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Interpolation of functionals of stochastic sequences with stationary increments from observations with noise

Maksym Luz, Mykhailo Moklyachuk

TL;DR

This work addresses optimal linear interpolation of the functional $A_N\xi$ for stochastic sequences with stationary $n$-th increments, using noisy observations. It develops a spectral-decomposition framework for the increments, turns the interpolation task into a projection problem in Hilbert space, and derives the corresponding spectral characteristics and mean-squared errors under spectral certainty. When spectral densities are unknown, the paper adopts a minimax approach, characterizing least-favorable densities and minimax spectral characteristics for several admissible density sets, and providing explicit constructions via linear systems and spectral constraints. The results unify interpolation and filtering perspectives for processes with stationary increments and offer practical formulas to compute robust estimates, with applications to time series and actuarial/financial modeling. The framework advances robust estimation for linear functionals of nonstationary-origin processes by leveraging spectral methods and convex optimization over density sets.

Abstract

The problem of optimal estimation of linear functional ${{A}_{N}}ξ=\sum\limits_{k=0}^{N}{a(k)ξ(k)}\,$ depending on the unknown values of a stochastic sequence $ξ(m)$ with stationary $n$-th increments from observations of the sequence $ξ(k)$ at points $k=-1,-2,\ldots $ and of the sequence $ξ(k)+η(k)$ at points of time $k=N+1,N+2,\ldots $ is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed under condition of spectral certainty, where spectral densities of the sequences $ξ(m)$ and $η(m)$ are exactly known. Minimax (robust) method of estimation is applied in the case where the spectral densities are not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some specific sets of admissible densities.

Interpolation of functionals of stochastic sequences with stationary increments from observations with noise

TL;DR

This work addresses optimal linear interpolation of the functional for stochastic sequences with stationary -th increments, using noisy observations. It develops a spectral-decomposition framework for the increments, turns the interpolation task into a projection problem in Hilbert space, and derives the corresponding spectral characteristics and mean-squared errors under spectral certainty. When spectral densities are unknown, the paper adopts a minimax approach, characterizing least-favorable densities and minimax spectral characteristics for several admissible density sets, and providing explicit constructions via linear systems and spectral constraints. The results unify interpolation and filtering perspectives for processes with stationary increments and offer practical formulas to compute robust estimates, with applications to time series and actuarial/financial modeling. The framework advances robust estimation for linear functionals of nonstationary-origin processes by leveraging spectral methods and convex optimization over density sets.

Abstract

The problem of optimal estimation of linear functional depending on the unknown values of a stochastic sequence with stationary -th increments from observations of the sequence at points and of the sequence at points of time is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed under condition of spectral certainty, where spectral densities of the sequences and are exactly known. Minimax (robust) method of estimation is applied in the case where the spectral densities are not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some specific sets of admissible densities.
Paper Structure (8 sections, 11 theorems, 122 equations)

This paper contains 8 sections, 11 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Stationary increments. Spectral decomposition
  3. The interpolation problem
  4. The relationship between of the problem of interpolation from observations with noise and the problem of filtering
  5. Minimax (robust) method of interpolation from noisy observations
  6. The least favorable densities in the class $D_{0,\mu }^{-}\times D_{0,\mu }^{-}$
  7. The least favorable densities in the class $D_{M,\mu }^{-}\times D_{M,\mu }^{-}.$
  8. Conclusions

Key Result

Theorem 2.1

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

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Theorem 2.1
  • Theorem 3.1
  • Corollary 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Definition 3
  • Definition 4
  • Lemma 5.1
  • ...and 5 more