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Minimax Estimation Problem for Periodically Correlated Stochastic Processes

Iryna Dubovets'ka, Mykhailo Moklyachuk

TL;DR

This work addresses robust linear estimation of a functional $A\zeta$ of a periodically correlated process from pre-zero observations under spectral-density uncertainty. It develops a minimax framework that leverages the canonical factorization of the spectral density operator $f(\lambda)$ to relate the PC process to a regular stationary sequence, yielding a worst-case error of $P\nu^{2}$ where $\nu^{2}$ is the top eigenvalue of an associated operator. The least-favorable process is shown to be a one-sided moving-average sequence, which provides a constructive path to the minimax estimator. By connecting spectral and operator-theoretic tools, the results supply explicit, robust estimation formulas for PC processes, guiding extrapolation/interpolation under spectral uncertainty with practical applicability.

Abstract

The problem of optimal linear estimation of linear functionals depending on the unknown values of a periodically correlated stochastic process from observations of the process with additive noise is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functionals are proposed in the case where spectral densities are exactly known and in the case where the spectral densities are unknown while a class of admissible spectral densities is given. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics are proposed for a given class of admissible spectral densities.

Minimax Estimation Problem for Periodically Correlated Stochastic Processes

TL;DR

This work addresses robust linear estimation of a functional of a periodically correlated process from pre-zero observations under spectral-density uncertainty. It develops a minimax framework that leverages the canonical factorization of the spectral density operator to relate the PC process to a regular stationary sequence, yielding a worst-case error of where is the top eigenvalue of an associated operator. The least-favorable process is shown to be a one-sided moving-average sequence, which provides a constructive path to the minimax estimator. By connecting spectral and operator-theoretic tools, the results supply explicit, robust estimation formulas for PC processes, guiding extrapolation/interpolation under spectral uncertainty with practical applicability.

Abstract

The problem of optimal linear estimation of linear functionals depending on the unknown values of a periodically correlated stochastic process from observations of the process with additive noise is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functionals are proposed in the case where spectral densities are exactly known and in the case where the spectral densities are unknown while a class of admissible spectral densities is given. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics are proposed for a given class of admissible spectral densities.
Paper Structure (4 sections, 2 theorems, 63 equations)

This paper contains 4 sections, 2 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Periodically correlated processes and generated vector stationary sequences
  3. Minimax estimation of linear functional
  4. Conclusions

Key Result

Theorem 1

Let the coefficients $\{{{\vec{a}}_{j}},j=0,...,N\}$ which determine the functional ${{A}_{N}}\zeta$ satisfy the condition The function $\Delta (\zeta ,{{\hat{A}}_{N}})$ has a saddle point on the set $\mathbf{Y}\times \Lambda$ and where $\nu _{N}^{2}$ is the greatest eigenvalue of the self-adjoint compact operator ${{Q}_{N}}=\{{{Q}_{N}}(p,q)\}_{p,q=0}^{N}$ in the space ${{\ell }_{2}}$ determined

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2