#### 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

Maksym Luz^{1} and Mikhail Moklyachuk^{2
1}BNP Paribas Cardif, Kyiv, Ukraine

^{2}Taras Shevchenko National University of Kyiv, Ukraine

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

**Chapter DOI: **https://doi.org/10.52305/JBSU8922

We consider the problem of optimal estimation of linear functionals constructed from unobserved values of a stochastic sequence with periodically stationary increments based on observations of the sequence with a periodically stationary noise. For sequences with known spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas defining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are proposed in the case where the spectral densities of the sequences are not exactly known while some sets of admissible spectral densities are given.

**Keywords: **periodically stationary increments, PARIMA, minimax-robust estimate, least

favorable spectral density, minimax spectral characteristic

[1] Porter-Hudak S., An application of the seasonal fractionally differenced model to the

monetary aggegrates, Journal of the American Statistical Association, vol. 85, no. 410,

pp. 338–344, 1990.

[2] Burridge P. and Wallis K. F., Seasonal adjustment and kalman filtering: Extension to

periodic variances, Journal of Forecasting, vol. 9, no. 2, p. 109–118, 1990.

[3] Reisen V. A., Monte E. Z., Franco G. C., Sgrancio A. M., Molinares F. A. F., Bondon P., Ziegelmann F. A. and Abraham B., Robust estimation of fractional seasonal

processes: Modeling and forecasting daily average SO2 concentrations, Mathematics

and Computers in Simulation, vol. 146, pp. 27–43, 2018.

[4] Solci C. C., Reisen V. A., Sarnaglia A. J. Q. and Bondon P., 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.

[5] Gardner W. A., Cyclostationarity in communications and signal processing, New

York: IEEE Press, 504 p.,1994.

[6] Dudek G., Forecasting time series with multiple seasonal cycles using neural networks

with local learning, In: Rutkowski L., Korytkowski M., Scherer R., Tadeusiewicz R.,

Zadeh L.A., Zurada J.M. (eds) Artificial Intelligence and Soft Computing. ICAISC

2013. Lecture Notes in Computer Science, vol. 7894. Springer, Berlin, Heidelberg,

pp. 52–63, 2013.

[7] Box G. E. P., Jenkins G. M., Reinsel G. C. and Ljung G. M., Time series analysis.

Forecasting and control. 5rd ed., Hoboken, NJ: John Wiley & Sons, 712 p., 2016.

[8] Hassler U., Time series analysis with Long Memory in view, Wiley, Hoboken, NJ, 288

p., 2019.

[9] Granger C. W. J., Cointegrated variables and error correction models UCSD Discussion paper 83-13a, 1983.

[10] Baillie R. T., Kongcharoen C. and Kapetanios G., Prediction from ARFIMA models:

Comparisons between MLE and semiparametric estimation procedures, International

Journal of Forecasting, vol. 28, pp. 46–53, 2012.

[11] Gould P. G., Koehler A. B., Ord J. K., Snyder R. D., Hyndman R. J. and VahidAraghi F., Forecasting time-series with multiple seasonal patterns, European Journal

of Operational Research, vol. 191, pp. 207-222, 2008.

[12] Hassler U. and Pohle M. O., Forecasting under long memory and nonstationarity,

arXiv:1910.08202, 2019.

[13] Johansen S. and Nielsen M. O., The role of initial values in conditional sum-ofsquaresestimation of nonstationary fractional time series models, Econometric Theory, vol. 32, no. 5, pp. 1095–1139, 2016.

[14] Reisen V. A., Zamprogno B., Palma W. and Arteche J., A semiparametric approach

to estimate two seasonal fractional parameters in the SARFIMA model, Mathematics

and Computers in Simulation, vol. 98, pp. 1–17, 2014.

[15] Tsai H., Rachinger H. and Lin E. M. H., Inference of seasonal long-memory time

series with measurement error, Scandinavian Journal of Statistics, vol. 42, no. 1,

pp. 137–154, 2015.

[16] Lund R., Choosing seasonal autocovariance structures: PARMA or SARMA, In: Bell

WR, Holan SH, McElroy TS (eds) Economic time series: modelling and seasonality.

