2000 character limit reached
Prediction problem for continuous time stochastic processes with periodically correlated increments observed with noise (2401.08642v1)
Published 12 Dec 2023 in math.ST and stat.TH
Abstract: We propose solution of the problem of the mean square optimal estimation of linear functionals which depend on the unobserved values of a continuous time stochastic process with periodically correlated increments based on observations of this process with periodically stationary noise. To solve the problem, we transform the processes to the sequences of stochastic functions which form an infinite dimensional vector stationary sequences. In the case of known spectral densities of these sequences, 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.
- I.V. Basawa, R. Lund, and Q. Shao, First-order seasonal autoregressive processes with periodically varying parameters, Statistics & Probability Letters, vol. 67, no. 4, pp. 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 processes from observations 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, Introduction to the theory of random processes. Moskva: Gosudarstv. Izdat. Fiz.-Mat. Lit. 1965.
- 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, and V. Mandrekar,: Spectral theory of stationary H-valued processes, J. Multivariate Analysis, vol. 1, pp. 1–16, 1971.
- S. A. Kassam, and H. V. Poor, Robust techniques for signal processing: A survey, Proceedings of the IEEE No.73, pp. 433-481, 1985.
- K. Karhunen, Uber lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales Academiae Scientiarum Fennicae. Ser. A I, no. 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.
- 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.
- M. Luz, and M. Moklyachuk, Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences, Statistics, Optimization and Information Compututing, 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 observed 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. Luz, and M. Moklyachuk, Estimation problem for continuous time stochastic processes with periodically correlated increments, Statistics, Optimization and Information Computing, vol.11, no. 4, pp. 811 – 828, 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, pp. 1–82, 1981.
- M. P. Moklyachuk, On a problem of game theory and the extrapolation of stochastic processes with values in a Hilbert space, Theory Probability and Mathematical Statistics, vol. 24, pp. 115–124, 1981.
- M. P. Moklyachuk, On an antagonistic game and prediction of stationary random sequences in a Hilbert space, Theory Probability and Mathematical Statistics, vol. 25, pp. 107–113, 1982.
- M. P. Moklyachuk, Robust estimations of functionals of stochastic processes, Kyïv: Vydavnychyj Tsentr “Kyïvs’kyĭ Universytet”, 320 p., 2008. (in Ukrainian)
- 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, Saarbrücken: LAP Lambert Academic Publishing, 308 p., 2016.
- M.P. Moklyachuk, and A.Yu. Masyutka, Minimax-robust estimation technique: For stationary stochastic processes, LAP Lambert Academic Publishing, 296 p. 2012.
- M. Moklyachuk, M. Sidei, and O. Masyutka, Estimation of stochastic processes with missing observations, Mathematics Research Developments. New York, NY: Nova Science Publishers, 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 n𝑛nitalic_nth 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, NJ: Princeton University Press, 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, An analysis of the effects of spectral uncertainty on Wiener filtering, Automatica vol.28, 289-293, 1983.
- 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 n𝑛nitalic_nth increments, Mat. Sbornik, vol. 37, no. 1, pp. 141–196, 1955.
- A. M. Yaglom, Correlation theory of stationary and related random functions. Vol. 1: Basic results; Vol. 2: Suplementary notes and references, Springer Series in Statistics, Springer-Verlag, New York etc., 1987.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.