Filtration of linear functionals of periodically correlated sequences

Authors:
I. I. Dubovets′ka and M. P. Moklyachuk

Translated by:
N. Semenov

Journal:
Theor. Probability and Math. Statist. **86** (2013), 51-64

MSC (2010):
Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11

DOI:
https://doi.org/10.1090/S0094-9000-2013-00888-6

Published electronically:
August 20, 2013

MathSciNet review:
2986449

Full-text PDF Free Access

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Abstract: The problem of the optimal estimation is considered for the linear functional \[ A{\zeta }=\sum _{j=0}^\infty {a}(j){\zeta }(-j) \] that depends on unknown values of a periodically correlated stochastic sequence $\zeta (j)$; the estimator is constructed from observations of the sequence $\zeta (j)+\theta (j)$, $j\leq 0$, where $\theta (j)$ is a periodically correlated noise. We obtain the mean square error and spectral characteristic of the optimal linear estimate of the functional $A{\zeta }$ in the case where the spectral densities of the sequences that generate $\zeta (j)$ and $\theta (j)$ are known. For the case where these spectral densities are unknown but a set of admissible spectral densities is given, we find the least favorable spectral density and minimax spectral characteristic for the optimal estimate of the functional $A{\zeta }$.

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Additional Information

**I. I. Dubovets′ka**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4-E, Kiev 03127, Ukraine

Email:
idubovetska@gmail.com

**M. P. Moklyachuk**

Affiliation:
Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue, 4-E, Kiev 03127, Ukraine

Email:
mmp@univ.kiev.ua

Keywords:
Periodically correlated sequence,
robust estimate,
mean square error,
least favorable spectral density,
minimax spectral characteristic

Received by editor(s):
November 21, 2011

Published electronically:
August 20, 2013

Article copyright:
© Copyright 2013
American Mathematical Society