On minimax estimation problem for stationary stochastic sequences from observations in special sets of points

TL;DR

This paper addresses the problem of optimally estimating linear functionals of stationary stochastic sequences from observations at specific points, deriving formulas for error and spectral characteristics under spectral certainty and uncertainty.

Contribution

It introduces formulas for optimal estimation and minimax spectral densities when the spectral density is known or uncertain, advancing robust estimation methods.

Findings

Derived formulas for mean-square error and spectral characteristics under spectral certainty.

Established minimax estimation formulas for uncertain spectral densities.

Identified least favorable spectral densities for specific admissible sets.

Abstract

The problem of the mean-square optimal estimation of the linear functionals which depend on the unknown values of a stochastic stationary sequence from observations of the sequence in special sets of points is considered. Formulas for calculating the mean-square error and the spectral characteristic of the optimal linear estimate of the functionals are derived under the condition of spectral certainty, where the spectral density of the sequence is exactly known. The minimax (robust) method of estimation is applied in the case where the spectral density of the sequence is not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics are derived for some special sets of admissible densities.