# Estimation problem for continuous time stochastic processes with periodically correlated increments

### Abstract

We deal with the problem of optimal estimation of the linear functionals constructed from unobserved values of a continuous time stochastic process with periodically correlated increments based on past observations of this process. To solve the problem, we construct a corresponding to the process sequence of stochastic functions which forms an infinite dimensional vector stationary increment sequence. In the case of known spectral density of the stationary increment sequence, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given.### References

I.V. Basawa, R. Lund, and Q. Shao, First-order seasonal autoregressive processes with periodically varying parameters, Statistics and Probability Letters, vol. 67, no. 4, p. 299–306, 2004.

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, Extrapolation of periodically correlated stochastic processes observed with noise, Theory of Probability and Mathematical Statistics, vol. 88, pp. 67–83, 2014.

A. Dudek, H. Hurd, and W. Wojtowicz, PARMA methods based on Fourier representation of periodic coefficients, Wiley Interdisciplinary Reviews: Computational Statistics, vol. 8, no. 3, pp. 130–149, 2016.

J. Franke, and H. V. Poor, Minimax-robust filtering and finite-length robust predictors, In: Robust and Nonlinear Time Series Analysis, Lecture Notes in Statistics, Springer-Verlag, No.26, pp. 87–126, 1984.

I. I. Gikhman, and A. V. Skorokhod, The theory of stochastic processes. I., Springer, Berlin, 574 p. 2004.

E. G. Gladyshev, Periodically and almost-periodically correlated random processes with continuous time parameter. Theory Probab. Appl. vol. 8, pp. 173 –177, 1963.

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

Y. Hosoya, Robust linear extrapolations of second-order stationary processes, Annals of Probability, vol. 6, no. 4, pp. 574–584, 1978.

G. Kallianpur, V. Mandrekar, Spectral theory of stationary H-valued processes. J. Multivariate Analysis, vol. 1, pp. 1–16, 1971.

K. Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae. Ser. A I, no. 37, 1947.

S. A. Kassam, and H. V. Poor, Robust techniques for signal processing: A survey, Proceedings of the IEEE, vol. 73, no. 3, pp. 1433–481, 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.

P. S. Kozak, and M. P. Moklyachuk, Estimates of functionals constructed from random sequences with periodically stationary increments, Theory Probability and Mathematical Statistics, vol. 97, pp. 85–98, 2018.

P. S. Kozak, M. M. Luz and M. P. Moklyachuk, Minimax prediction of sequences with periodically stationary increments, Carpathian Mathematical Publications, vol.13, iss.2 pp. 352–376, 2021.

M. Luz, and M. Moklyachuk, 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.

M. Luz, and M. Moklyachuk, Estimation of stochastic processes with stationary increments and cointegrated sequences, London: ISTE; Hoboken, NJ: John Wiley & Sons, 282 p., 2019.

M. Luz and M. Moklyachuk, 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.

M. Luz and M. Moklyachuk, Minimax prediction of sequences with periodically stationary increments observes with noise and cointegrated sequences, In: M. Moklyachuk (ed.) Stochastic Processes: Fundamentals and Emerging Applications. Nova Science Publishers, New York, pp. 189–247, 2023.

M. P. Moklyachuk, Estimation of linear functionals of stationary stochastic processes and a two-person zero-sum game. Stanford University Technical Report, no. 169, 82 p., 1981.

M. P. Moklyachuk, Stochastic autoregressive sequences and minimax interpolation,

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

M. P. Moklyachuk, Robust estimations of functionals of stochastic processes,

{Ky"{i}v: Vydavnychyj Tsentr ``Ky"{i}vs'kyu{i} Universytet''}, 320 p., 2008.

M. P. Moklyachuk, Minimax-robust estimation problems for stationary stochastic sequences,

Statistics, Optimization and Information Computing, vol. 3, no. 4, pp. 348--419, 2015.

M. P. Moklyachuk, and I. I. Golichenko, Periodically correlated processes estimates, Saarbr"ucken: LAP Lambert Academic Publishing. 308 p., 2016.

M.P. Moklyachuk and A.Yu. Masyutka, Minimax-robust estimation technique: For stationary stochastic processes, Saarbr"ucken: LAP Lambert Academic Publishing, 296 p., 2012.

M. Moklyachuk, M. Sidei, and O. Masyutka, Estimation of stochastic processes with missing observations, Mathematics Research Developments. Nova Science Publishers, New York, NY, 336 p., 2019.

A.Napolitano, Cyclostationarity: New trends and applications, Signal Processing, vol. 120, pp. 385–408, 2016.

M. S. Pinsker and A. M. Yaglom, On linear extrapolaion of random processes with $n$th stationary incremens, Doklady Akademii Nauk SSSR, n. Ser. vol. 94, pp. 385--388, 1954.

V. A. Reisen, E. Z. Monte, G. C. Franco, A. M. Sgrancio, F. A. F. Molinares, P. Bondond, F. A. Ziegelmann and B. Abraham, Robust estimation of fractional seasonal processes: Modeling and forecasting daily average SO2 concentrations, Mathematics and Computers in Simulation, vol. 146, pp. 27–43, 2018.

R. T. Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 451 p., 1997.

C. C. Solci, V. A. Reisen, A. J. Q. Sarnaglia, and P. Bondon, Empirical study of robust estimation methods for PAR models with application to the air quality area, Communication in Statistics - Theory and Methods, vol. 48, no. 1, pp. 152–168, 2020.

S. K. Vastola, and H. V. Poor, Robust Wiener-Kolmogorov theory, IEEE Trans. Inform. Theory, vol. 30, no. 2, pp. 316-327, 1984.

N. Wiener, Extrapolation, interpolation, and smoothing of stationary time series. With engineering applications. Cambridge, Mass.: The M. I. T. Press, Massachusetts Institute of Technology, 163 p. 1966.

A. M. Yaglom, Correlation theory of stationary and related random processes with stationary nth increments. American Mathematical Society Translations: Series 2, vol. 8, pp. 87–141, 1958.

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

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*11*(4), 811-828. https://doi.org/10.19139/soic-2310-5070-1792

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