Modern Stochastics: Theory and Applications logo


  • Help
Login Register

  1. Home
  2. Issues
  3. Volume 3, Issue 1 (2016)
  4. Minimax interpolation of sequences with ...

Modern Stochastics: Theory and Applications

Submit your article Information Become a Peer-reviewer
  • Article info
  • Full article
  • Cited by
  • More
    Article info Full article Cited by

Minimax interpolation of sequences with stationary increments and cointegrated sequences
Volume 3, Issue 1 (2016), pp. 59–78
Maksym Luz   Mikhail Moklyachuk  

Authors

 
Placeholder
https://doi.org/10.15559/16-VMSTA51
Pub. online: 1 April 2016      Type: Research Article      Open accessOpen Access

Received
11 March 2016
Revised
16 March 2016
Accepted
17 March 2016
Published
1 April 2016

Abstract

We consider the problem of optimal estimation of the linear functional $A_{N}\xi ={\sum _{k=0}^{N}}a(k)\xi (k)$ depending on the unknown values of a stochastic sequence $\xi (m)$ with stationary increments from observations of the sequence $\xi (m)+\eta (m)$ at points of the set $\mathbb{Z}\setminus \{0,1,2,\dots ,N\}$, where $\eta (m)$ is a stationary sequence uncorrelated with $\xi (m)$. We propose formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional in the case of spectral certainty, where spectral densities of the sequences are exactly known. We also consider the problem for a class of cointegrated sequences. We propose relations that determine the least favorable spectral densities and the minimax spectral characteristics in the case of spectral uncertainty, where spectral densities are not exactly known while a set of admissible spectral densities is specified.

