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Robust extrapolation problem for stochastic sequences with stationary increments

Maksym Luz, Mykhailo Moklyachuk

TL;DR

This work addresses robust extrapolation for stochastic sequences with stationary $n$th increments by formulating mean-square optimal estimates of linear functionals $A\xi$ and $A_N\xi$ from data at negative indices and, when the spectral density is uncertain, develops minimax-robust estimators. The authors employ a Hilbert-space projection approach under spectral certainty to derive spectral characteristics and mean-squared errors, and extend to minimax-robust estimation by identifying least-favorable densities and minimax spectral characteristics across several admissible classes $D_0$, $D_M$, $D_v^u$, and $D_\varepsilon$. Each class yields explicit representations of the least-favorable density (often via a squared-magnitude form $f^{0}(\lambda)=|\cdot|^{2}$) and associated optimal spectral filter, along with conditions to compute the relevant parameters from moment or SNR constraints. The results provide robust extrapolation techniques for functionals of processes with stationary increments, with potential applications in extrapolation, interpolation, and filtering when exact spectral information is unavailable.

Abstract

The problem of optimal estimation of functionals $Aξ=\sum\nolimits_{k=0}^{\infty }{}a(k)ξ(k)$ and ${{A}_{N}}ξ=\sum\nolimits_{k=0}^{N}{}a(k)ξ(k)$ which depend on the unknown values of stochastic sequence $ξ(k)$ with stationary $n$th increments is considered. Estimates are based on observations of the sequence $ξ(m)$ at points of time $m=-1,-2,\ldots$. Formulas for calculating the value of the mean square error and the spectral characteristic of the optimal linear estimates of the functionals are derived in the case where spectral density of the sequence is exactly known. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case where the spectral density of the sequence is not known but a set of admissible spectral densities is given.

Robust extrapolation problem for stochastic sequences with stationary increments

TL;DR

This work addresses robust extrapolation for stochastic sequences with stationary th increments by formulating mean-square optimal estimates of linear functionals and from data at negative indices and, when the spectral density is uncertain, develops minimax-robust estimators. The authors employ a Hilbert-space projection approach under spectral certainty to derive spectral characteristics and mean-squared errors, and extend to minimax-robust estimation by identifying least-favorable densities and minimax spectral characteristics across several admissible classes , , , and . Each class yields explicit representations of the least-favorable density (often via a squared-magnitude form ) and associated optimal spectral filter, along with conditions to compute the relevant parameters from moment or SNR constraints. The results provide robust extrapolation techniques for functionals of processes with stationary increments, with potential applications in extrapolation, interpolation, and filtering when exact spectral information is unavailable.

Abstract

The problem of optimal estimation of functionals and which depend on the unknown values of stochastic sequence with stationary th increments is considered. Estimates are based on observations of the sequence at points of time . Formulas for calculating the value of the mean square error and the spectral characteristic of the optimal linear estimates of the functionals are derived in the case where spectral density of the sequence is exactly known. Formulas that determine the least favorable spectral densities and minimax (robust) spectral characteristic of the optimal linear estimates of the functionals are proposed in the case where the spectral density of the sequence is not known but a set of admissible spectral densities is given.
Paper Structure (9 sections, 20 theorems, 110 equations)

This paper contains 9 sections, 20 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Stationary stochastic increment sequence. Spectral representation
  3. Hilbert space projection method of extrapolation of linear functionals
  4. Minimax-robust method of extrapolation
  5. Least favorable spectral densities in the class ${{D}_{0}}$
  6. Least favorable spectral densities in the class ${{D}_{M}}$
  7. Least favorable spectral densities in the class $D_{v}^{u}$
  8. Least favorable spectral densities in the class ${{D}_{\varepsilon }}$
  9. Conclusions

Key Result

Theorem 2.1

The mean value ${{c}^{(n)}}(\mu )$ and the structural function ${{D}^{(n)}}(m,{{\mu }_{1}},{{\mu }_{2}})$ of the stochastic stationary $n$th increment sequence ${{\xi }^{(n)}}(m,\mu )$ can be represented in the following forms where $c$ is a constant, $F(\lambda )$ is a left-continuous nondecreasing bounded function with $F(-\pi )=0.$ The constant $c$ and the function $F(\lambda )$ are determined

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Definition 2.3
  • Theorem 2.2
  • Definition 2.4
  • Theorem 2.3
  • Corollary 2.1
  • Theorem 3.1
  • Corollary 3.1
  • ...and 19 more