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Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

 
 

 

Estimates of functionals constructed from random sequences with periodically stationary increments


Authors: P. S. Kozak and M. P. Moklyachuk
Translated by: N. N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 97 (2017).
Journal: Theor. Probability and Math. Statist. 97 (2018), 85-98
MSC (2010): Primary 60G10, 60G25, 60G35; Secondary 62M20, 93E10, 93E11
DOI: https://doi.org/10.1090/tpms/1050
Published electronically: February 21, 2019
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Abstract: The problem of the optimal estimation of the linear functional

$\displaystyle A_N{\xi }=\sum _{k=0}^Na(k)\xi (k)$    

is studied. The functional depends on unknown values of a random sequence $ \xi (k)$ with periodically stationary increments. The estimate is constructed from observations of this sequence at points $ \mathbb{Z}\setminus \{0,1,\dots ,N\}$. Expressions for evaluating the mean square error and spectral characteristics are found for the optimal estimate of the functional in the case where the spectral density of the sequence is known. For a given set of admissible spectral densities, the sets of least favorable spectral densities are found and the spectral characteristics of the optimal estimate of the functional are determined.

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Additional Information

P. S. Kozak
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: petrokozak91@gmail.com

M. P. Moklyachuk
Affiliation: Department of Probability Theory, Statistics, and Actuarial Mathematics, Faculty for Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email: mmp@univ.kiev.ua

DOI: https://doi.org/10.1090/tpms/1050
Keywords: Sequence with periodically stationary increments, robust estimate, mean square error, least favorable spectral density, minimax spectral characteristics
Received by editor(s): March 30, 2017
Published electronically: February 21, 2019
Dedicated: Dedicated to the blessed memory of Mykhailo Yosypovych Yadrenko
Article copyright: © Copyright 2019 American Mathematical Society