations, Carpathian Mathematical Publications, vol. 11, no.2, pp. 361 – 378, 2019.

[55] Moklyachuk M., Masyutka O. and Sidei M., Interpolation problem for multidimensional stationary sequences with missing observations, Stochastic Modeling and Applications, vol. 22, no. 23, pp. 85–103, 2018

[56] Moklyachuk M., Sidei M. and Masyutka O., Estimation of stochastic processes with

missing observations, Mathematics Research Developments. New York, NY: Nova

Science Publishers, 336 p., 2019.

[57] Bondon P., 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.

[58] Bondon P., Prediction with incomplete past of a stationary process, Stochastic Process

and their Applications. vol. 98, pp. 67–76, 2002.

[59] Kasahara Y., and Pourahmadi M. and Inoue A., 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.

[60] Pourahmadi M., Inoue A. and Kasahara Y. A prediction problem in L²(w). Proceedings of the American Mathematical Society. vol. 135, no. 4, pp. 1233-1239, 2007.

[61] Moklyachuk M. P., Masyutka A. Yu., Golichenko I. I. Estimates of periodically correlated isotropic random fields, Nova Science Publishers Inc. New York, 309 p., 2018.

[62] Yadrenko M. I., Spectral theory of random fields, New York: Optimization Software,

Inc., Publications Division, Springer-Verlag, New York etc., 259 p., 1983.

[63] Gikhman I. I. and Skorokhod A. V., The theory of stochastic processes. I., Berlin:

Springer, 574 p., 2004.

[64] Karhunen K., Uber lineare methoden in der wahrscheinlichkeitsrechnung ¨ , Annales

Academiae Scientiarum Fennicae. Ser. A I, vol. 37, 1947.

[65] Ioffe A. D. and Tihomirov V. M., Theory of extremal problems, Studies in Mathematics and its Applications, Vol. 6. Amsterdam, New York, Oxford: North-Holland

Publishing Company. XII, 460 p., 1979.

[66] Pshenichnyj B. N., Necessary conditions of an extremum, Pure and Applied mathematics. 4. New York: Marcel Dekker, 230 p., 1971.

[67] Rockafellar R. T., Convex Analysis, Princeton University Press, 451 p., 1997.

[68] Krein M. G. and Nudelman A. A., The Markov moment problem and extremal problems, Translations of Mathematical Monographs. Vol. 50. Providence, R.I.: American

Mathematical Society, 417 p., 1977.

[69] Liu Y., Xue Yu. and Taniguchi M., Robust linear interpolation and extrapolation of

stationary time series in Lp, Journal of Time Series Analysis, vol. 41, no. 2, pp. 229–

248, 2020.

[70] Salehi H., Algorithms for linear interpolator and interpolation error for minimal stationary stochastic processes, The Annals of Probability, vol. 7, no. 5, pp. 840–846,

1979.

We publish over 800 titles annually by leading researchers from around the world. Submit a Book Proposal Now!