Extrapolation problem for periodically correlated stochastic sequences with missing observations

Abstract

The problem of optimal estimation of the linear functionals $A{\zeta}=\sum_{j=1}^{\infty}{a}(j){\zeta}(j),$ 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 $j\in\{...,-n,...,-2,-1,0\}\setminus S$, $S=\bigcup _{l=1}^{s-1}\{-M_l\cdot T+1,\dots,-M_{l-1}\cdot T-N_{l}\cdot T\}$, 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.

Mykhailo Moklyachuk – to the 75th anniversary of his birth

“…Mykhailo Moklyachuk -to the 75th anniversary of his birth 1,2,3,4,5,6 Taras Shevchenko National University of Kyiv, 64/13 Volodymyrska st., Kyiv, Ukraine, 01601, e-mail: 1 borysenko_oleksandr@knu.ua, 2 volodymyr.zubchenko@knu.ua, 3 yuliyamishura@knu.ua, 4 pmo@univ.kiev.ua, 5 rostyslav.yamnenko@knu.ua, 6 In 2016, Professor Mokliachuk was elected an academician of the Academy of Sciences of Higher Education of Ukraine. In 2019, he was awarded the badge of the Ministry of Education and Science of Ukraine "For Scientific and Educational Achievements".…”

“…The monograph [5] investigates methods for estimating linear functionals from unknown values of stochastic processes. The formulas for calculating the root mean square error of the optimal estimate are given.…”