Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences

  • Maksym Luz Taras Shevchenko National University of Kyiv
  • Mikhail Moklyachuk Taras Shevchenko National University of Kyiv
Keywords: Stochastic sequence with stationary increments, cointegrated sequence, minimax-robust estimate, mean square error, least favorable spectral density, minimax-robust spectral characteristic

Abstract

The problem of optimal estimation of the linear functionals $A {\xi}=\sum_{k=0}^{\infty}a (k)\xi(k)$ and $A_N{\xi}=\sum_{k=0}^{N}a (k)\xi(k)$ which depend on the unknown values of a stochastic sequence $\xi(m)$ with stationary $n$th increments is considered. Estimates are obtained which are based on observations of the sequence $\xi(m)+\eta(m)$ at points of time $m=-1,-2,\ldots$, where the sequence $\eta(m)$ is stationary and uncorrelated with the sequence $\xi(m)$. Formulas for calculating the mean-square errors and spectral characteristics of the optimal estimates of the functionals are derived in the case of spectral certainty, where spectral densities of the sequences $\xi(m)$ and $\eta(m)$ are exactly known. These results are applied for solving extrapolation problem for cointegrated sequences. In the case where spectral densities of the sequences are not known exactly, but sets of admissible spectral densities are given, the minimax-robust method of estimation is applied. Formulas that determine the least favorable spectral densities and minimax spectral characteristics are proposed for some special classes of admissible densities. 

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Published
2015-05-12
How to Cite
Luz, M., & Moklyachuk, M. (2015). Minimax-robust prediction problem for stochastic sequences with stationary increments and cointegrated sequences. Statistics, Optimization & Information Computing, 3(2), 160-188. https://doi.org/10.19139/soic.v3i2.132
Section
Research Articles