Minimax-robust estimation problems for sequences with periodically stationary increments observed with noise

TL;DR

This paper develops minimax-robust estimation methods for stochastic sequences with periodically stationary increments observed with noise, providing formulas for optimal estimates and robustness when spectral densities are uncertain.

Contribution

It introduces formulas for calculating optimal estimates and their errors, and proposes minimax-robust spectral characteristics for uncertain spectral densities.

Findings

Formulas for mean square errors of optimal estimates

Spectral characteristics of optimal estimates

Minimax-robust spectral density formulas

Abstract

The problem of optimal estimation of linear functionals constructed from the unobserved values of a stochastic sequence with periodically stationary increments based on observations of the sequence with stationary noise is considered. For sequences with known spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas that determine the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal linear estimates of functionals are proposed in the case where spectral densities of the sequence are not exactly known while some sets of admissible spectral densities are specified.