Minimax extrapolation problem for periodically correlated stochastic sequences with missing observations

TL;DR

This paper develops minimax extrapolation methods for periodically correlated stochastic sequences with missing data, providing formulas for optimal estimation, error calculation, and robustness under spectral uncertainty.

Contribution

It introduces new formulas for optimal estimation and robustness of periodically correlated sequences with missing observations, considering spectral uncertainty.

Findings

Derived formulas for mean square error and spectral characteristics of optimal estimates.

Established minimax-robust spectral characteristics under spectral density uncertainty.

Provided methods for optimal estimation with missing data in periodically correlated sequences.

Abstract

The problem of optimal estimation of the linear functionals which depend on the unknown values of a periodically correlated stochastic sequence ${\zeta}(j)$ from observations of the sequence ${\zeta}(j)+{\theta}(j)$ at points $j\in\{\dots,-n,\dots,-2,-1,0\}\setminus S$, $S=\bigcup _{l=1}^{s-1}\{-M_l\cdot T+1,\dots,-M_{l-1}\cdot T-N_{l}\cdot T\}$, is considered, where ${\theta}(j)$ is an uncorrelated with ${\zeta}(j)$ periodically correlated stochastic sequence. Formulas for calculation the mean square error and the spectral characteristic of the optimal estimate of the functional $A\zeta$ are proposed in the case where spectral densities of the sequences are exactly known. Formulas that determine the least favorable spectral densities and the minimax-robust spectral characteristics of the optimal estimates of functionals are proposed in the case of spectral uncertainty, where the spectral densities are not exactly known while some sets of admissible spectral densities are specified.