# On estimation problem for continuous time stationary processes from observations in special sets of points

## DOI:

https://doi.org/10.17721/1812-5409.2022/1.2## Keywords:

stationary stochastic process, minimax-robust estimate, least favorable spectral density, minimax-robust spectral characteristics## Abstract

*The problem of the mean-square optimal estimation of the linear functionals which depend on the unknown values of a stochastic stationary process from observations of the process with missings is considered. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functionals are derived under the condition of spectral certainty, where the spectral density of the process is exactly known. The minimax (robust) method of estimation is applied in the case where the spectral density of the process 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.*

* Pages of the article in the issue*: 20 - 33

**Language of the article**: English

## References

BONDON, P. (2005) Influence of missing values on the prediction of a stationary time series. J. Time Ser. Anal. 26(4). p. 519–525.

BONDON, P. (2002) Prediction with incomplete past of a stationary process. Stoch. Process Their Appl. 98. p. 67–76.

CHENG, R., MIAMEE, A.G. and POURAHMADI, M. (1998) Some extremal problems in Lp(w). Proc. Am. Math. Soc. 126. p. 2333–2340.

DUBOVETS’KA, I. I., MASYUTKA, A. YU. and MOKLYACHUK, M. P. (2012) Interpolation of periodically correlated stochastic sequences. Theory Probab. Math. Stat. 84. p. 43–55.

FRANKE, J. (1985) Minimax robust prediction of discrete time series. Z. Wahrscheinlichkeitstheor. Verw. Geb. 68. p. 337–364.

FRANKE, J. and POOR, H.V. (1984) Minimax-robust filtering and finite-length robust predictors. Lecture Notes in Statist. 26. p. 87–126.

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

GOLICHENKO, I. I. and MOKLYACHUK, M. P. (2016) Periodically correlated processes estimates. LAP Lambert Academic Publishing.

GOLICHENKO, I. I., MASYUTKA, A. YU. and MOKLYACHUK, M. P. (2021) Extrapolation problem for periodically correlated stochastic sequences with missing observations. Visn., Ser. Fiz.-Mat. Nauky, Kyiv. Univ. Im. Tarasa Shevchenka. 2. p. 39–52.

GRENANDER, U. (1957) A prediction problem in game theory. Ark. Mat. 6. p. 371–379.

HANNAN, E. J. (1970) Multiple time series. Wiley, New York.

IOFFE, A.D. and TIHOMIROV, V.M. (1979) Theory of extremal problems. North-Holland Publishing Company.

KARHUNEN, K. (1947) ¨Uber lineare Methoden in der Wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fenn., Ser. A I. 37. p. 1–79.

KASAHARA, Y., POURAHMADI, M. and INOUE, A. (2009) Duals of random vectors and processes with applications to prediction problems with missing values. Statist. Probab. Lett. 79(14). p. 1637–1646.

KASSAM, S. A. and POOR, H. V. (1985) Robust techniques for signal processing: A survey. Proc. IEEE. 73(3). p. 433–481.

KOLMOGOROV, A. N. (1992) In: Shiryayev A. N. (Ed.) Selected works by A. N. Kolmogorov. Vol. II: Probability theory and mathematical statistics Kluwer, Dordrecht etc.

LUZ, M. and MOKLYACHUK, M. (2014) Robust extrapolation problem for stochastic processes with stationary increments. Math. Stat. 1(2). p. 78–88.

LUZ, M. and MOKLYACHUK, M. (2015) Minimax interpolation problem for random processes with stationary increments. Stat. Optim. Inf. Comput. 3. p. 30–41.

LUZ, M., MOKLYACHUK, M. (2019) Estimation of Stochastic Processes with Stationary Increments and Cointegrated Sequences. London: ISTE; Hoboken, NJ: John Wiley & Sons .

LUZ, M., MOKLYACHUK, M. (2020) Minimax-robust estimation problems for sequences with periodically stationary increments observed with noise. Visn., Ser. Fiz.-Mat. Nauky, Kyiv. Univ. Im. Tarasa Shevchenka. 3. p. 68–83.

MASYUTKA, A. YU. and MOKLYACHUK, M. P. (2022) On a problem of minimax interpolation of stationary sequences. Cybernet. Systems Anal. 58(2). p. 128–142.

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

MOKLYACHUK, M. P. (2001) Game theory and convex optimization methods in robust estimation problems. Theory Stoch. Process. 7(1–2). p. 253–264.

MOKLYACHUK, M. P. (2008) Robust estimations of functionals of stochastic processes. Kyiv University, Kyiv.

MOKLYACHUK, M. P. (2015) Minimax-robust estimation problems for stationary stochastic sequences. Stat., Optim. Inf. Comput. 3(4). p. 348–419.

MOKLYACHUK, M. P. and MASYUTKA, A. YU. (2007) Robust filtering of stochastic processes. Theory Stoch. Process. 13(1–2). p. 166–181.

MOKLYACHUK, M. P. and MASYUTKA, A. YU. (2011) Minimax prediction problem for multidimensional stationary stochastic processes. Commun. Stat., Theory Methods. 40(19–20). p. 3700–3710.

MOKLYACHUK, M. P. and MASYUTKA, A. YU. (2012) Minimax-robust estimation technique for stationary stochastic processes. LAP LAMBERT Academic Publishing.

MOKLYACHUK, M. P. and SIDEI, M. I. (2016) Interpolation of stationary sequences observed with a noise. Theory Probab. Math. Statist. 93. p.153–167.

MOKLYACHUK, M. P. and SIDEI, M. I. (2015) Interpolation problem for stationary sequences with missing observations. Stat., Optim. Inf. Comput. 3(3). p. 259–275.

MOKLYACHUK, M. P. and SIDEI, M. I. (2016) Filtering problem for stationary sequences with missing observations. Stat., Optim. Inf. Comput. Computing. 4(4). p. 308–325.

MOKLYACHUK, M. P., SIDEI, M. I. and MASYUTKA, O. YU. (2019) Estimation of stochastic processes with missing observations. New York, NY: Nova Science Publishers.

NAKAZI, T. (1984) Two problems in prediction theory. Studia Math. 78. p. 7–14.

POURAHMADI, M., INOUE, A. and KASAHARA, Y. (2007) A prediction problem in L2(w). Proc. Amer. Math. Soc. 135(4). p. 1233–1239.

PSHENICHNYI, B.N. (1971) Necessary conditions of an extremum. New York: Marcel Dekker.

ROCKAFELLAR, R. T. (1997) Convex Analysis. Princeton University Press.

ROZANOV, YU.A. (1967) Stationary stochastic processes. San Francisco-Cambridge-London-Amsterdam: Holden-Day.

SALEHI, H. (1979) Algorithms for linear interpolator and interpolation error for minimal stationary stochastic processes. Ann. Probab. 7(5). p. 840–846.

VASTOLA, S. K. and POOR, H. V. (1984) Robust Wiener-Kolmogorov theory. IEEE Trans. Inform. Theory. 30(2). p. 316–327.

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

WOLD, H. (1938) A study in the analysis of stationary time series. Thesis University of Stockholm, 1938.

WOLD, H. (1948) On prediction in stationary time series / H. Wold // Ann. Math. Stat. 19(4). p. 558–567.

YAGLOM, A. M. (1987) Correlation theory of stationary and related random functions. Vol. 1: Basic results; Vol. 2: Supplementary notes and references. Springer-Verlag, New York etc.

## Downloads

## Published

## How to Cite

*Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences*, (1), 20–33. https://doi.org/10.17721/1812-5409.2022/1.2

## Issue

## Section

## License

Authors who publish with this journal agree to the following terms:

- 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).