Interpolation of functionals of stochastic sequences with stationary increments from observations with noise

TL;DR

This paper develops optimal and minimax estimation methods for linear functionals of stochastic sequences with stationary increments, using noisy observations, providing formulas for mean square error and spectral characteristics under spectral certainty and uncertainty.

Contribution

It introduces formulas for optimal linear estimation and minimax robust estimation of functionals of stochastic sequences with stationary increments, considering spectral uncertainty.

Findings

Formulas for mean square error and spectral characteristics under spectral certainty.

Minimax estimation formulas for unknown spectral densities.

Identification of least favorable spectral densities for robust estimation.

Abstract

The problem of optimal estimation of linear functional ${{A}_{N}}\xi =\sum\limits_{k=0}^{N}{a(k)\xi (k)}\,$ depending on the unknown values of a stochastic sequence $\xi (m)$ with stationary $n$-th increments from observations of the sequence $\xi (k)$ at points $k=-1,-2,\ldots $ and of the sequence $\xi (k)+\eta (k)$ at points of time $k=N+1,N+2,\ldots $ is considered. Formulas for calculating the mean square error and the spectral characteristic of the optimal linear estimate of the functional are proposed under condition of spectral certainty, where spectral densities of the sequences $\xi (m)$ and $\eta (m)$ are exactly known. Minimax (robust) method of estimation is applied in the case where the spectral densities are not known exactly while some sets of admissible spectral densities are given. Formulas that determine the least favorable spectral densities and the minimax spectral characteristics are proposed for some specific sets of admissible densities.