Chapman and Hall, London, pp. 63–80, 2011.

[17] Osborn D., The implications of periodically varying coefficients for seasonal timeseries processes, Journal of Econometrics, vol. 48, no. 3, pp. 373–384, 1991.

[18] Gladyshev E. G., Periodically correlated random sequences, Sov. Math. Dokl. vol. 2,

pp. 385–388, 1961.

[19] Gardner W. A., Napolitano A. and Paura L., Cyclostationarity: Half a century of

research, Signal Processing, vol. 86, pp. 639–697, 2006.

[20] Hurd H. and Miamee A., Periodically correlated random sequences, John Wiley &

Sons, Inc., Publication, 353 p., 2007.

[21] Hurd H. and Pipiras V., Modeling periodic autoregressive time series with multiple

periodic effects, In: Chaari F., Leskow J., Zimroz R., Wylomanska A., Dudek A.

(eds) Cyclostationarity: Theory and Methods – IV. CSTA 2017. Applied Condition

Monitoring, vol 16. Springer, Cham, pp. 1–18, 2020.

[22] Napolitano A., Generalizations of cyclostationary signal processing. Spectral analysis

and applications., Hoboken, NJ: John Wiley & Sons, 480 p., 2012

[23] Napolitano A., Cyclostationarity: Limits and generalizations, Signal processing, vol.

120, pp. 323–347, 2016

[24] Napolitano A., Cyclostationarity: New trends and applications, Signal Processing,

vol. 120, pp. 385–408, 2016.

[25] Napolitano A., Cyclostationary processes and time series: Theory, applications, and

generalizations, London: Elsevier/Academic Press, 626 p., 2019.

[26] Chaari F., Leskow J., Napolitano A., Zimroz R. and Sanchez-Ramirez A. Eds., Cyclostationarity: Theory and methods, Cham: Springer, 190 p., 2014.

[27] Chaari F., Leskow J., Napolitano A., Zimroz R., Wylomanska A. and Dudek A. Eds.,

Cyclostationarity: theory and methods II. Contributions to the 7th workshop on cyclostationary systems and their applications, Grodek, Poland, February 2014, Cham:

Springer, 210 p., 2015.

[28] Chaari F., Leskow J., Napolitano A., Zimroz R. and Wylomanska A. Eds., Cyclostationarity: theory and methods III. Contributions to the 9th workshop on cyclostationary systems and their applications, Grodek, Poland, February 2016, Cham: Springer,

257 p., 2017.

[29] Chaari F., Leskow J., Napolitano A., Zimroz R., Wylomanska A. and Dudek A. Eds.,

Cyclostationarity: theory and methods IV. Contributions to the 10th workshop on cyclostationary systems and their applications, Grodek, Poland, February 2017, Cham:

Springer, 226 p., 2020.

[30] Makagon A., Theoretical prediction of periodically correlated sequences, Probab.

Math. Statist., vol. 19, no. 2, pp. 287–322, 1999.

[31] Makagon A., Miamee A. G., Salehi H. and Soltani A. R., Stationary sequences associated with a periodically correlated sequence, Probab. Math. Statist., vol. 31, no. 2,

pp. 263–283, 2011.

[32] Baek C., Davis R. A. and Pipiras V., Periodic dynamic factor models: estimation

approaches and applications, Electronic Journal of Statistics, vol. 12, no. 2, pp. 4377–

4411, 2018.

[33] Basawa I.V., Lund R. and Shao Q., First-order seasonal autoregressive processes with

periodically varying parameters, Statistics and Probability Letters, vol. 67, no. 4,

pp. 299–306, 2004.

[34] Ansley C. F. and Kohn R., Estimation, filtering and smoothing in state space models with incompletely specified initial conditions, The Annals of Statistics, vol. 13,

pp. 1286–1316, 1985.

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

pp. 371-379, 1957.

[36] Hosoya Y., Robust linear extrapolations of second-order stationary processes, Annals

of Probability, vol. 6, no. 4, pp. 574–584, 1978.

