Minimax Extrapolation Problem For Harmonizable Stable Sequences With Noise Observations

TL;DR

This paper addresses the problem of optimally estimating a linear functional of a harmonizable stable sequence from noisy observations, deriving formulas for error and spectral characteristics under both known and uncertain spectral densities.

Contribution

It introduces a minimax extrapolation framework for stable sequences with noise, providing explicit formulas for estimation error and spectral characteristics under spectral uncertainty.

Findings

Derived formulas for optimal linear estimates with known spectral densities.

Established relations for least favorable spectral densities under uncertainty.

Extended minimax estimation theory to harmonizable stable sequences.

Abstract

We consider the problem of optimal linear estimation of the functional $A \xi~=~\sum_{j = 0}^{\infty} a_j \xi_j$ that depends on the unknown values $\xi_j,j=0,1,\dots, $ of a random sequence $\{\xi_j,j\in\mathbb Z\}$ from observations of the sequence $\{\xi_k+\eta_k,k\in\mathbb Z\}$ at points $k = -1, -2, \dots$, where $\{\xi_k,k\in\mathbb Z\}$ and $\{\eta_k,k\in\mathbb Z\}$ are mutually independent harmonizable symmetric $\alpha$-stable random sequences which have the spectral densities $f(\theta)>0$ and $g(\theta)>0$ satisfying the minimality condition. The problem is investigated under the condition of spectral certainty as well as under the condition of spectral uncertainty. Formulas for calculating the value of the error and the spectral characteristic of the optimal linear estimate of the functional are derived under the condition of spectral certainty where spectral densities of the sequences are exactly known. In the case of spectral uncertainty where spectral densities of the sequences are not exactly known, while a class of admissible spectral densities is given, relations that determine the least favorable spectral densities and the minimax spectral characteristic are derived.