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On Minimax Estimation Problems for Periodically Correlated Stochastic Processes

Iryna Dubovets'ka, Mykhailo Moklyachuk

TL;DR

The article develops a complete framework for mean-square optimal estimation of linear functionals of periodically correlated processes from noisy observations, including both spectral certainty and spectral uncertainty. By transforming PC processes into infinite-dimensional vector-valued stationary sequences and applying Hilbert-space projection, it derives explicit spectral characteristics $h(f,g)$ and mean-square errors $\Delta(h;f,g)$ for interpolation, extrapolation, and filtering. When spectral densities are unknown, the minimax-robust approach identifies least-favorable densities and minimax spectral characteristics $h^{0}$ within prescribed classes $D$, with concrete results showing least-favorable models as AR/MA-type processes and one-sided moving averages. These results yield practical robust estimators for PC signals and provide insight into the structure of worst-case spectral scenarios, facilitating robust design in applications with periodic statistical structure.

Abstract

The aim of this article is to overview the problem of mean square optimal estimation of linear functionals which depend on unknown values of periodically correlated stochastic process. Estimates are based on observations of this process and noise. These problems are investigated under conditions of spectral certainty and spectral uncertainty. Formulas for calculating the main characteristics (spectral characteristic, mean square error) of the optimal linear estimates of the functionals are proposed. The least favorable spectral densities and the minimax-robust spectral characteristics of optimal estimates of the functionals are presented for given sets of admissible spectral densities.

On Minimax Estimation Problems for Periodically Correlated Stochastic Processes

TL;DR

The article develops a complete framework for mean-square optimal estimation of linear functionals of periodically correlated processes from noisy observations, including both spectral certainty and spectral uncertainty. By transforming PC processes into infinite-dimensional vector-valued stationary sequences and applying Hilbert-space projection, it derives explicit spectral characteristics and mean-square errors for interpolation, extrapolation, and filtering. When spectral densities are unknown, the minimax-robust approach identifies least-favorable densities and minimax spectral characteristics within prescribed classes , with concrete results showing least-favorable models as AR/MA-type processes and one-sided moving averages. These results yield practical robust estimators for PC signals and provide insight into the structure of worst-case spectral scenarios, facilitating robust design in applications with periodic statistical structure.

Abstract

The aim of this article is to overview the problem of mean square optimal estimation of linear functionals which depend on unknown values of periodically correlated stochastic process. Estimates are based on observations of this process and noise. These problems are investigated under conditions of spectral certainty and spectral uncertainty. Formulas for calculating the main characteristics (spectral characteristic, mean square error) of the optimal linear estimates of the functionals are proposed. The least favorable spectral densities and the minimax-robust spectral characteristics of optimal estimates of the functionals are presented for given sets of admissible spectral densities.
Paper Structure (12 sections, 17 theorems, 133 equations)

This paper contains 12 sections, 17 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Periodically correlated continuous processes and generated vector-valued stationary sequences
  3. Hilbert space projection method of estimation of PC processes
  4. Interpolation problem
  5. Extrapolation problem
  6. Filtering problem
  7. Minimax-robust estimation method
  8. Minimax-robust spectral characteristics for given classes $D$
  9. Interpolation problem in the class $D_{M}^{-}$
  10. Minimax extrapolation in the class $D_{0}^{1}$
  11. Filtering problem in the class $D_{0}^{2}\times {{D}_{\varepsilon }}$
  12. Conclusions

Key Result

Theorem 3.1

Let $\{\zeta (t),\,t\in \mathbb{R}\}$ and $\{\theta (t),\,t\in \mathbb{R}\}$ be uncorrelated PC stochastic processes such that the generated stationary sequences $\{{{\zeta }_{j}},\,j\in \mathbb{Z}\}$ and $\{{{\theta }_{j}},\,j\in \mathbb{Z}\}$ have spectral densities $f(\lambda )$ and $g(\lambda )$

Theorems & Definitions (22)

  • Definition 2.1
  • Theorem 3.1
  • Corollary 3.1
  • Theorem 3.2
  • Corollary 3.2
  • Definition 3.1
  • Theorem 3.3
  • Theorem 3.4
  • Corollary 3.3
  • Theorem 3.5
  • ...and 12 more