References

[1] 
Bell, W.: Signal extraction for nonstationary time series. Ann. Stat. 12(2), 646–664 (1984). MR0740918. doi:10.1214/aos/1176346512
[2] 
Box, G.E.P., Jenkins, G.M., Reinsel, G.C.: Time Series Analysis. Forecasting and Control. 3rd edn. Englewood Cliffs, NJ, Prentice Hall (1994). MR1312604
[3] 
Chigira, H., Yamamoto, T.: Forecasting in large cointegrated processes. J. Forecast. 28(7), 631–650 (2009). MR2744389. doi:10.1002/for.1076
[4] 
Clements, M.P., Hendry, D.F.: Forecasting in cointegrated systems. J. Appl. Econom. 10, 127–146 (1995)
[5] 
Engle, R.F., Granger, C.W.J.: Co-integration and error correction: Representation, estimation and testing. Econometrica 55, 251–276 (1987). MR0882095. doi:10.2307/1913236
[6] 
Franke, J.: Minimax robust prediction of discrete time series. Z. Wahrscheinlichkeitstheor. Verw. Geb. 68, 337–364 (1985). MR0771471. doi:10.1007/BF00532645
[7] 
Gikhman, I.I., Skorokhod, A.V.: The Theory of Stochastic Processes. I. Springer, Berlin (2004). MR2058259
[8] 
Granger, C.W.J.: Cointegrated variables and error correction models. UCSD Discussion paper 83-13a (1983)
[9] 
Gregoir, S.: Fully modified estimation of seasonally cointegrated processes. Econom. Theory 25(5), 1491–1528 (2010). MR2684793. doi:10.1017/S026646660999065X
[10] 
Grenander, U.: A prediction problem in game theory. Ark. Mat. 3, 371–379 (1957). MR0090486
[11] 
Ioffe, A.D., Tihomirov, V.M.: Theory of Extremal Problems. North–Holland Publishing Company, Amsterdam, New York, Oxford (1979). MR0528295
[12] 
Johansen, S.: Representation of cointegrated autoregressive processes with application to fractional processes. Econom. Rev. 28, 121–145 (2009). MR2487849. doi:10.1080/07474930802387977
[13] 
Kolmogorov, A.N.: Selected Works by A.N. Kolmogorov. Vol. II: Probability Theory and Mathematical Statistics. A.N. Shiryayev (ed.) Math Appl. Sov. Ser., vol.  26, Kluwer Academic Publishers, Dordrecht, etc. (1992). MR1153022
[14] 
Luz, M., Moklyachuk, M.: Minimax-robust filtering problem for stochastic sequences with stationary increments and cointegrated sequences. Stat. Optim. Inf. Comput. 2(3), 176–199 (2014). MR3351379. doi:10.19139/56
[15] 
Luz, M., Moklyachuk, M.: Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences. Stat. Optim. Inf. Comput. 3(2), 160–188 (2015). MR3352757. doi:10.19139/132
[16] 
Luz, M.M., Moklyachuk, M.P.: Interpolation of functionals of stochastic sequences with stationary increments. Theory Probab. Math. Stat. 87, 117–133 (2013). MR3241450. doi:10.1090/S0094-9000-2014-00908-4
[17] 
Luz, M.M., Moklyachuk, M.P.: Minimax-robust filtering problem for stochastic sequence with stationary increments. Theory Probab. Math. Stat. 89, 127–142 (2014). MR3235180. doi:10.1090/S0094-9000-2015-00940-6
[18] 
Moklyachuk, M.: Minimax-robust estimation problems for stationary stochastic sequences. Stat. Optim. Inf. Comput. 3(4), 348–419 (2015). MR3435278
[19] 
Moklyachuk, M., Luz, M.: Robust extrapolation problem for stochastic sequences with stationary increments. Contemp. Math. Stat. 1(3), 123–150 (2013)
[20] 
Moklyachuk, M.P.: Robust Estimations of Functionals of Stochastic Processes. Kyivskyi University, Kyiv (2008)
[21] 
Pinsker, M.S.: The theory of curves with nth stationary increments in Hilbert spaces. Izv. Akad. Nauk SSSR, Ser. Mat. 19(5), 319–344 (1955). MR0073957
[22] 
Pshenichnyi, B.N.: Necessary Conditions of an Extremum. Nauka, Moskva (1982). MR0686452
[23] 
Rockafellar, R.T.: Convex Analysis. Princeton University Press (1997). MR1451876
[24] 
Rozanov, Y.A.: Stationary Stochastic Processes. Holden-Day, San Francisco (1967). MR0214134
[25] 
Salehi, H.: Algorithms for linear interpolator and interpolation error for minimal stationary stochastic processes. Ann. Probab. 7(5), 840–846 (1979). MR0542133
[26] 
Vastola, K.S., Poor, H.V.: An analysis of the effects of spectral uncertainty on Wiener filtering. Automatica 28, 289–293 (1983). MR0740656
[27] 
Wiener, N.: Extrapolation, Interpolation and Smoothing of Stationary Time Series: With Engineering Applications. MIT Press, Cambridge (1966)
[28] 
Yaglom, A.M.: Correlation theory of stationary and related random processes with stationary nth increments. Mat. Sb. 37(79)(1), 141–196 (1955). MR0071672
[29] 
Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions. Vol. 1: Basic Results. Springer, New York etc. (1987). MR0893393
[30] 
Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions. Vol. 2: Supplementary Notes and References. Springer, New York, etc. (1987). MR0915557

Full article Cited by PDF XML
Full article Cited by PDF XML

Copyright
© 2016 The Author(s). Published by VTeX
by logo by logo
Open access article under the CC BY license.

Keywords
Stochastic sequence with stationary increments cointegrated sequences minimax-robust estimate mean square error least favorable spectral density minimax-robust spectral characteristic

MSC2010
60G10 60G25 60G35 62M20 93E10 93E11

Metrics
since March 2018
529

Article info
views

562

Full article
views

304

PDF
downloads

140

XML
downloads

Export citation

Copy and paste formatted citation
Placeholder

Download citation in file


Share


RSS

MSTA

MSTA

  • Online ISSN: 2351-6054
  • Print ISSN: 2351-6046
  • Copyright © 2018 VTeX

About

  • About journal
  • Indexed in
  • Editors-in-Chief

For contributors

  • Submit
  • OA Policy
  • Become a Peer-reviewer

Contact us

  • ejournals-vmsta@vtex.lt
  • Mokslininkų 2A
  • LT-08412 Vilnius
  • Lithuania
Powered by PubliMill  •  Privacy policy