Minimax-robust estimation problems for stationary stochastic sequences

TL;DR

This survey reviews optimal linear estimation methods for stationary stochastic sequences, focusing on spectral certainty and minimax-robust approaches under spectral uncertainty, including formulas for spectral characteristics and errors.

Contribution

It provides a comprehensive overview of minimax-robust estimation techniques for stationary sequences, including formulas for least favorable spectral densities and spectral characteristics.

Findings

Formulas for spectral characteristics under spectral certainty.

Minimax estimation methods for spectral uncertainty.

Identification of least favorable spectral densities.

Abstract

This survey provides an overview of optimal estimation of linear functionals which depend on the unknown values of a stationary stochastic sequence. Based on observations of the sequence without noise as well as observations of the sequence with a stationary noise, estimates could be obtained. Formulas for calculating the spectral characteristics and the mean-square errors of the optimal estimates of functionals are derived in the case 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 known exactly while sets of admissible spectral densities are given, the minimax-robust method of estimation is applied. Formulas that determine the least favourable spectral densities and the minimax spectral characteristics of estimates are presented for some special classes of admissible spectral densities.