[37] Kassam S. A., Robust hypothesis testing and robust time series interpolation and regression, Journal of Time Series Analysis, vol. 3, no. 3, pp. 185–194, 1982.

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

Proceedings of the IEEE, vol. 73, no. 3, pp. 1433–481, 1985.

[39] Franke J., Minimax-robust prediction of discrete time series, Z. Wahrscheinlichkeitstheor. Verw. Gebiete, vol. 68, no. 3, pp. 337–364, 1985.

[40] Vastola S. K. and Poor H. V., Robust Wiener-Kolmogorov theory, IEEE Trans. Inform.

Theory, vol. 30, no. 2, pp. 316–327, 1984.

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

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

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

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

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

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

419, 2015.

[45] 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.

[46] Moklyachuk M. P. and Masyutka A. Yu., Extrapolation of multidimensional stationary

processes Random Operators and Stochastic Equations, vol. 14, no. 3, pp. 233–244,

2006.

[47] Moklyachuk M. P. and Masyutka A. Yu., Minimax prediction of stochastic sequences,

Theory of Stochastic Processes, vol. 14, no. 3-4, pp. 89–103, 2008.

[48] 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.

[49] Moklyachuk M. P. and Masyutka A. Yu., Minimax-robust estimation technique: For

stationary stochastic processes, LAP Lambert Academic Publishing, 296 p., 2012.

[50] 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.

[51] 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.

[52] Dubovets’ka I. I. and Moklyachuk M. P., Minimax estimation problem for periodically

correlated stochastic processes, Journal of Mathematics and System Science, vol. 3,

no. 1, pp. 26–30, 2013.

[53] Dubovets’ka I. I. and Moklyachuk M. P., Extrapolation of periodically correlated

stochastic processes observed with noise, Theory of Probability and Mathematical

Statistics, vol. 88, pp. 67–83, 2014.

[54] Dubovets’ka I. I. and Moklyachuk M. P., On minimax estimation problems for periodically correlated stochastic processes, Contemporary Mathematics and Statistics,

vol.2, no. 1, pp. 123–150, 2014.

[55] 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

[56] Golichenko I. I., Yu Masyutka O. 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.

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

LAP Lambert Academic Publishing, 308 p., 2016.

[58] 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.

[59] 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.

[60] 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.

[61] 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.

[62] 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

[63] 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.

[64] 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.

[65] 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.

[66] 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.

[67] 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.

[68] 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.

[69] 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.

[70] 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.

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

periodically stationary increments, Carpathian Math. Publ., vol.13, no.2 pp. 352–376,

2021

[72] 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.

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

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

pp. 259–275, 2015.

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

observations, Statistics, Optimization & Information Computing, Vol. 4, No. 4, pp.

308–325, 2016.

[75] 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.

[76] 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.

[77] 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.

[78] 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.

[79] 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.

[80] 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.

[81] 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.

[82] 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.

[83] 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

[84] 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.

[85] 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.

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

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

[87] Pinsker M. S. and Yaglom A. M., On linear extrapolaion of random processes with

nth stationary incremens, Doklady Akademii Nauk SSSR, vol. 94, pp. 385–388, 1954.

[88] Yaglom A. M., Correlation theory of stationary and related random processes with

stationary nth increments. American Mathematical Society Translations: Series 2,

vol. 8, 87–141, 1958.

[89] Yaglom A. M., 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.

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

Springer, 574 p., 2004.

[91] Karhunen K., Uber lineare Methoden in der Wahrscheinlichkeitsrechnung, Annales

Academiae Scientiarum Fennicae. Ser. A I, no. 37, 1947.

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

[93] Hannan E. J., Multiple time series. 2nd rev. ed., John Wiley & Sons, New York, 536 p.,

2009.

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

and mathematical statistics. Ed. by A. N. Shiryayev. Mathematics and its Applications.

Soviet Series. 26. Dordrecht etc. Kluwer Academic Publishers, 597 p., 1992.

[95] Rockafellar R. T., Convex Analysis, Princeton Landmarks in Mathematics. Princeton,

NJ: Princeton University Press, 451 p., 1997.

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