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Robust Forecasting of Sequences with Periodically Stationary Long Memory Multiplicative Seasonal Increments Observed with Noise and Cointegrated Sequences

Maksym Luz, Mykhailo Moklyachuk

TL;DR

The paper tackles robust forecasting of linear functionals of unobserved values in sequences with periodically stationary GM increments observed amidst periodic noise. It develops a Hilbert-space projection framework, leveraging MA representations and spectral factorization to obtain optimal linear forecasts and their mean-square errors when spectral densities are known. When densities are uncertain, it introduces minimax (robust) strategies by defining admissible density classes and characterizing least-favorable densities, including extensions to cointegrated and seasonally cointegrated vector sequences. The results yield explicit formulas for spectral characteristics and MSEs, along with saddle-point properties, enabling robust forecasting in long-memory, periodically varying environments. This has practical implications for accurate prediction and filtering in economic, climatological, and signal-processing contexts where periodicity and long-range dependence coexist.

Abstract

The problem of optimal estimation of linear functionals constructed from unobserved values of stochastic sequence with periodically stationary increments based on observations of the sequence with a periodically stationary noise is considered. 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 that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.

Robust Forecasting of Sequences with Periodically Stationary Long Memory Multiplicative Seasonal Increments Observed with Noise and Cointegrated Sequences

TL;DR

The paper tackles robust forecasting of linear functionals of unobserved values in sequences with periodically stationary GM increments observed amidst periodic noise. It develops a Hilbert-space projection framework, leveraging MA representations and spectral factorization to obtain optimal linear forecasts and their mean-square errors when spectral densities are known. When densities are uncertain, it introduces minimax (robust) strategies by defining admissible density classes and characterizing least-favorable densities, including extensions to cointegrated and seasonally cointegrated vector sequences. The results yield explicit formulas for spectral characteristics and MSEs, along with saddle-point properties, enabling robust forecasting in long-memory, periodically varying environments. This has practical implications for accurate prediction and filtering in economic, climatological, and signal-processing contexts where periodicity and long-range dependence coexist.

Abstract

The problem of optimal estimation of linear functionals constructed from unobserved values of stochastic sequence with periodically stationary increments based on observations of the sequence with a periodically stationary noise is considered. 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 that determine the least favorable spectral densities and minimax (robust) spectral characteristics of the optimal linear estimates of functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are given.

Paper Structure

This paper contains 17 sections, 32 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. Stochastic sequences with periodically stationary generalized multiple increments
  3. Definition and spectral representation of a periodically stationary GM increment
  4. Definition and spectral representation of stochastic sequences with periodically stationary GM increment
  5. Moving average representation of periodically stationary GM increment
  6. Stochastic sequences with GM fractional increments
  7. Hilbert space projection method of forecasting
  8. Forecasting of vector-valued stochastic sequences with stationary GM increments
  9. Forecasting based on factorizations of the spectral densities
  10. Forecasting of stochastic sequences with periodically stationary increment
  11. Forecasting of one class of cointegrated vector stochastic sequences
  12. Minimax (robust) method of forecasting
  13. Least favorable spectral density in classes $\mathcal{D}_0 \times \mathcal{D}_V^U$
  14. Least favorable spectral density in classes $\mathcal{D}_0 \times \mathcal{D}_V^U$ for cointegrated vector sequences
  15. Least favorable spectral density in classes $\mathcal{D}_{\varepsilon}\times \mathcal{D}_{1\delta}$
  16. ...and 2 more sections

Key Result

Theorem 2.1

1. The mean value and the structural function of the vector-valued stochastic stationary GM increment sequence $\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$ can be represented in the form where $c$ is a vector, $F(\lambda)$ is the matrix-valued spectral function of the stationary stochastic sequence $\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$. The vector $c$ and the matri

Theorems & Definitions (45)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Theorem 2.1
  • Definition 2.4
  • Theorem 2.2
  • Theorem 2.3
  • Definition 2.5
  • Lemma 2.1
  • ...and 35 more