Extrapolation Problem for Multidimensional Stationary Sequences with Missing Observations

TL;DR

This paper develops methods for optimal estimation of linear functionals of multidimensional stationary sequences with missing data, addressing both spectral certainty and uncertainty cases, and deriving formulas for errors and spectral characteristics.

Contribution

It introduces new formulas for optimal estimates and errors in the spectral certainty case and proposes minimax estimation methods under spectral uncertainty.

Findings

Formulas for mean-square errors under spectral certainty

Methods for calculating spectral characteristics of estimates

Minimax estimates for spectral uncertainty cases

Abstract

This paper focuses on the problem of the mean square optimal estimation of linear functionals which depend on the unknown values of a multidimensional stationary stochastic sequence. Estimates are based on observations of the sequence with an additive stationary noise sequence. The aim of the paper is to develop methods of finding the optimal estimates of the functionals in the case of missing observations. The problem is investigated in the case of spectral certainty where the spectral densities of the sequences are exactly known. Formulas for calculating the mean-square errors and the spectral characteristics of the optimal linear estimates of functionals are derived under the condition of spectral certainty. The minimax (robust) method of estimation is applied in the case of spectral uncertainty, where spectral densities of the sequences are not known exactly while sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics of the optimal estimates of functionals are proposed for some special sets of admissible densities.